1. Introduction and statement of results

In his book on thermodynamic formalism [Reference RuelleRue04], Ruelle introduced the concept of Smale spaces, which intended to axiomatically generalize the concept of a smooth hyperbolic map to the setting of compact metric spaces. This general framework was a unified theory under which many types of chaotic systems could be included. Some of the systems included under this umbrella are Anosov systems, basic sets of Axiom A systems, mixing shifts of finite type, hyperbolic toral automorphisms, etc. One of the many features shared by these systems is the existence of stable and unstable sets, generalizing the concept of stable and unstable manifolds found in some smooth systems. Later, Wieler [Reference WielerWie14] characterized Smale spaces with zero-dimensional stable sets as inverse limits satisfying certain conditions.

This paper studies various aspects of inverse limits of locally expanding affine linear maps on flat branched manifolds, which I call flat Wieler solenoids since they are a subclass of the types of Smale spaces classified by Wieler. They are spaces which are in a sense of intermediate type: on one extreme, Anosov maps are defined on smooth manifolds whereas mixing subshifts of finite type are defined on totally disconnected sets. Flat Wieler solenoids lie somewhere between these two extremes and it is this intermediate position that makes them particularly interesting. Although this is a subclass of Wieler’s more general classification, it is a class with a lot of rich examples. In particular, it includes the tiling spaces of self-similar tilings, or tilings built using a substitution rule. They are also related to primitive substitution systems in symbolic dynamics.

Let me be more precise (all the following terms are defined in §2): let $\Gamma $ be a flat branched manifold of dimension $d>0$ and $\gamma :\Gamma \rightarrow \Gamma $ be a locally expanding affine linear map. This means that $\gamma $ has constant derivative $D_\gamma \in GL(d,\mathbb {R})$ outside the branch points of $\Gamma $ , with smallest eigenvalue $\unicode{x3bb} _0$ satisfying $|\unicode{x3bb} _0|>1$ . The inverse limit of the pair $(\Gamma ,\gamma )$ is

$$ \begin{align*} \Omega_\gamma := \{ (z_0,z_1,z_2,\ldots)\in \Gamma^\infty: z_i = \gamma(z_{i+1})\mbox{ for all }i\geq 0\}. \end{align*} $$

This is a flat Wieler solenoid. It has a local product structure of the form $B_{\varepsilon }\times \mathcal {C}$ , where $B_{\varepsilon }$ is the $\varepsilon $ -ball of dimension d and $\mathcal {C}$ is a Cantor set. It becomes a Smale space when paired with the ‘hyperbolic’ map $\Phi :\Omega _\gamma \rightarrow \Omega _\gamma $ defined by applying $\gamma $ to every coordinate of $z\in \Omega _\gamma $ . Under this map, the Cantor sets in the local structure become stable sets and the Euclidean components become the unstable sets. Under suitable conditions, $\Omega _\gamma $ is a foliated space with each leaf dense in $\Omega _\gamma $ and homeomorphic to $\mathbb {R}^d$ .

Although $\Omega _\gamma $ is not a manifold, it still has a rich topological and geometric structure. First, the Čech cohomology can be computed as the direct limit of the cohomologies of $\Gamma $ under the induced map $\gamma ^{*}$ . Another way to recover this topological information, à la de Rham, is to consider the set of functions with the highest possible regularity on $\Omega $ , consider differential forms with coefficients in this space of functions, and then compute the cohomology of this complex of forms defined by the natural leafwise de Rham differential operator [Reference Kellendonk and PutnamKP06]. The space of functions which serves this role are the leafwise $C^\infty $ , transversally locally constant functions, denoted by $C^\infty _{tlc}(\Omega )$ . These functions have the highest regularity in the smooth direction ( $C^\infty $ ) as well as the highest regularity in the transverse, totally disconnected direction (locally constant).

One of the motivations behind the paper is a lack of good spaces of functions on $\Omega _\gamma $ of intermediate regularity. The usual tricks afforded by Sobolev spaces on smooth manifolds are not immediately available in this setting since it is not clear how they can afford an accounting of the regularity of a function in the totally disconnected direction. I propose that looking at the Hölder regularity in the totally disconnected direction can get us started in working with spaces of functions with intermediate regularity. Unlike Euclidean spaces, the space of $\alpha $ -Hölder functions on a Cantor set is non-trivial for all $\alpha>0$ and thus $\alpha $ -Hölder functions do supplant the notion of regularity in this setting.

To get a grasp on a good space of smooth functions on $\Omega $ , in §2.1, I define the set $\mathcal {S}^r_{\alpha }$ of functions on $\Omega $ which are roughly described as functions which are $C^r$ in the leaf direction and $\alpha $ -Hölder in the transverse direction. This is how one can handle different degrees of regularity in the smooth and non-smooth directions of the space $\Omega $ . It will be shown that $\mathcal {S}^r_{\alpha }$ is, in a sense, dense in the space $C^r_{\alpha }$ of leafwise $C^r$ functions which are $\alpha $ -Hölder in the transverse direction up to r derivatives.

The concept of $\alpha $ -Hölder functions implicitly makes reference to a metric on $\Omega $ and so a function which is $\alpha $ -Hölder in one metric may be $\alpha '$ -Hölder in another metric for some $\alpha '\neq \alpha $ . In these types of spaces, there is a natural metric in reference to which all statements will be made. One of the properties of this metric is that, for any leafwise first-order differential operator $\partial $ and $f\in \mathcal {S}^r_{\alpha }$ , the derivative satisfies $\partial f \in \mathcal {S}^{r-1}_{\alpha +1}$ . In other words, leafwise differentiation reduces leafwise regularity, but increases transverse regularity. It seems to me like this has gone unnoticed, and there is much to gain from this observation.

Let $H^{*}_{r,\alpha }$ be the cohomology of the complex of leafwise smooth forms with coefficients in $\mathcal {S}^r_{\alpha }$ . The first result gives a lower bound on the regularity of functions in this complex, above which the usual real cohomology of $\Omega $ can be recovered.

Theorem 1.1. Let $\Omega $ be a flat Wieler solenoid and $H^{*}_{r,\alpha }(\Omega )$ the cohomology of tangential forms on $\Omega $ with coefficients in $\mathcal {S}^r_{\alpha }(\Omega )$ . If $r\in \mathbb {N}$ and $\alpha>1$ , then

$$ \begin{align*} H^{*}_{r,\alpha}(\Omega)\cong \check H^{*}(\Omega;\mathbb{R}). \end{align*} $$

This theorem should remind one of de Rham regularization. Indeed, part of the proof uses de Rham regularization (to find bounds for r). However, more needs to be done to ensure that $\alpha>1$ guarantees that the cohomology is finite dimensional. The first immediate application of this theorem is for the speed of ergodicity of functions in $\mathcal {S}^r_{\alpha }$ for $r\in \mathbb {N}$ and $\alpha>1$ . This essentially follows from the arguments in [Reference Schmieding and TreviñoST18] and it is discussed in §3.1. It improves previous results on deviations of ergodic integrals in that it increases the set of functions for which the deviation results hold.

As mentioned above, there is a self-homeomorphism $\Phi :\Omega _\gamma \rightarrow \Omega _\gamma $ which makes a flat Wieler solenoid a Smale space. This map preserves an absolutely continuous probability measure $\mu $ and is topologically mixing [Reference Anderson and PutnamAP98, Proposition 3.1]. The second application of the careful study of transverse Hölder regularity is to the speed of mixing of the map $\Phi :\Omega \rightarrow \Omega $ . To do this, the notion of the Ruelle spectrum has to be defined.

Definition 1.2. Let $\Phi :\Omega \rightarrow \Omega $ be a map preserving a probability measure $\mu $ and $\mathcal {F}$ a space of bounded functions on $\Omega $ . Let I be a finite or countable set, $\Lambda = \{\unicode{x3bb} _i\}_{i\in I}$ a set of complex numbers with $|\unicode{x3bb} _i|\in (0,1]$ such that for any $\varepsilon>0$ , there are only finitely many i such that $|\unicode{x3bb} _i|>\varepsilon $ , and let $\{N_i\}_{i\in I}$ be non-negative integers. Then $\Phi $ has Ruelle spectrum $\Lambda $ with Jordan block dimension $\{N_i\}$ on $\mathcal {F}$ if, for any $f,g\in \mathcal {F}$ and $\varepsilon>0$ , there is an asymptotic expansion

where $c_{i,j}(f,g)$ are non-zero bilinear functions of f and g of finite rank.

Note that these asymptotics give precise information on the rates of mixing. Moreover, if $f\in \mathcal {F}$ is an eigenfunction for $\Phi $ with eigenvalue $\nu $ and the essential spectrum for $\Phi ^{*}$ is reduced to $\{0\}$ , then $\nu \in \Lambda $ . Thus, the search for the Ruelle spectrum reduces to the search of generalized eigenfunctions for the pullback operator defined by $\Phi $ , called the transfer operator $\mathcal {L} = \Phi ^{*}$ , on a good space of functions $\mathcal {F}$ . To find a good $\mathcal {F}$ , the use of the so-called anisotropic Banach spaces will be employed.

Computing the Ruelle spectra of systems has become fashionable in the last half-decade, especially through the use of anisotropic Banach spaces. In [Reference Faure, Gouëzel and LanneauFGL19], using transfer operator techniques and anisotropic Banach spaces, the authors noticed that the Ruelle spectrum for linear pseudo-Anosov maps on Riemann surfaces is composed entirely from cohomological information. Using different techniques, this was soon reproved in [Reference ForniFor22a], and that point of view was extended to the case of nonlinear pseudo-Anosov actions on surfaces [Reference ForniFor22b]. Those works were followed by [Reference Butterley, Kiamari and LiveraniBKL22] and the recent PhD dissertation of D. Galli—both using transfer operators and anisotropic Banach spaces—where the focus has been to extract resonances from the cohomology spectrum in a larger class of nonlinear systems. They concluded that the Ruelle spectrum contains cohomological information, but is not necessarily made up exclusively of cohomological information. The results here are of that type.

I will define anisotropic Banach spaces $\mathcal {B}^{r,\alpha }_m$ as the completion of the space of tangential m-forms with coefficients in $\mathcal {S}_{\alpha }^r$ with respect to an anisotropic norm, and then study the spectrum of the transfer operator on these spaces. The spaces of functions for which part of the Ruelle spectrum can be identified will be $\mathcal {B}_0^{\infty ,\alpha } = \bigcap _{r>0}\mathcal {B}_0^{r,\alpha }$ for $\alpha $ large enough.

Let $h_{\mathrm {top}}$ be the topological entropy of $\gamma :\Gamma \rightarrow \Gamma $ and $\chi _-$ the smallest Lyapunov exponent of $\gamma $ . Let $\sigma ^-$ be the set of eigenvalues $\nu $ of $\Phi ^{-1*}:\check H^d(\Omega ;\mathbb {R})\rightarrow \check H^d(\Omega ;\mathbb {R})$ which satisfy $\log |\nu |< \chi _- - h_{\mathrm {top}}$ . Note that when $d=1$ , this is the set of all contracting eigenvalues.

Finally, another application of the study of transverse Hölder regularity in these spaces is to the solution of the cohomological equation in the setting of primitive substitution subshifts. Let $\mathcal {A}$ be a finite set and let $\varrho $ be a primitive substitution rule on $\mathcal {A}$ (all of these terms are defined in §5). Let $\sigma :X_{\varrho }\rightarrow X_{\varrho }$ be the minimal subshift defined by this substitution rule. Let $H_{\alpha }(X_{\varrho })$ be the space of $\alpha $ -Hölder functions on $X_{\varrho }$ (this is with respect to some natural ultrametric; the following results are stated with respect to a specific, natural ultrametric). Let $H^0_{\alpha }(X_{\varrho })$ be the quotient of $H_{\alpha }(X_{\varrho })$ with respect to the equivalence relation $f\sim g$ if there exists a $u\in H_{\alpha -2}(X_{\varrho })$ such that $f-g = u\circ \sigma - u$ . This is the $\alpha $ -Hölder cohomology of $X_{\varrho }$ .

This paper is organized as follows. Background material is covered in §2, including the function spaces $\mathcal {S}^r_{\alpha }$ , which, as far as I know, are new. Section 3 deals with proving that the $r,\alpha $ cohomology is isomorphic to the usual real cohomologies (Theorem 1.1) for $r,\alpha $ large enough. Section 4 is devoted to the study of the Ruelle spectrum: anisotropic Banach spaces of forms are introduced and the action of the transfer operator on them is studied leading to Theorem 1.3. Finally, in §5, the solutions of the cohomological equation on $X_{\varrho }$ are studied by relating them to solutions of the cohomological equation on one-dimensional solenoids, proving Theorem 1.5.

2. Wieler solenoids of flat branched manifolds

Let $\Gamma $ be a connected, flat branched manifold of dimension d. This means that $\Gamma $ is obtained by gluing several polytopes of dimension d along their faces in such a way that every k-dimensional face meets another of the same dimension. This branched manifold has a natural CW structure and flat metric; denote by $I^k$ its $k\mathrm {th}$ -skeleton. Let B be the branching set, that is, the set of points x such that a small ball around x is not homeomorphic to an open Euclidean ball. If $B=\varnothing $ , then $\Gamma $ is a manifold. The focus here will be on cases where $B\neq \varnothing $ . Setting $B_k := (I^k\setminus I^{k-1})\cap B$ , it is worth pointing out that every point on $B_k$ is a k-dimensional flat manifold and, as such, it has well-defined tangent and cotangent spaces.

Let $\gamma :\Gamma \rightarrow \Gamma $ be a locally expanding and surjective map with constant compatible derivative. This means that the derivative map is non-singular and on $I^d\setminus I^{d-1}$ , the derivative $D_\gamma $ can be identified with some $D_\gamma \in GL(d,\mathbb {R})$ . Let $\unicode{x3bb} _1,\ldots ,\unicode{x3bb} _d>1$ be the eigenvalues of $D_\gamma $ and set $\unicode{x3bb} _0 = \min _i|\unicode{x3bb} _i|$ and $\unicode{x3bb} = \mathrm {det}\, D_\gamma $ .

The solenoid defined by $\gamma $ is the space

(1) $$ \begin{align} \Omega=\Omega_\gamma=\{\bar{z} = (z_0,z_1,\ldots)\in \Gamma^\infty:\gamma(z_{i+1}) = z_i\mbox{ for all }i\geq 0\}. \end{align} $$

It can also be defined as an inverse/projective limit: the inverse limit

(2) $$ \begin{align} \Omega_\gamma = \lim_{\leftarrow}(\Gamma,\gamma) \end{align} $$

can easily be seen to match the first definition. This is a flat Wieler solenoid since it is part of Wieler’s classification of Smale spaces with totally disconnected stable sets (this structure will be examined below). The $\gamma $ -solenoid comes equipped with a probability measure which is compatible with the inverse limit structure. To describe this more precisely, denote by $\pi _k:\bar {z}\mapsto z_k$ the projection onto the $k\mathrm {th}$ coordinate, and denote by $\Gamma _k$ to be the $k\mathrm {th}$ copy of $\Gamma $ : $\Gamma _k = \pi _k(\Omega )$ . Let $\mu _k$ be the normalized Lebesgue measure on $\Gamma _k$ induced from the flat metric. Since $\gamma $ preserves Lebesgue measure, it follows that $\gamma _*\mu _k = \mu _{k-1}$ . Thus, if we equip $\Omega $ with $\mu := \otimes _k\mu _k$ , we have that $\pi _{k*}\mu = \mu _k$ for every k.

Two assumptions need to be made on the pair $(\Gamma ,\gamma )$ .

1. (i) The map $\gamma :\Gamma \rightarrow \Gamma $ is primitive: there exists a $K>0$ such that for any two faces $F_1,F_2\subset \Gamma $ , there is a point $x\in F_1$ such that $\gamma ^K(x)\in F_2$ .

2. (ii) The map $\gamma :\Gamma \rightarrow \Gamma $ forces the border. Although I will not discuss what is meant here, an implication of this assumption will be pointed out below.

3. (iii) The map $\gamma $ is recognizable: roughly speaking, all d-cells have distinct images.

The solenoid $\Omega $ has a local product structure of $B_{\varepsilon }(0)\times \mathcal {C}$ , where $B_{\varepsilon }(0)$ is an $\varepsilon $ -ball in $\mathbb {R}^d$ and $\mathcal {C}$ is a Cantor set. Indeed, if $z = (z_0,z_1,\ldots )\in \Omega $ , then points close to z come from either varying the $z_0$ coordinate by a small amount (this is parameterized by $B_{\varepsilon }$ ) or by varying in $\pi _0^{-1}(z_0)$ , which amounts to picking one of (potentially several) points in $\gamma ^{-1}(z_0)$ , then one of (potentially several) points in the $\gamma $ -preimage of that point, and so on. It follows that since $\unicode{x3bb}>1$ , this sequence of choices naturally gives a Cantor set.

There are two complementary dynamical systems defined on $\Omega $ . First, there is a self-homeomorphism $\Phi :\Omega \rightarrow \Omega $ which preserves the special measure $\mu $ . In coordinates, the map is defined by

$$ \begin{align*}\Phi:(z_0,z_1,z_2,\ldots)\mapsto (\gamma(z_0),z_0,z_1,\ldots),\end{align*} $$

which, by construction, preserves the measure $\mu $ . The inverse is obtained by deleting the first coordinate.

For $z\in \Omega $ , define its $k\mathrm {th}$ transversal set as

$$ \begin{align*}C_k^{\perp}(z) := \{z'\in \Omega: z_i^{\prime} = z_i\mbox{ for all }i\leq k \}.\end{align*} $$

The sets $C_0^{\perp }(x)$ are all ultrametric sets which will be endowed with the metric

(3)

where $k(y,z)$ is the smallest integer i such that $y_i\neq z_i$ ; equivalently, the smallest integer i such that $C_i^{\perp }(y)\neq C_i^{\perp }(z)$ . These transversal sets can be seen to be the local stable set of x: if $y\in C_0^{\perp }(x)$ , then $d(\Phi ^n(x), \Phi ^n(y))\rightarrow 0$ as $n\rightarrow \infty $ . The local unstable sets can also be seen, in the coordinates $B_{\varepsilon }\times \mathcal {C}$ around z, to be the Euclidean balls $B_{\varepsilon }\times \{c\}$ for $c\in \mathcal {C}$ .

Second, there is the $\mathbb {R}^d$ action on $\Omega $ given by translating along unstable sets. More precisely, for $z\in \Omega $ with $z_0=\pi _0(z)$ not in a branch of $\Gamma $ and $t\in \mathbb {R}^d$ of small norm, the translation of $z = (z_0,z_1,z_2,\ldots )$ by t is

(4) $$ \begin{align} \varphi_t(z):= (z_0-t, z_1-D_\gamma^{-1}t, z_2-D_\gamma^{-2}t,\ldots), \end{align} $$

which is seen to preserve the condition in equation (1). The assumption that $(\Gamma ,\gamma )$ forces the border implies that this extends to an action of $\mathbb {R}^d$ on $\Omega $ ; the primitivity condition implies that this action is minimal, that is, every orbit is dense. Finally, the recognizability condition implies that the action is free and so every orbit is homeomorphic to $\mathbb {R}^d$ . Under these conditions, the action of $\mathbb {R}^d$ is uniquely ergodic [Reference SolomyakSol97, Theorem 3.1], where the unique invariant probability measure is $\mu $ .

Since the solenoid $\Omega $ has a local product structure of $B_{\varepsilon }(0)\times C_k^{\perp }(x)$ , where $B_{\varepsilon }(0)$ is an $\varepsilon $ -ball, the $\Phi $ -invariant measure $\mu $ has a local product structure of $\mbox {Leb}\times \nu _{x,k}$ , where $\mbox {Leb}$ is Lebesgue measure and $\nu _{x,k}$ is a measure on a local transversal $C_k^{\perp }(x)$ . This measure assigns to each local transversal $C^{\perp }_k(x)$ the measure $\nu _{x,k}(C_k^{\perp }(x))$ . Since $\{\nu _{x,k}\}$ forms a system of transverse invariant measures for the $\mathbb {R}^d$ action in the sense of [Reference Bowen and MarcusBM77], they will all be denoted by $\nu $ unless not doing so leads to ambiguity. Define

$$ \begin{align*} \hat\nu_{x,k} := \nu_{x,k}(C_k^{\perp}(x)). \end{align*} $$

Lemma 2.1. There exists a $C_\mu>1$ such that for any $x\in \Omega $ and $k\in \mathbb {N}_0$ ,

$$ \begin{align*} C_\mu^{-1}\unicode{x3bb}^{-k}\leq \hat\nu_{x,k} \leq C_\mu\unicode{x3bb}^{-k}. \end{align*} $$

2.1. Function spaces

Let $C^r(\Gamma )$ be the space of functions on the branched manifold which are $C^r$ smooth at the branch set. More precisely, let $i_k:B_k\rightarrow \Gamma $ be the inclusion of the $k\mathrm {th}$ -dimensional part of the branched set. A first order differential operator X on $\Gamma $ is one for which there exists a first order differential operator $X_k$ on $B_k$ which can be extended to X on $\Gamma $ , that is, so that $X_k i^{*}_k (f) = i_k^{*}(Xf)$ for all k. In other words, $C^r(\Gamma )$ is the set of functions $f\in C^r(\Gamma )$ such that

(5)

for some $X_k$ and for all $k>0$ . Thus, a first order differential operator X can be identified with a d-tuple $\{X_1,\ldots , X_d\}$ of first order differential operators, where $X_k$ on $B_k$ satisfies equation (5).

Another way to characterize $C^r(\Gamma )$ is as follows: since $\mathbb {R}^d$ acts on $\Omega $ by translation along unstable leaves, any $v\in \mathbb {R}^d$ defines the leafwise differential operators $\partial _{v}$ as

(6) $$ \begin{align} \partial_{v}f(z) = \lim_{s\rightarrow 0^+}\frac{f(z+sv)- f(z)}{s}. \end{align} $$

Let $C^r(\Omega )$ be the set of leafwise- $C^r$ functions on $\Omega $ with respect to the family of operators $\partial _{e_1},\ldots , \partial _{e_d}$ . Thus, $f\in C^r(\Gamma _k)$ if and only if $\pi ^{*}_kf\in C^r(\Omega )$ . Let

$$ \begin{align*}C^r_{tlc}(\Omega) := \bigcup_{k}\pi_k^{*}C^r(\Gamma_k)\quad\mbox{and}\quad C^\infty_{tlc}(\Omega) := \bigcup_{k}\pi_k^{*}C^\infty(\Gamma_k). \end{align*} $$

For an $h\in C^\infty _{tlc}(\Omega )$ and fixed $p\in \Omega $ , the function $f_h(t):=h\circ \varphi _t(p):\mathbb {R}^d\rightarrow \mathbb {R}$ is called p-equivariant.

For a function f on $\Omega $ , define the transversal Hölder seminorm

(7) $$ \begin{align} |f|^{\perp}_{\alpha} = \sup_{x\in \Omega_\gamma} \sup_{x\neq y\in C^{\perp}_0(x)}\frac{| f(x)-f(y)|}{d(x,y)^\alpha}, \end{align} $$

and let $H_{\alpha }^{\perp }(\Omega )$ be the Banach space of transversally $\alpha $ -Hölder functions with norm

(8) $$ \begin{align} \|f\|^{\perp}_{\alpha} = \|f\|_{C^0} + |f|^{\perp}_{\alpha}. \end{align} $$

Note that for any $0<\beta <\alpha $ , there is a compact inclusion $H^{\perp }_{\alpha } \subset H^{\perp }_\beta $ .

Remark 2.2. Since the local transversals $C^{\perp }_k$ are totally disconnected sets, the spaces $H_{\alpha }^{\perp }(\Omega )$ are non-trivial for every real $\alpha>0$ . This is a significant difference from stable sets which are smooth, and $\alpha $ here controls the analogous quantity for smoothness in a totally disconnected direction of $\Omega $ .

Let $C^r(\Omega )$ denote the set of functions f on $\Omega $ such that $\partial _{v_{i_1}}\cdots \partial _{v_{i_r}} f$ is continuous for any choice of r vectors $v_{i_j}$ in an orthonormal basis $\{v_1,\ldots v_d\}$ of $\mathbb {R}^d$ . Finally, define for $r\in \mathbb {N}$ and $\alpha>0$ ,

(9) $$ \begin{align} C^r_{\alpha}(\Omega) = \bigg\{f\in C^r(\Omega)\cap H^{\perp}_{\alpha}(\Omega): \sum_{0\leq p \leq r}\sum_{|i|=p} \|\partial^if\|_{C^0}+|\partial^i f|^{\perp}_{\alpha} < \infty \bigg\} \end{align} $$

to be the space of functions which not only are $C^r$ smooth in the leaf direction but also whose derivatives up to order r are transversally $\alpha $ -Hölder. Here the traditional multiindex notation $i = (i_1,\ldots , i_d)$ has been used, with $|i|=i_1+\cdots + i_d$ . This is a Banach space under the norm

(10) $$ \begin{align} \|f\|_{r,\alpha}:= \sum_{0\leq p \leq r}\sum_{|i|=p} \|\partial^if\|_{C^0}+|\partial^i f|^{\perp}_{\alpha}. \end{align} $$

Given a function $f\in L^1(\Omega )$ and $k\in \mathbb {N}_0:= \mathbb {N}\cup \{0\}$ , set

(11) $$ \begin{align} \Pi_k f(x):= \hat\nu_{k,x}^{-1}\int_{C_k^{\perp}(x)} f(z)\, d\nu_{k,x},\quad \Delta_kf := f-\Pi_k f,\quad\mbox{and}\quad\delta_kf := \Pi_kf-\Pi_{k-1}f. \end{align} $$

These functions will be crucial for most results of this paper, so let me take some time to discuss how one can think of them. Any $f\in L^1_\mu (\Omega )$ is a function of infinitely many variables, since this is how the solenoid is defined. The function $\Pi _k f$ is obtained by integrating along local transversals and consequently $\Pi _kf$ is transversally locally constant. This means that $\Pi _k f$ only depends on finitely many coordinates, that is, there is a function $g_k\in L^1(\Gamma )$ such that $\Pi _k f = \pi ^{*}_k g_k$ . Thus, $\Pi _kf$ can be thought of as the best approximation to f if we can only consider the first k coordinates. We will see below that $\Pi _kf\rightarrow f$ in a satisfying sense as $k\rightarrow \infty $ .

If $\Pi _k f$ is an approximation to f using the first k coordinates, then $\Delta _k f$ is the error in this approximation. If $\Pi _kf\rightarrow f$ in some sense, then one should expect that $\Delta _kf\rightarrow 0$ in the same sense (this will come up later). Finally, if $\Pi _kf$ is an approximation to f using the first k coordinates, then $f_k:=\delta _k f$ is the best approximation to f using only the $k\mathrm {th}$ coordinate. So after setting $\Pi _{-1}f =0$ , the approximation $\Pi _k f$ can be written as the finite sum

$$ \begin{align*} \Pi_k f = \sum_{i=0}^k \delta_k f = \sum_{i=0}^k f_k \end{align*} $$

and so if $\Pi _kf\rightarrow f$ , then f can be written as a sum $f = \sum f_k$ in a canonical way. These of course are rough descriptions of how one can think of these functions but since they will appear regularly in this paper, one should have a good way to think of them.

A more precise description of what is happening involves conditional expectation. For $k\in \mathbb {N}_0$ , let $\mathcal {A}_k$ be the $\sigma $ -algebra generated by the preimages $\pi ^{-1}_k(A)$ of Borel sets $A\subset \Gamma _k$ . This is an increasing sequence of sub- $\sigma $ -algebras of the Borel $\sigma $ -algebra $\mathcal {A}$ of $\Omega $ . The conditional expectation $E(\cdot |\mathcal {A}_k):L^1(\Omega ,\mathcal {A},\mu )\rightarrow L^1(\Omega ,\mathcal {A}_k,\mu )$ map coincides with $\Pi _k$ , that is, for any $f\in L^1(\Omega ,\mu )$ , $E(f|\mathcal {A}_k) = \Pi _kf$ , and $\nu _{k,x}$ is the conditional measure of this conditional expectation. By the increasing martingale theorem [Reference Einsiedler and WardEW11, Theorem 5.5], $\Pi _kf\rightarrow f$ almost everywhere and in $L^1$ .

Using the notation $f_k = \delta _k f$ from above, for $r,\alpha \geq 0$ , let

Let me make two comments which motivate the definition of these function spaces. The first one comes from the algebraic setting: if S is the inverse limit of locally expanding affine linear maps of $\mathbb {T}^d$ , then $L^2_\mu $ is spanned by a Fourier basis, and so one needs to quantify the decay rates of the Fourier coefficients to capture degrees of regularity. The generalization of this idea leads to the space $\mathcal {S}^r_{\alpha }$ as defined above. The second reason is that the representation of a function as a sum of pullbacks of distinct functions on approximants is canonical: if $f = \pi ^{*}_0f^{(0)}$ , then f is transversally locally constant, so $\delta _kf = 0$ for all $k>0$ , and so the f is uniquely represented as a sum of finitely many terms.

Proposition 2.3. For the spaces $\mathcal {S}^r_{\alpha }(\Omega )$ and $C^r_{\alpha }(\Omega )$ defined above with $r\in \mathbb {N},\alpha>0$ :

1. (i) for any $v\in \{v_1,\ldots , v_d\}$ , if $f\in \mathcal {S}^r_{\alpha }(\Omega )$ , then $\partial _{v}f\in \mathcal {S}^{r-1}_{\alpha +1}(\Omega )$ ,

2. (ii) for any $\varepsilon \in (0,\alpha )$ , $\mathcal {S}^r_{\alpha }(\Omega )\subset C^r_{\alpha }(\Omega )$ densely with respect to the norm of $C^r_{\alpha -\varepsilon }(\Omega )$ .

Remark 2.4. I would like to remark on the surprising feature in item (i): taking leafwise derivatives increases regularity in the transverse direction. Since this is the Hölder regularity, this depends on the metric used. The form adopted here is the one which corresponds to the natural choice of transversal metric in equation (3), as well as using the same number $\unicode{x3bb} _0$ in defining the spaces $\mathcal {S}^r_{\alpha }(\Omega )$ .

Proof of Proposition 2.3

For item (i), if $f\in \mathcal {S}^r_{\alpha }$ , then using equations (4) and (6),

$$ \begin{align*} \partial_v f(z) = \sum_{k\geq 0} \partial_v f_k(z) = \sum_{k\geq 0} \pi^{*}_k( \nabla f^{(k)} \cdot D^{-k}_\gamma v)(z) \end{align*} $$

and

So $\partial _v f \in \mathcal {S}_{\alpha +1}^{r-1}(\Omega )$ .

For item (ii), for $f\in C^r_{\alpha }$ , consider the approximations $\Pi _k f$ . For every k, these approximations are all in $\mathcal {S}^r_{\alpha }$ for any $\alpha>0$ since they are transversally local functions. What needs to be proved is that for any multiindex i with $|i|\leq r$ ,

$$ \begin{align*} \|\partial^i\Pi_kf - \partial^if\|_{C^0}+ |\partial^i\Pi_kf - \partial^if|^{\perp}_{\alpha-\varepsilon}\rightarrow 0 \end{align*} $$

as $k\rightarrow \infty $ .

Let i be one such multiindex. Then

(12)

and so $\|\partial ^i\Pi _kf - \partial ^if\|_{C^0}\rightarrow 0$ . Here it was used that not only f but its derivatives are transversally $\alpha $ -Hölder.

Let $x\in \Omega $ and suppose that $a,b\in C^{\perp }_\ell (x)$ . Then if $\ell \leq k$ ,

where equation (12) was used in simplifying the numerator. If $\ell>k$ ,

And so $|\partial ^i\Pi _kf - \partial ^if|^{\perp }_{\alpha -\varepsilon }\rightarrow 0$ , so the proof is concluded.

3. Transversal Hölder cohomology for Wieler solenoids

Wieler solenoids, being defined as an inverse limit of surjective and locally expaning maps have the advantage that their (Čech) cohomology can be explicitly computed. At the most basic level, this can be done with coefficients in $\mathbb {Z}$ through

$$ \begin{align*}\check{H}^{*}(\Omega;\mathbb{Z}) = \lim_{n\rightarrow\infty} (\check{H}^{*}(\Gamma;\mathbb{Z}),\gamma^{*}).\end{align*} $$

By the universal coefficient theorem and universality of inverse limits, the cohomology with real coefficients is

(13) $$ \begin{align} \check{H}^{*}(\Omega;\mathbb{R}) = \lim_{n\rightarrow\infty} (\check H^{*}(\Gamma;\mathbb{R}),\gamma^{*}). \end{align} $$

Likewise, since there is a free $\mathbb {R}^d$ action on $\Omega $ , there is an associated Lie-algebra cohomology defined as follows. Let $X_1,\ldots , X_d$ be an orthonormal frame of $\mathbb {R}^d$ . Define the operators $\partial _i$ using $X_i$ as in equation (6). Let $C^\infty (\Omega ;\Lambda \mathbb {R}^{d*})$ be the space of leafwise smooth sections of $\Omega $ to $\Lambda \mathbb {R}^{d*}$ , the graded exterior algebra of $\mathbb {R}^{d*}$ . In other words, $C^\infty (\Omega ;\Lambda \mathbb {R}^{d*})$ is the space of functions $f:\Omega \rightarrow \Lambda \mathbb {R}^{d*}$ such that $\partial _{i_1}\cdots \partial _{i_k}f\in C^\infty (\Omega ;\Lambda \mathbb {R}^{d*})$ for any finite collection of indices $i_1,\ldots , i_k$ .

Let $d:C^\infty (\Omega ;\Lambda ^k\mathbb {R}^{d*})\rightarrow C^\infty (\Omega ;\Lambda ^{k+1}\mathbb {R}^{d*}) $ be the exterior differential operator defined as follows. Let $\{dx_1,\ldots , dx_d\}$ denote the dual frame to $\{X_1,\ldots , X_d\}$ . Then

$$ \begin{align*}d(f dx_{i_1}\wedge\cdots \wedge dx_{i_k}) = \sum_{i=1}^d \partial_if dx_i\wedge dx_{i_1}\wedge \cdots \wedge dx_{i_k}.\end{align*} $$

It is immediate to verify that this operator satisfies $d^2=0$ . Moreover, there is natural subcomplex $C^\infty _{tlc}(\Omega ;\Lambda ^k\mathbb {R}^{d*})$ of sections with tlc coefficients.

Definition 3.1. The Lie-algebra cohomology $H^{*}(\Omega )$ of $\Omega $ is the cohomology of the differential complex $(C^\infty (\Omega ;\Lambda \mathbb {R}^{d*}),d)$ . That is,

$$ \begin{align*} H^{\bullet}(\Omega):= \frac{\ker \{d:C^\infty(\Omega;\Lambda^{\bullet}\mathbb{R}^{d*})\rightarrow C^\infty(\Omega;\Lambda^{\bullet+1}\mathbb{R}^{d*})\}}{\mathrm{Im}\, \{d:C^\infty(\Omega;\Lambda^{\bullet-1}\mathbb{R}^{d*})\rightarrow C^\infty(\Omega;\Lambda^{\bullet}\mathbb{R}^{d*})\}}. \end{align*} $$

The transversally locally constant (tlc) Lie-algebra cohomology is the cohomology of the subcomplex of tlc functions:

$$ \begin{align*} H^{\bullet}_{tlc}(\Omega):= \frac{\ker \{d:C^\infty_{tlc}(\Omega;\Lambda^{\bullet}\mathbb{R}^{d*})\rightarrow C^\infty_{tlc}(\Omega;\Lambda^{\bullet+1}\mathbb{R}^{d*})\}}{\mathrm{Im}\, \{d:C^\infty_{tlc}(\Omega;\Lambda^{\bullet-1}\mathbb{R}^{d*})\rightarrow C^\infty_{tlc}(\Omega;\Lambda^{\bullet}\mathbb{R}^{d*})\}}. \end{align*} $$

Finally, there is one more type of cohomology which is relevant here: since for $p\in \Omega $ a form $\eta \in C^\infty (\Omega ;\Lambda ^{*}\mathbb {R}^{d*})$ defines a p-equivariant form $\omega _\eta (t) := \varphi ^{*}_t\eta :\mathbb {R}^d\rightarrow \Lambda ^{*}\mathbb {R}^{d*}$ , the p-equivariant cohomology $H^{*}_p(\Omega ;\mathbb {R})$ is defined as the cohomology of this subcomplex of the de Rham complex of $\mathbb {R}^d$ .

Kellendonk and Putnam [Reference Kellendonk and PutnamKP06] proved that the p-equivariant cohomology is isomorphic to the tlc Lie-algebra cohomology, which in turn is isomorphic to the Čech cohomology in equation (13) with coefficients in $\mathbb {R}$ . In general, it is not true that $H^{\bullet }_{tlc}(\Omega )$ is isomorphic to $H^{\bullet }(\Omega )$ . These two cohomologies capture two extremes of regularity in the transverse direction: functions in $C^\infty $ satisfy $|f|^{\perp }_0<\infty $ , whereas $g\in C^\infty _{tlc}$ has $|g|_{\alpha }^{\perp }<\infty $ for all $\alpha \geq 0$ . Thus, it is natural to ask how much regularity is needed in the transverse direction to recover the real Čech cohomology. This is the transversally Hölder cohomology.

The rest of the section is devoted to proving that instead of looking at the cohomology of $C^\infty _{tlc}$ sections, one could consider the cohomology of sections with coefficients in $\mathcal {S}^r_{\alpha }(\Omega )$ or $C^r_{\alpha }$ . To do this, a name for this type of cohomology is needed.

Definition 3.2. The space $\Psi ^m_{r,\alpha }$ is the space of m-forms $\eta :\Omega \rightarrow \Lambda ^m\mathbb {R}^{d*}$ with coefficients in $\mathcal {S}^{r}_{\alpha }$ . That is, an element $\Psi ^m_{r,\alpha }$ can be written as

$$ \begin{align*}\eta = \sum_{I\in I_m} \eta_I dx_I,\end{align*} $$

where each $dx_I$ is an m-form of the form $dx_{i_1}\wedge \cdots \wedge dx_{i_m}$ , and $\eta _I\in \mathcal {S}^{r}_{\alpha }$ for all $I\in I_m$ . Let

$$ \begin{align*} \mathcal{Z}^m_{r,\alpha}:= \ker \{d:\Psi^m_{r,\alpha}\rightarrow \Psi^{m+1}_{r-1,\alpha+1}\}\quad\mbox{and}\quad\mathcal{B}^m_{r,\alpha}:= \mathrm{Im}\, \{d:\Psi^{m-1}_{r+1,\alpha-1}\rightarrow \Psi^{m}_{r,\alpha} \}, \end{align*} $$

and define the $\mathcal {S}^{r}_{\alpha }$ -cohomology of $\Omega $ as

$$ \begin{align*}\mathcal{H}_{r,\alpha}^{*}(\Omega):= \mathcal{Z}^{*}_{r,\alpha}/\mathcal{B}^{*}_{r,\alpha},\end{align*} $$

which is the Lie algebra cohomology with coefficients in $\Psi _{r,\alpha }^{*}$ .

Theorem 3.3. Fix $r\in \mathbb {N}$ and $\alpha>1$ , and let $\eta \in \mathcal {Z}^m_{r,\alpha }$ . Then there exists $\eta '\in C^\infty _{tlc}(\Omega ;\Lambda ^m\mathbb {R}^{d*})$ and $\omega \in \Psi ^{m-1}_{r+1,\alpha -1} $ such that $\eta -\eta ' = d\omega $ . That is, for $r\geq 1$ and $\alpha>1$ ,

$$ \begin{align*}\mathcal{H}^{*}_{r,\alpha}(\Omega)\cong H^{*}_{tlc}(\Omega)\cong \check{H}^{*}(\Omega;\mathbb{R}). \end{align*} $$

The rest of this section is devoted to the proof of this theorem. First, we start by reviewing de Rham cohomology for branched manifolds, then de Rham regularization for branched manifolds, and finally we put all of this together in the inverse limit structure.

Let $H^{*}_{r,tlc}(\Omega )$ be the Lie-algebra cohomology with $C^r_{tlc}$ coefficients. That is, two forms $\eta _1,\eta _2$ are in the same cohomology class if there exists an $\omega \in C^r_{tlc}$ such that $\eta _1-\eta _2 = d\omega $ .

Let $\Gamma $ be a flat branched manifold of dimension d, where the set of branches is denoted by $B\subset \Gamma $ . Denote by $I^k$ the k-skeleton of $\Gamma $ . For each $x\in B_k:= B\cap (I^k\setminus I^{k-1})$ , there is a natural tangent space $T_x\Gamma $ of dimension k and a corresponding cotangent space. Let $\Delta _{\ell }^{k}$ be the set of smooth $\ell $ -forms on $B_k$ , on which there is the usual de Rham differential operator $d_k: \Delta _{\ell }^{k}\rightarrow \Delta _{\ell +1}^{k}$ and which are included in $\Gamma $ using maps $i_k:B_k \rightarrow \Gamma $ . To give the entire branched space a smooth structure, consider the space of smooth maps

$$ \begin{align*}\Delta_k(\Gamma)= \{\omega:\Gamma\rightarrow \Lambda^k\mathbb{R}^{d*}\}\end{align*} $$

with coboundary operators $d:\Delta _k(\Gamma )\rightarrow \Delta _{k+1}(\Gamma )$ satisfying $d_k\circ i_k^{*} = i_k^{*}\circ d$ for all $k\geq 0$ , that is, the operators which are conjugated to $d_k$ by $i_k^{*}$ for all k:

(14)

Another way to characterize $\Delta _m(\Gamma )$ is the set of m-forms $\eta $ on $\Gamma $ such that $\pi ^{*}_k\eta $ is a $C^\infty $ m-form on $\Omega $ .

Sadun [Reference SadunSad07] proved that $H_{dR}^{*}(\Gamma )$ is isomorphic to the real Čech cohomology of $\Gamma $ .

Lemma 3.6. For $\rho \in [1,\infty ]$ , let $\|\cdot \|':H^{*}_{\rho ,tlc}(\Omega )\rightarrow \mathbb {R}$ be a norm. Then there exists a K depending on the norm and on $\gamma $ such that

$$ \begin{align*}\|[\pi^{*}_k \eta]\|'\leq K \|\eta\|_{C^r(\Gamma_k)}\end{align*} $$

for any closed $\eta \in \Delta _*(\Gamma _k) $ which is $C^r$ , $k\geq 0$ , and $r\in [1,\rho ]$ .

Proof. Since $H^i_{tlc}(\Omega )$ is finite dimensional, by Proposition 3.4, it will suffice to prove it for some norm. Let $\|\cdot \|$ be some norm on $H_i(\Gamma _k;\mathbb {R})$ and denote by $\|\cdot \|$ the dual norm on $H^i(\Gamma _k;\mathbb {R})$ . That is, for a closed i-form $\eta \in \Delta _i(\Gamma _k)$ ,

$$ \begin{align*}\|[\eta]\| = \sup_{0\neq [c] \in H_i(\Gamma_k;\mathbb{R})} \bigg|\frac{[\eta]([c])}{\|[c]\|}\bigg| = \sup_{0\neq [c] \in H_i(\Gamma_k;\mathbb{R})} \|[c]\|^{-1}\bigg|\!\int_c\eta\bigg|.\end{align*} $$

The $i\mathrm {th}$ skeleton of $\Gamma _k$ is finitely generated and so there is a collection of representative cycles $c_1,\ldots , c_m$ of a basis of $H_i(\Gamma _k;\mathbb {Z})$ . Thus, for an integral class $[c] = \sum _{j=1}^m a_j(c) c_j\in H_i(\Gamma _k;\mathbb {Z})$ , the absolute value of $\int _c\eta $ can be bounded as

(15) $$ \begin{align} \bigg|\!\int_c\eta\bigg|&= \bigg|\sum_{j=1}^m a_j(c) \int_{c_j}\eta\bigg|\leq \sum_{j=1}^m |a_j(c)| \bigg|\!\int_{c_j}\eta\bigg| \leq \|\eta\|_{C^r(\Gamma_k)}\sum_{j=1}^m |a_j(c)\bigg| \mathrm{Vol}_i(c_j)\nonumber \\ &\leq K' (\max_j\mathrm{Vol}_i(c_j)) \|[c]\|\|\eta\|_{C^r(\Gamma_k)} \leq K"\|[c]\|\|\eta\|_{C^r(\Gamma_k)}, \end{align} $$

where $K'$ comes from the equivalence of the $L^1$ norm and $\|\cdot \|$ in $H_i(\Gamma _k;\mathbb {R})$ , and $\mathrm {Vol}_i(c_j)$ is the i-dimensional volume of the cycle $c_j$ . Here, it was used that $\int _{c_j}\eta $ can be bounded by the volume of the cycle times the $C^1$ norm of $\eta $ . Thus, it follows that

$$ \begin{align*} \|[\pi^{*}_k\eta]\| = \|[\eta]\| = \sup_{0\neq c \in H_i(\Gamma_k;\mathbb{R})} \|c\|^{-1}\bigg|\!\int_c\eta\bigg| \leq K \|\eta\|_{C^r(\Gamma_k)}\end{align*} $$

for any $r\geq 1$ .

The following completes the proof of Theorem 3.3.

Proof. Let $\eta :\Omega \rightarrow \Lambda ^{*}\mathbb {R}^{d*}$ be a closed form with coefficients in $\mathcal {S}^r_{\alpha }(\Omega )$ with $r\geq 1$ and $\alpha>1$ . By Proposition 3.4, it suffices to show that there is a form $\eta '\in H^{*}_{r,tlc}(\Omega )$ such that $\eta -\eta ' = d\omega $ for some $\omega $ with coefficients in $\mathcal {S}^{r+1}_{\alpha -1}$ .

By the definition of $\mathcal {S}^r_{\alpha }(\Omega )$ for each $k\geq 0$ , there exists $\eta _k\in \Delta _*(\Gamma _k)$ such that $\eta $ is canonically expressed as $\eta = \sum \nolimits _{k\geq 0} \delta _k\eta _k= \sum _{k\geq 0} \pi ^{*}_k\eta _k$ (see §2.1), where there are infinitely many non-zero terms, as that would otherwise make $\eta $ a tlc form. Note that this expression for $\eta $ has the property that $\eta _k\neq \gamma ^{*} \eta '$ for some $\eta '\in \Delta _*(\Gamma _{k-1})$ . Now define the sequence of forms

$$ \begin{align*}\eta^{(n)}:= \sum_{k=0}^n \pi^{*}_k\eta_k.\end{align*} $$

Each of these forms is a closed tlc form. Indeed,

$$ \begin{align*} 0=d\eta = d\bigg(\sum_{k\geq 0} \pi^{*}_k\eta_k \bigg)= \sum_{k\geq 0} d\pi^{*}_k\eta_k = \sum_{k\geq 0} d\delta_k \eta = \sum_{k\geq 0} \delta_k d\eta, \end{align*} $$

since $d\Pi _k\eta = \Pi _k d\eta $ by the preservation of the transverse measure. Thus, $d\delta _k\eta = \delta _kd\eta = \delta _k 0 = 0$ for all k. It follows that $\eta ^{(n)}$ has a tlc cohomology class $[\eta ^{(n)}]\in H^{*}_{r,tlc}(\Omega )$ . Observe that $[\eta ^{(n)}]$ is a convergent sequence. Indeed, by Lemma 3.6, for $n>m>0$ ,

Thus, $[\eta ]$ should be assigned the cohomology class $\lim _{n\rightarrow \infty }[\eta ^{(n)}] \in H^{*}_{r,tlc}(\Omega )$ and so the goal is to find a tlc representative of this class and show that $\eta $ is cohomologous to it.

Recall the eventual range

$$ \begin{align*} ER_i(\Omega):= (\gamma^{*})^{\beta_i}\check{H}^i(\Gamma;\mathbb{R})\subset \check H^i(\Gamma;\mathbb{R}), \end{align*} $$

where $\beta _i = \dim H^i(\Gamma ;\mathbb {R})$ . Any class $c\in \check H^i(\Omega ;\mathbb {R})$ is represented by a class in $ER_i(\Omega )\subset \check H^i(\Gamma _{\beta _i};\mathbb {R})$ . Since $\check H^i(\Gamma ;\mathbb {R})$ is isomorphic to $H^i_{dR}(\Gamma )$ [Reference SadunSad07, Appendix A], any class $c\in H^i_{tlc}(\Omega )$ has a representative $ \pi _{\beta _i}^{*} \eta _c$ coming from the $\beta _i^{\mathrm {th}}$ projection map.

Let $\pi ^{*}_{\beta _i}\eta ^{\prime }_k$ be the form cohomologous to $\pi _k^{*}\eta _k$ and let

$$ \begin{align*} \eta_{(n)} = \sum_{k=0}^n \pi^{*}_{\beta_i}\eta^{\prime}_k = \pi^{*}_{\beta_i}\bigg(\sum_{k=0}^n \eta^{\prime}_k\bigg). \end{align*} $$

Then $\eta _{(n)} - \eta ^{(n)} = d\omega _n$ for all n. In addition,

$$ \begin{align*} \eta_{(\infty)} := \lim_{n\rightarrow \infty}\sum_{k=0}^n \pi^{*}_{\beta_i}\eta^{\prime}_k = \pi^{*}_{\beta_i}\bigg(\lim_{n\rightarrow \infty}\sum_{k=0}^n \eta^{\prime}_k\bigg) \end{align*} $$

is a $C^r$ tlc function which is cohomologous to $\eta $ .

3.1. Application: deviations of ergodic averages

The spectrum of $\Phi ^{*}:\check H^d(\Omega ;\mathbb {R})\rightarrow \check H^d(\Omega ;\mathbb {R})$ gives rates of convergence of ergodic integrals. This was first proved in [Reference SadunSad11] in the self-similar case and later in [Reference Schmieding and TreviñoST18] in the self-affine case. The class of functions used in those results were $C^\infty _{tlc}$ . Theorem 3.3 implies that the same rates of convergence can now be given for functions in $\mathcal {S}^r_{\alpha }$ for $r\in \mathbb {N}$ and $\alpha>1$ . The statement will be provided here without proof as Theorem 3.3 allows the argument in [Reference Schmieding and TreviñoST18] to carry over verbatim.

Before stating the theorem, some notation needs to be established. Denote by $|\nu _1|> \cdots > |\nu _r| > 0$ the norms of the r distinct eigenvalues of the map $\Phi ^{*}$ acting on $H^d$ . Let $E_i$ be the generalized eigenspaces for the action of $\Phi $ on $H^d(\Omega ;\mathbb {R})$ induced by the map $\Phi ^{*}$ corresponding to the eigenvalue $\nu _i$ . The subspaces $E_i$ are decomposed as

$$ \begin{align*} E_i = \bigoplus_{j=1}^{\kappa(i)} E_{i,j}, \end{align*} $$

where $\kappa (i)$ is the size of the largest Jordan block associated with $\nu _i$ , as follows. For each i, we choose a basis of classes $\{[\eta _{i,j,k}]\}$ with the property that $\langle [\eta _{i,j,1}],[\eta _{i,j,2}],\ldots , [\eta _{i,j,s(i,j)}]\rangle = E_{i,j}$ and

(16) $$ \begin{align} \Phi^{*} [\eta_{i,j,k}] = \left\{\!\!\begin{array}{ll} \nu_i [\eta_{i,j,k}] + [\eta_{i,j-1,k}] &\mbox{ for } j>1, \\ \nu_i [\eta_{i,j,k}] &\mbox{ for } j=1. \end{array} \right. \end{align} $$

Definition 3.8. The rapidly expanding subspace $E^+(\Omega ) \subset H^d(\Omega )$ is the direct sum of all generalized eigenspaces $E_i$ of $\Phi ^{*}$ such that the corresponding eigenvalues $\nu _i$ of $\Phi ^{*}$ satisfy

(17)

The subspace $E^{++}\subset E^+$ consists of all vectors for which the inequality in equation (17) is strict.

We order the indices of distinct subspaces of $E^+(\Omega )$ as follows. First, we set $I^+ = I^{+,>} \cup I^{+,=}$ to be the index set of classes $[\eta _{i,j,k}]$ which form a generalized eigenbasis for $E^+(\Omega )$ , where the indices in $I^{+,>}$ contain vectors corresponding to a strict inequality in equation (17) and the indices in $I^{+,=}$ correspond to vectors associated to eigenvalues which give an equality in equation (17). The set $I^{+,=}$ can be empty but $I^{+,>}$ always has at least one element. The set $I^+$ is partially ordered: $(i,j,k)\leq (i',j',k')$ if $L(i,j,T)T^{ds_i}\geq L(i',j',T)T^{ds_{i'}}$ for $T>1$ , where

(18) $$ \begin{align} L(i,j,T) = \left\{\!\!\begin{array}{ll} (\log T)^{j-1} &\mbox{if } \nu_i \mbox{ satisfies equation~(17) strictly,} \\ (\log T)^{j} &\mbox{if } \nu_i \mbox{ satisfies equality in equation~(17),} \end{array} \right. \end{align} $$

and $s_i = {\log |\nu _i|}/{\log \nu _1}$ . The order does not depend on the indices k.

By passing to a power, we can assume that $D_\gamma \in GL^+(d,\mathbb {R}) = \mathrm {exp}( \mathfrak {gl}(d,\mathbb {R}))$ . Let $a\in \mathfrak {gl}(d,\mathbb {R})$ be the matrix which satisfies $\exp (a) =D_\gamma $ and let $g_t = \exp (at)$ . Letting $B_1$ denote the unit ball, define the averaging family $\{B_T\}_{T\geq 1}$ by

(19) $$ \begin{align} B_T = g_{\sigma \log T} B_1, \end{align} $$

where $\sigma = d/\log \det A$ . As such, we have that $\mathrm {Vol}(B_T) = \mathrm {Vol}(B_1) T^d$ . Let $\rho = \mathrm {dim}\, E^+(\Omega _\Lambda )$ .

Theorem 3.9. For $r\in \mathbb {N}$ and $\alpha>1$ , there exist a constant $C_{\gamma }$ and $\rho $ $\mathbb {R}^d$ -invariant distributions $\{\mathcal {D}_{i,j,k}\}_{(i,j,k)\in I^+}$ such that, for any $f \in \mathcal {S}^r_{\alpha }(\Omega )$ , if there is an index $(i,j,k)$ such that $\mathcal {D}_{i',j',k'} (f) = 0$ for all $(i',j',k')<(i,j,k)$ but $\mathcal {D}_{i,j,k} (f) \neq 0$ , then for $T>3$ and any $x \in \Omega _{\gamma }$ ,

$$ \begin{align*} \bigg| \!\int_{B_T} f\circ \varphi_s(x)\,ds \bigg| \leq C_{\gamma,f} L(i,j,T)T^{d ({\log|\nu_{i}|}/{\log \unicode{x3bb}})}. \end{align*} $$

Moreover, if $\mathcal {D}_{i,j,k}(f) = 0$ for all $(i,j,k)\in I^+_\Lambda $ , then

(20)

for all $T>1$ .

4. Ruelle spectrum and quantitative mixing

An application of transverse Hölder cohomology is the construction of anisotropic Banach spaces for solenoids. This section is dedicated to proving the quantitative mixing results for $\Phi $ . To do this, it is necessary to introduce so-called anisotropic spaces of functions. I am particularly inspired by [Reference Butterley, Kiamari and LiveraniBKL22, Reference Faure, Gouëzel and LanneauFGL19] and the PhD dissertation of D. Galli, and so I will follow some of the ideas there.

4.1. Anisotropic Banach spaces for flat Wieler solenoids

Let $\varphi :\Omega \rightarrow \wedge ^kT^{*}\mathbb {R}^d$ be an m-form. Analogous to equation (7), let

(21) $$ \begin{align} |\varphi|^{\perp}_{\alpha,m} = \sup_{x\in \Omega_\gamma} \sup_{x\neq y\in C^{\perp}_0(x)}\frac{\| \varphi(x)-\varphi(y)\|}{d(x,y)^\alpha}. \end{align} $$

At first, it may seem like $\varphi (x)-\varphi (y)$ is not defined, as each summand lives on a different fiber of the bundle. However, since the leaves of the foliation are flat and dense in $\Omega $ , parallel transport makes this operation unambiguous. Define the space $H_{\alpha ,m}^{\perp }$ of $\alpha $ -Hölder m-forms to be those for which

(22) $$ \begin{align} \|\varphi\|^{\perp}_{\alpha,m} := \|\varphi\|_{C^0}+|\varphi|^{\perp}_{\alpha,m}<\infty. \end{align} $$

This is a Banach space, and if $\alpha>\beta $ , we have that $H^{\perp }_{\alpha ,m}\subset H^{\perp }_{\beta ,m}$ compactly. Denote by $B_{\varepsilon ,m}^{\perp }(\alpha )\subset H_{\alpha ,m}^{\perp }$ the $\varepsilon $ -ball with respect to equation (22). Note that $B_{1,m}^{\perp }(\alpha )\subset B_{1,m}^{\perp }(\alpha -\delta )$ for all $\delta $ small enough.

For the sake of convenience, it will be assumed that $D_\gamma $ has no Jordan blocks. As such, let $v_1,\ldots , v_d$ be a normalized basis of $\mathbb {R}^d$ which are also eigenvectors for $D_\gamma $ : $D_\gamma v_i = \unicode{x3bb} _i v_i$ with $\unicode{x3bb} _i>1$ and $\unicode{x3bb} = \unicode{x3bb} _1\cdots \unicode{x3bb} _d$ . Given this choice, for any $i\in \{1,\ldots , d\}$ , $\partial _i$ will denote the differential operator $\partial _{v_i}$ and for a multiindex $i = (i_1,\ldots , i_d)$ , we will denote by $\partial ^i= \partial ^{i_1}_{1}\cdots \partial ^{i_d}_d$ and $|i| = i_1+\cdots +i_d$ . With this notation, let

and let $\mathcal {B}^{r,\alpha }_{m}$ be the completion of $\Psi ^m_{r,\alpha }$ with respect to and set $\Lambda = {\log \unicode{x3bb} }/{\log \unicode{x3bb} _0}$ .

To the uninitiated reader, it is worth pointing out that functions in the anisotropic Banach spaces $\mathcal {B}_m^{r,\alpha }$ play two simultaneous roles, which become evident from the way the norm was defined: first, they serve as functions in the Euclidean variable; whereas in the transversal variable, they serve the roles of currents.

Recall the Hodge- $\star $ operator which sends m-forms to $(d-m)$ forms $\star : \Psi ^m_{r,\alpha }\rightarrow \Psi ^{d-m}_{r,\alpha }$ . The invariant probability measure $\mu $ gives a canonical choice of the Lebesgue volume element on the $\mathbb {R}^d$ -leaves of $\Omega $ , and thus there is a canonical volume element $dt = \star 1$ . With this choice, the Hodge- $\star $ operator gives a canonical bijection between functions and tangential d-forms: for $h\in \mathcal {S}^r_{\alpha }$ , $\star h = h(\star 1) = h\, dt\in \Psi ^d_{r,\alpha }$ .

Proposition 4.1. $\mathcal {B}^{r',\alpha '}_{m}\subset \mathcal {B}^{r,\alpha }_{m}$ if $r'\geq r$ and $\alpha '\leq \alpha $ , and the inclusion is compact if $r'>r$ and $\alpha>\alpha '+\Lambda >\Lambda >0$ .

The inclusions follow from the definitions of the norms so what is left to prove is the compactness of the inclusion. The following compactness criterion [Reference Faure, Gouëzel and LanneauFGL19, §2.2] will be used: let $\mathcal {B}\subset \mathcal {C}$ be two Banach spaces and assume that for any $\varepsilon>0$ , there exist finitely many continuous linear forms $L_1,\ldots , L_m$ on $\mathcal {B}$ such that for any $x\in \mathcal {B}$ ,

$$ \begin{align*}\|x\|_{\mathcal{C}}\leq \varepsilon \|x\|_{\mathcal{B}}+ \sum_{i\leq m}|L_i(x)|.\end{align*} $$

Then the inclusion of $\mathcal {B}$ in $\mathcal {C}$ is compact.

To apply the criterion, several estimates will need to be obtained. For an m-form $\varphi :\Omega \rightarrow \wedge ^m T^{*}\mathbb {R}^d$ and $k\in \mathbb {N}_0$ , set $\hat \nu _{k,x}:=\nu _{k,x}(C^{\perp }_k(x))$ , recall equation (11):

(23) $$ \begin{align} \Pi_k\varphi(x):= \hat\nu_{k,x}^{-1}\int_{C_k^{\perp}(x)} \varphi\,d\nu_{k,x}\quad \mbox{and}\quad\Delta_k:= \varphi-\Pi_k\varphi, \end{align} $$

both of which are m-forms. Recall that $\Pi _k\varphi $ is transversally locally constant. That is, $\Pi _k\varphi (z)$ depends only on the first k coordinates $z_i$ of z.

Lemma 4.2. For $r\in \mathbb {N}_0$ , let $\varphi :\Omega \rightarrow \wedge ^m T^{*}\mathbb {R}^d$ with $\|\varphi \|^{\perp }_{\alpha +r,m}\leq 1$ . If $\alpha>\Lambda +\alpha '>0$ , then for any $\varepsilon>0$ , there exists a $k>0$ such that

Proof. Using Lemma 2.1,

and so $\|\Delta _k\varphi \|_{C^0}\leq 2 C_\varphi \unicode{x3bb} _0^{-(\alpha +r)k}$ .

Now $\|\Delta _k\varphi \|_{\alpha ',m}^{\perp }$ will be bound. If $a,b\in C^{\perp }_\ell (x)$ with $\ell \geq k$ , then $C^{\perp }_k(a) = C_k^{\perp }(b)$ and so

If $\ell <k$ and $a,b\in C^{\perp }_\ell (x)$ ,

Thus, if $a,b\in C_\ell ^{\perp }(x)$ ,

(24)

Putting everything together:

Thus, if

(25)

then $\|\Delta _k\varphi \|^{\perp }_{\alpha '+r,m}\leq \varepsilon /2$ , which proves the first estimate.

To obtain the second estimate, first note that for $a,b\in C^{\perp }_\ell (x)$ , $\|\Pi _k\varphi (a)-\Pi _k\varphi (b)\| = 0$ if $\ell \geq k$ and if $\ell <k$ ,

where the first estimate follows from the fact that $\|\varphi \|^{\perp }_{r+\alpha ,m}\leq 1$ and the second from the estimate leading to equation (24). Thus,

and so it follows that

(26)

Using in equation (26) the choice for k in equation (25), the second estimate follows.

Proof of Proposition 4.1

To apply the compactness criterion, let $\varepsilon>0$ . Since $H^{\perp }_{\alpha +r,m}$ is compactly embedded in $H^{\perp }_{\alpha '+r,m}$ , let $\{\varphi _j\}_{j\leq K_{\varepsilon }}\subset H^{\perp }_{\alpha +r,m}$ be a finite family of forms such that for any $\varphi \in H^{\perp }_{\alpha +r,m}$ with $\|\varphi _j\|^{\perp }_{\alpha +r,m}\leq 4\unicode{x3bb} _0^\Lambda C_\mu ({8C_\mu }/{\varepsilon })^{{\Lambda }/{r+\alpha -\Lambda -\alpha '}}$ , there is a $\varphi _{j^{*}}$ in the family such that $\|\varphi -\varphi _{j^{*}}\|_{\alpha '+r,m}^{\perp }\leq \varepsilon /2$ . Define the finite family of linear forms $L_{j,i}:\mathcal {B}^{r',\alpha '}_m\rightarrow \mathbb {R}$ by

$$ \begin{align*}L_{j,i}(\eta) = \int_{\Omega}\langle\varphi_j,\partial^i\eta\rangle\, d\mu,\end{align*} $$

where i is a multiindex of length at most $r'$ and $j\leq K_{\varepsilon }$ .

For $\varphi $ with $\|\varphi \|^{\perp }_{\alpha +r,m}\leq 1$ , let $k\in \mathbb {N}$ be the one given by Lemma 4.2, so that $\|\Delta _k\varphi \|^{\perp }_{\alpha '+r,m}\leq \varepsilon /2$ . Let $j^{*}\leq K_{\varepsilon }$ be such that $\|\Pi _k\varphi -\varphi _{j^{*}}\|^{\perp }_{\alpha '+r,m}\leq \varepsilon /2$ . Thus, for $\eta \in \mathcal {B}^{r',\alpha '}_m$ and a multiindex i with $|i|=r$ , it follows that

$$ \begin{align*}\bigg|\!\int_\Omega \langle \varphi, \partial^i \eta\rangle\, d\mu\bigg|\leq \bigg|\!\int_\Omega\langle \Delta_k\varphi ,\partial^i\eta\rangle\, d\mu\bigg| + \bigg|\!\int_\Omega \langle\Pi_k\varphi-\varphi_{j^{*}},\partial^i\eta\rangle\, d\mu\bigg| + \bigg|\!\int_\Omega \langle \varphi_{j^{*}},\partial^i\eta\rangle\,d\mu\bigg|, \end{align*} $$

and so it follows that

and so compactness follows from the criterion.

It will be useful below to have a version of cohomology with coefficients in the anisotropic spaces $\mathcal {B}_*^{r,\alpha }$ . To that end, let $H_{\mathcal {B}^{r,\alpha }}^{*}(\Omega )$ be the cohomology of tangentially smooth forms with $\mathcal {B}^{r,\alpha }_0$ coefficients.

Proposition 4.3. Let $r\in \mathbb {N}$ and $\alpha>1$ . Then $H_{\mathcal {B}^{r,\alpha }}^{*}(\Omega )\cong \mathcal {H}^{*}_{r,\alpha }(\Omega ) \cong \check H(\Omega ;\mathbb {R})$ .

Proof. Let $\eta \in \mathcal {B}_m^{r,\alpha }$ be a closed m-form. It will be shown that there is a form $\eta '\in \Psi ^m_{r,\alpha }$ such that $\eta -\eta ' = d\omega $ .

Let $\{\eta _k\}\subset \Psi ^m_{r,\alpha }$ be a sequence of closed forms such that $\eta _k\rightarrow \eta $ in $\mathcal {B}_m^{r,\alpha }$ . As in the proof of Proposition 3.7, for each k, there is an $\eta ^{\prime }_k\in \pi ^{*}_{\beta _m} C^r(\Gamma )$ such that $\eta _k - \eta _k^{\prime } = d\omega _k$ . Thus,

$$ \begin{align*} \int_{\Omega}\langle\varphi,\eta-\eta_k\rangle\,d\mu = \int_{\Omega}\langle\varphi,\eta-\eta_k^{\prime}-d\omega_k\rangle\,d\mu\rightarrow 0 \end{align*} $$

for any $\varphi \in H^{\perp }_{\alpha }(\Omega )$ . Thus, $[\eta -\eta ^{\prime }_k] = [\eta ]-[\eta ^{\prime }_k]\rightarrow 0$ . However, $\eta ^{\prime }_k\in \pi ^{*}_{\beta _m}C^r$ for all k, and so $[\eta ^{\prime }_k]\in \pi ^{*}_{\beta _m}\check H^m(\Gamma ;\mathbb {R})\subset \check H^m(\Omega ;\mathbb {R})$ for all k, meaning that $[\eta ]$ defines a class in $\check H^m(\Omega ;\mathbb {R})$ .

Proposition 2.3 and the definition of the spaces $\mathcal {B}_m^{r,\alpha }$ implies the following.

Lemma 4.4. The differential operator d on m forms is a bounded operator from $\mathcal {B}^{r,\alpha }_m$ to $\mathcal {B}^{r-1,\alpha +1}_{m+1}$ .

4.2. The transfer operator

Define the transfer operator to be

$$ \begin{align*}\mathcal{L}f := f\circ \Phi^{-1},\end{align*} $$

where f is an m-form and note that for any $y\in C_0^{\perp }(x)$ ,

and so

(27)

The following estimates, similar to those of Lemma 4.2, will be needed in the proof for the Lasota–Yorke inequalities below.

Lemma 4.5. If $\|\varphi \|^{\perp }_{\beta ,m}\leq 1$ , then for $\alpha>\beta $ ,

(28)

Proof. The estimate for $\|\Delta _k\varphi \|^{\perp }_{\beta ,m}$ follows essentially from the computations up to and including equation (24), and combined with equation (27), the estimate for $\|(\Delta _k\varphi )\circ \Phi ^n\|^{\perp }_{\beta ,m}$ follows. Now, if $a,b\in C^{\perp }_\ell (x)$ and $\ell \geq k$ , then $\|\Pi _k\varphi (a)-\Pi _k\varphi (b)\|=0$ . If $\ell <k$ ,

where the last term is obtained in the same way as the estimates leading to equation (24), and so $|\Pi _k\varphi |^{\perp }_{\alpha ,m}\leq 3C_\mu \unicode{x3bb} _0^{(\alpha -\beta )k}$ , which combined with equation (27) proves the estimate for $\|\Pi _k\varphi \circ \Phi ^n\|^{\perp }_{\alpha ,m}$ .

Proposition 4.6. (Lasota–Yorke inequalities)

For each $r,n\in \mathbb {N}$ , $\alpha ,\alpha '$ with $\alpha>\alpha '>0$ , $\nu \in (\unicode{x3bb} _0^{-r},1)$ , and $m\in \{0,\ldots , d\}$ , there are $D,E>0$ such that

Proof. First observe that

(29)

This combined with equation (27) yields the first inequality. To address the second, first note that

and so equation (29) implies that

(30)

Now, if $\|\varphi \|^{\perp }_{\alpha '+p,m}\leq 1$ and $|i|=p\leq r$ , again by equation (29),

and so using Lemma 4.5,

Combining these estimates with equation (30),

from which the result follows.

4.3. The spectrum of $\mathcal {L}$

Propositions 4.1 and 4.6, combined with Hennion’s theorem [Reference Demers, Kiamari and LiveraniDKL21, Theorem B.14], yields the following corollary.

Corollary 4.7. For $\alpha>\Lambda $ and $r\in \mathbb {N}$ , the spectrum of $\mathcal {L}:\mathcal {B}^{r,\alpha }_m\rightarrow \mathcal {B}^{r,\alpha }_m$ is contained in the closed unit ball in $\mathbb {C}$ and the essential spectrum is contained in the closed ball of radius $\unicode{x3bb} _0^{-r}$ in $\mathbb {C}$ .

The following Proposition is a consequence of a theorem of Baladi and Tsuji, and shows that to a certain extent, the spectrum is independent of Banach spaces used.

Proposition 4.8. The discrete spectrum is independent of the Banach space $\mathcal {B}_0^{r,\alpha }$ : for $\alpha>\Lambda $ and $r'>r>0$ , then the discrete part of the spectrum of $\mathcal {L}$ of norm greater than $\unicode{x3bb} _0^{-r}$ coincides for $\mathcal {L}|_{\mathcal {B}_0^{r,\alpha }}$ and $\mathcal {L}|_{\mathcal {B}_0^{r',\alpha }}$ . In addition, the corresponding generalized eigenspaces are contained in $\mathcal {B}_0^{r,\alpha }\cap \mathcal {B}_0^{r',\alpha }$ .

Proof. The results will follow from [Reference Baladi, Tsujii, Burns, Dolgopyat and PesinBT08, Lemma A.1] as long as the inclusion $\mathcal {S}_{\alpha }^{r'}\subset \mathcal {S}^r_{\alpha }$ is shown to be dense with respect to .

Write $f = \sum _{k\geq 0}f^{(k)}\in \mathcal {S}^r_{\alpha }$ in the canonical way, where $f^{(k)} = \pi ^{*}_k g_k$ for some $g_k\in C^r(\Gamma _k)$ as described in §2.1. Since smooth functions are dense in $C^r(\Gamma _k)$ , for every $n\in \mathbb {N}$ and $k\in \mathbb {N}_0$ , pick a $g_{n,k}\in C^{r'}(\Gamma _k)$ such that $\|g_k-g_{n,k}\|_{C^r(\Gamma _k)}\leq \unicode{x3bb} _0^{-(n+\alpha k)}$ . Let $f_n:= \sum _{k\geq 0} \pi ^{*}_kg_{n,k}$ . It now needs to be shown that $f_n\rightarrow f$ in $\mathcal {B}_0^{r,\alpha }$ .

If $\varphi \in L^1$ and $\|\varphi \|_\infty \leq 1$ ,

(31)

and thus and the statement follows from [Reference Baladi, Tsujii, Burns, Dolgopyat and PesinBT08, Lemma A.1].

Denote by $\Sigma _m^+$ the spectrum of $\Phi ^{*}:\check H^m(\Omega ;\mathbb {R})\rightarrow \check H^m(\Omega ;\mathbb {R})$ consisting of expanding eigenvalues, $\Sigma _m^-$ the spectrum of $\Phi ^{-1*}:\check H^m(\Omega ;\mathbb {R})\rightarrow \check H^m(\Omega ;\mathbb {R})$ consisting of contracting eigenvalues, and set

to be the discrete spectrum of $\mathcal {L}$ on $\mathcal {B}^{r,\alpha }_m$ , assuming $r\in \mathbb {N}$ and $\alpha>\Lambda $ by Corollary 4.7. Note that the map $x\mapsto x^{-1}$ gives a bijection between $\Sigma _m^+$ and $\Sigma _m^-$ .

Lemma 4.9. For $\alpha>\Lambda $ , $r\in \mathbb {N}$ , and $\sigma _{r,\alpha , m}$ as above:

1. (i) $1\in \sigma _{r,\alpha ,0}$ ; it is the unique eigenvalue of modulus 1 and it has multiplicity one;

2. (ii) $\nu \in \sigma _{r,\alpha ,0}$ if and only if $\unicode{x3bb} ^{-1}\nu \in \sigma _{r,\alpha ,d}$ ;

3. (iii) for $r\in \mathbb {N}$ and $\alpha>\Lambda +1$ , $\Sigma _d^-\setminus \sigma _{r+1,\alpha -1,d-1}\subset \sigma _{r,\alpha ,d}$ ;

4. (iv) $\Sigma _d^-\setminus \{\unicode{x3bb} ^{-1}\}\subset \sigma _{r,\alpha , d-1}$ ;

5. (v) $\sigma _{r,\alpha , m}\subset (\sigma _{r+1,\alpha -1,m-1}\cup \Sigma _m^-\cup \sigma _{r-1,\alpha +1,m+1})$ .

Proof. The map $\Phi $ is topologically mixing [Reference Anderson and PutnamAP98, Proposition 3.1]. Property (i) follows from a standard argument depending on this mixing hypothesis; see [Reference DemersDem18, §4.1] or [Reference BaladiBal18, §7.1.1]. Item (ii) follows from the duality between 0 and d forms given by the Hodge- $\star $ operator.

For property (iii), it needs to be shown that if $\nu \in \Sigma _d^-$ and $\nu \not \in \sigma _{r+1,\alpha -1,d-1}$ , then there exists a $\eta \in \Psi ^d_{r,\alpha }$ such that $(\mathcal {L}-\nu \cdot \mathrm {Id})^k\eta = 0$ for some $k\geq 0$ . Now, if $\nu \in \Sigma _d^-\setminus \sigma _{r+1,\alpha -1,d-1}$ , then by Theorem 3.3, there exists $\eta \in \Psi ^d_{r,\alpha }$ such that $(\mathcal {L}-\nu \cdot \mathrm {Id})^k[\eta ] = 0$ for all k large enough and so there is an $\omega _\eta \in \Psi ^{d-1}_{r+1,\alpha -1}$ such that $(\mathcal {L}-\nu \cdot \mathrm {Id})^k\eta = d\omega _\eta $ . If $\theta = (\mathcal {L}-\nu \cdot \mathrm {Id} )^{-k}\omega _\eta \in \Psi ^{d-1}_{r+1,\alpha -1}$ , then $\eta ' = \eta -d\theta $ satisfies $(\mathcal {L}-\nu \cdot \mathrm {Id})^k\eta ' = 0$ .

For item (iv), note that the smallest element (in norm) of $\Sigma ^-_d$ is $\unicode{x3bb} ^{-1}$ . Suppose that for some $\nu \in \Sigma ^-_d$ with $\unicode{x3bb} ^{-1}< \nu $ , $\nu \not \in \sigma _{r,\alpha , d-1}$ . Then by property (iii), $\nu \in \sigma _{r-1,\alpha +1, d}$ , which by property (ii) means that $\unicode{x3bb} \nu \in \sigma _{r-1,\alpha +1, 0}$ . However, $|\unicode{x3bb} \nu |>1$ , contradicting that the spectral radius is 1. So $\nu \in \sigma _{r,\alpha , d-1}$ .

Item (v) is due to Daniele Galli, but the proof is included here for completeness. Let $\nu \in \sigma _{r,\alpha , m}$ . Then there exists an $\eta \in \Psi ^m_{r,\alpha }$ and J such that $(\mathcal {L}-\nu )^J\eta = 0$ . If $\eta $ is not closed, then $(\mathcal {L}-\nu )^Jd\eta = d(\mathcal {L}-\nu )^J\eta = 0$ , meaning that $\nu \in \sigma _{r-1,\alpha +1,m+1}$ . Now suppose that $\eta $ is closed and not exact. Then $(\mathcal {L}-\nu )^J\eta = 0$ implies that $\nu \in \Sigma _m^-$ . Now suppose that $\eta = d\theta \neq 0$ for some $\theta \in \Psi ^{m-1}_{r+1,\alpha -1}$ . Then $(\mathcal {L}-\nu )^J\eta = (\mathcal {L}-\nu )^Jd\theta = d(\mathcal {L}-\nu )^J\theta =0 $ , that is, either $\nu \in \sigma _{r+1,\alpha -1,m-1}$ or $(\mathcal {L}-\nu )^J\theta $ is closed. Suppose $(\mathcal {L}-\nu )^J\theta $ is closed, set $\omega :=(\mathcal {L}-\nu )^J\theta \in \Psi ^{m-1}_{r+1,\alpha -1}$ . If $(\mathcal {L}-\nu )^J$ is invertible on closed forms, then $\theta = (\mathcal {L}-\nu )^{-J}\omega $ is closed, and $d\theta = \eta = 0$ , which is a contradiction. So $(\mathcal {L}-\nu )^J$ is not invertible on closed forms, meaning that $\nu \in \sigma _{r+1,\alpha -1,m-1}$ .

Let $\sigma ^{-}$ be the eigenvalues associated to generalized eigenvectors of $\Phi ^{-1*}:E^{++}\rightarrow E^{++}$ . Note that when $d=1$ , this implies that $\Sigma _1^- = \sigma ^-$ . The following proposition gives Theorem 1.3.

Proposition 4.10. For $r\in \mathbb {N}$ and $\alpha>\Lambda $ , the following hold.

1. (i) If $d=1$ , then the set of eigenvalues for $\mathcal {L}$ acting on $\mathcal {B}_0^{r,\alpha }$ contains $\sigma ^-\setminus \{\unicode{x3bb} ^{-1}\}$ . In addition, if $\nu $ is an eigenvalue in $\mathcal {B}_0^{r,\alpha }$ and $k<\alpha -\Lambda $ , then $\unicode{x3bb} ^{-1}\nu $ is an eigenvalue in $\mathcal {B}_0^{r+k,\alpha -k}$ . It follows that if $\mathcal {F}:= \bigcap _{\alpha>0,r>0}\mathcal {S}^r_{\alpha }$ , then the Ruelle spectrum for functions in $\mathcal {F}$ contains the set of numbers of the form $\unicode{x3bb} ^{-k}\nu $ with $\nu \in \sigma ^-\setminus \{\unicode{x3bb} ^{-1}\}$ and $k\in \mathbb {N}$ ;

2. (ii) If $d=2$ , then the set of eigenvalues for $\mathcal {L}$ acting on $\mathcal {B}_0^{r,\alpha }$ contains $\sigma ^-\setminus \{\unicode{x3bb} ^{-1}\}$ . If $\mathcal {S}^{\infty }_{\alpha }:= \bigcap _{r>0}\mathcal {S}^r_{\alpha }$ , then the Ruelle spectrum for functions in $\mathcal {S}^{\infty }_{\alpha }$ contains the set $\sigma ^-\setminus \{\unicode{x3bb} ^{-1}\}$ .

Proof. Both of these are consequences of Lemma 4.9. For $d=1$ , that $\Sigma _1^-\setminus \{\unicode{x3bb} ^{-1}\}$ follows directly from part (iii) of Lemma 4.9. Now suppose that $\nu \in \sigma _{r,\alpha ,0}$ for some $|\nu |<1$ . Then by part (i), $\unicode{x3bb} ^{-1}\nu \in \sigma _{r,\alpha , 1}$ . So $(\mathcal {L}-\unicode{x3bb} ^{-1}\nu )^{J_1}\eta = 0$ for some $\eta \in \mathcal {B}_1^{r,\alpha }$ . Since $\unicode{x3bb} ^{-1}|\nu |< \unicode{x3bb} ^{-1}$ , $\eta $ has to be exact as $\Sigma _1^-$ is bounded from below by $\unicode{x3bb} ^{-1}$ . So $\eta = d\theta _1$ (where $\theta _1\in \mathcal {B}_0^{r+1,\alpha -1}$ is uniquely defined up to a closed $0$ -form, that is, a constant) and $(\mathcal {L}-\unicode{x3bb} ^{-1}\nu )^{J_1}\eta = (\mathcal {L}-\unicode{x3bb} ^{-1}\nu )^{J_1}d\theta _1 = d(\mathcal {L}-\unicode{x3bb} ^{-1}\nu )^{J_1}\theta _1 =0$ . Thus, either $(\mathcal {L}-\unicode{x3bb} ^{-1}\nu )^{J_1}\theta _1$ is closed or $(\mathcal {L}-\unicode{x3bb} ^{-1}\nu )^{J_1}\theta _1=0$ . If $(\mathcal {L}-\unicode{x3bb} ^{-1}\nu )^{J_1}\theta _1$ is closed, then it is constant, and denote by $c_1:= (\mathcal {L}-\unicode{x3bb} ^{-1}\nu )^{J_1}\theta _1$ . Letting $\theta _1^{\prime } = \theta _1 - c_1$ , it follows that $(\mathcal {L}-\unicode{x3bb} ^{-1}\nu )^{J_1}\theta _1^{\prime } = 0$ and thus it follows that $\unicode{x3bb} ^{-1}\nu \in \sigma _{r+1,\alpha -1,0}$ . So we are back where we started and the same argument gives that $\unicode{x3bb} ^{-2}\nu \in \sigma _{r+2,\alpha -2,0}$ and so on.

If $d=2$ , by parts (ii), (iv), and (v) of Lemma 4.9, it follows that

$$ \begin{align*} \Sigma_2^-\setminus\{\unicode{x3bb}^{-1}\}\subset \sigma_{r,\alpha, 1}\subset (\sigma_{r+1,\alpha-1,0}\cup \Sigma^-_1\cup \unicode{x3bb}^{-1}\cdot\sigma_{r-1,\alpha+1,0} ). \end{align*} $$

Let $\nu \in \sigma ^-\setminus \{\unicode{x3bb} ^{-1}\}\subset \Sigma _2^-\setminus \{\unicode{x3bb} ^{-1}\}$ . First, $\nu \not \in \Sigma ^-_1$ , since by definition, $\nu $ contracts faster than the smallest contracting eigenvalue in $\Sigma _1^-$ . Now, $\unicode{x3bb} |\nu |>1$ , so by part (ii), $\nu \not \in \sigma _{r-1,\alpha +1,2} = \unicode{x3bb} ^{-1}\sigma _{r-1,\alpha +1,0}$ . So it follows that $\nu \in \sigma _{r-1,\alpha +1,0}$ .

5. Applications to primitive substitution subshifts

Let $\mathcal {A}$ be a finite set (the alphabet) and $\mathcal {A}^{*}$ be the set of finite words on $\mathcal {A}$ . Let $\varrho :\mathcal {A}\rightarrow \mathcal {A}^{*}$ be a primitive substitution rule. This means there is an N such that for any $a,b\in \mathcal {A}$ , the symbol a appears in $\varrho ^N(b)$ . Without loss of generality (that is, by passing to a power), we will assume that there is a symbol $a\in \mathcal {A}$ such that $\varrho (a)$ begins with a. Let $\bar {a} = \lim _{N\rightarrow \infty }\varrho ^N(a) \in \mathcal {A}^{\mathbb {N}}$ be a fixed point of the substitution and define $X_{\varrho }$ to be the orbit closure of $\bar {a}$ under the shift map $\sigma :\mathcal {A}^{\mathbb {N}}\rightarrow \mathcal {A}^{\mathbb {N}}$ . The system $\sigma :X_{\varrho }\rightarrow X_{\varrho }$ is a minimal subshift.

Define the metric on $X_{\varrho }$ as

(32) $$ \begin{align} d(\bar{b},\bar{c}) = \unicode{x3bb}^{-k(\bar{b},\bar{c})} \end{align} $$

for $\bar {b},\bar {c}\in X_{\varrho }$ , where $k(\bar {b},\bar {c})\in \mathbb {N}$ is the smallest index i so that $c_i\neq b_i$ , and $\unicode{x3bb}>0$ is the Perron–Frobenius eigenvalue of the substitution matrix for $\varrho $ .

There is an associated solenoid to $\varrho $ constructed as follows. Let $r:X_{\varrho }\rightarrow \mathbb {R}^+$ be the function defined as $r(\bar {l}) = v_{l_1}$ , where $v\in \mathbb {R}^{|\mathcal {A}|}$ is a positive Perron–Frobenius eigenvector for the substitution matrix, and let $\Omega _{\varrho }$ be the suspension of $X_{\varrho }$ with roof function r. Then there exists a compact one-dimensional CW complex $\Gamma $ and a cellular affine (outside the zero-cells of $\Gamma $ ) map $\gamma :\Gamma \rightarrow \Gamma $ such that

$$ \begin{align*} \Omega_{\varrho} \cong \lim_{\leftarrow}(\Gamma,\gamma). \end{align*} $$

This is the Anderson–Putnam construction [Reference Anderson and PutnamAP98] and the CW complex $\Gamma $ is referred to as the AP-complex. Denote $\Gamma _k = \pi _k(\Omega _{\varrho })$ the ‘ $k\mathrm {th}$ ’ AP complex. The identification above is not only through a homeomorphism, but in fact an isometry. That is, there is a natural inclusion $i:X_{\varrho }\rightarrow \Omega _{\varrho }$ which, under the identification above, can be identified with $C^{\perp }_0(\bar {a})$ . This inclusion is an isometry with respect to the metrics in equations (3) and (32). Denote by $H_{\alpha }(X_{\varrho })$ the space of $\alpha $ -Hölder functions on $X_{\varrho }$ with respect to this metric.

Let $0<\varepsilon < \min _{l\in \mathcal {A}} |v_l| /4$ . With this choice of $\varepsilon $ , the $\varepsilon $ -neighborhood of $i(X_{\varrho }) = C^{\perp }_0(\bar {a})$ has the local coordinates $(t,c)\in (-\varepsilon ,\varepsilon )\times X_{\varrho }$ . If $u_{\varepsilon }:(-\varepsilon ,\varepsilon )\rightarrow \mathbb {R}$ is a smooth even bump function with compact support and of integral 1, then for any function $h:X_{\varrho }\rightarrow \mathbb {R}$ , let $h_{\varepsilon }:\Omega _{\varrho }\rightarrow \mathbb {R}$ be defined as $h_{\varepsilon }(t,c) = u_{\varepsilon }(t) h(c)$ for $(t,c)\in (-\varepsilon ,\varepsilon )\times X_{\varrho }$ and zero otherwise.

5.1. The cohomological equation for primitive substitution subshifts

This section is dedicated to the proof of Theorem 1.5 on the solutions of the cohomological equation for primitive substitution subshifts $\sigma :X_{\varrho }\rightarrow X_{\varrho }$ . That is, the goal here is to find a solution u to the equation $f = u\circ \sigma - u$ for a given f.

Lemma 5.1. If $h:X_{\varrho }\rightarrow \mathbb {R}$ is $\alpha $ -Hölder, then $h_{\varepsilon } \in \mathcal {S}^1_{\alpha -1}$ for $\alpha>1$ . If $r\in \mathbb {N}$ and $\alpha>2$ , then the cohomology class $[\star h_{\varepsilon }]\in H^1_{r,\alpha }(\Omega _{\varrho })$ is independent of $u_{\varepsilon }$ .

Proof. The goal is to write

$$ \begin{align*}h_{\varepsilon} = \sum_{k\geq 0}\pi^{*}_k g_k\end{align*} $$

in the canonical way described in §2.1 with the appropriate bounds on $\|g_k\|_{C^1(\Gamma _k)}$ . Using the notation of equation (23), define $h^k_{\varepsilon } := \Pi _kh_{\varepsilon }$ which, in the natural coordinates of the $\varepsilon $ -neighborhood of $C^{\perp }_0(\bar {a})$ , is defined as

$$ \begin{align*}h^k_{\varepsilon}(t,c) = \hat{\nu}_{k,(t,c)}^{-1}\int_{C^{\perp}_k(t,c)}h_{\varepsilon}(t,w)\, d\nu_{k,(t,c)} = u_{\varepsilon}(t)\hat{\nu}_{k,(t,c)}^{-1}\int_{C^{\perp}_k(c)}h(w)\, d\nu_{k,c}.\end{align*} $$

By the same calculation as in §4.1, $\|h^k_{\varepsilon } - h_{\varepsilon }\|_{C^0}\leq C_\mu \|u\|_\infty \unicode{x3bb} ^{-\alpha k}$ . Now, letting $h^{-1}_{\varepsilon } = 0$ , for any $k\geq 0$ , define $\delta _k h_{\varepsilon } = h^k_{\varepsilon } - h^{k-1}_{\varepsilon }$ . As such,

$$ \begin{align*}\sum_{j=0}^k \delta_jh_{\varepsilon} = h^k_{\varepsilon}\end{align*} $$

and, since $\delta _kh_{\varepsilon }$ only depends on at most the first k coordinates, $\delta _kh_{\varepsilon } = \pi ^{*}_k g_k$ for some $g_k:\Gamma _k\rightarrow \mathbb {R}$ . If $\alpha>1$ , then $\sum _{j=0}^k \pi ^{*}_kg_k \rightarrow h_{\varepsilon }$ pointwise. It remains to prove the $C^1$ bounds for $g_k$ .

Let $P_k = \gamma ^{-k}(\pi _0(\bar {a}))\subset \Gamma _k$ be the preimages of $\pi _0(\bar {a})$ under $\gamma ^k$ . These points can be of one of two types: flat points or branch points. Flat points have an Euclidean neighborhood whereas branch points do not.

Since $h_{\varepsilon }^k$ is transversally locally constant, $h_{\varepsilon }^k = \pi ^{*}_k H_k$ for some $H_k:\Gamma _k\rightarrow \mathbb {R}$ . Note that the task is to obtain $C^1$ bounds for $g_k = H_k - \gamma ^{*}H_{k-1}$ .

The function $H_k$ is supported in the $\unicode{x3bb} ^{-k}\varepsilon $ -neighborhood of $P_k$ as follows. If $z\in P_k$ is a flat point, then in the $\unicode{x3bb} ^{-k}\varepsilon $ -neighborhood of $z\in P_k$ , after identifying z with $0$ in these coordinates, $H_k(t) = u_{\varepsilon }(\unicode{x3bb} ^kt)h^k_{\varepsilon }(c_z)$ , where $c_z\in C^{\perp }_0(\bar {a})$ is a point in the clopen subset of $X_{\varrho }$ corresponding to z. From this, it follows that in these local coordinates,

(33) $$ \begin{align} \begin{aligned} &g_k(t) = u_{\varepsilon}(\unicode{x3bb}^kt)(h^k_{\varepsilon}(c_z) - h^{k-1}_{\varepsilon}(c_z))\\ &\quad=u_{\varepsilon}(\unicode{x3bb}^kt)\bigg[\hat{\nu}_{k,c_z}^{-1}\!\int_{C^{\perp}_k(c_z)} h(w)-h(c_z)\, d\nu_{k,c_z}\! -\! \hat{\nu}_{k-1,c_z}^{-1}\!\int_{C^{\perp}_{k-1}(c_z)} h(w)-h(c_z)\, d\nu_{k,c_z} \bigg] \end{aligned} \end{align} $$

in a $\unicode{x3bb} ^{-k}\varepsilon $ -neighborhood of $z\in P_k$ . Note that since h is $\alpha $ -Hölder, by rewriting it as in equation (33),

$$ \begin{align*} | h^k_{\varepsilon}(c_z) - h^{k-1}_{\varepsilon}(c_z)|\leq 2C_h\unicode{x3bb}^{-\alpha(k-1)} \end{align*} $$

and so $\|g_k\|_{C^0}\leq 2\unicode{x3bb} C_h\|u_{\varepsilon }\|_\infty \unicode{x3bb} ^{-k\alpha }$ . Moreover, in the neighborhood of $z\in P_k$ ,

$$ \begin{align*}g_k^{\prime} = \unicode{x3bb}^k u^{\prime}_{\varepsilon}(\unicode{x3bb}^kt)(h^k_{\varepsilon}(c_z) - h^{k-1}_{\varepsilon}(c_z))\end{align*} $$

and so

$$ \begin{align*}\|g_k^{\prime}\|_\infty = \unicode{x3bb}^k \|u^{\prime}_{\varepsilon}\|_\infty |h^k_{\varepsilon}(c_z) - h^{k-1}_{\varepsilon}(c_z)|\leq C_h \unicode{x3bb}^k \|u^{\prime}_{\varepsilon}\|_\infty \unicode{x3bb}^{-\alpha(k-1)}, \end{align*} $$

and so it follows that $\|g_k\|_{C^1} \leq C \unicode{x3bb} ^{-k(\alpha -1)}$ for all k, and thus $h_{\varepsilon } \in \mathcal {S}^1_{\alpha -1}$ . The case of ${z\in P_k}$ being a branched point is essentially treated in the same way: equation (33) needs to be written carefully to take into consideration the different branches coming out of z. Indeed, equation (33) treats two branches coming out of z in the flat case, and so equation (33) can be used to treat the branched case with minor modifications. The details are left to the reader.

That the cohomology class is independent of $u_{\varepsilon }$ follows from the fact that the compactly supported de Rham cohomology of the line is one-dimensional in top degree.

Remark 5.2. Note that the same proof can be modified to show that $h_{\varepsilon }\in \mathcal {S}^r_{\alpha -r}$ for any $r<\alpha $ . However, the focus is on $r=1$ since the transverse Hölder regularity is what needs to be optimized.

Definition 5.3. $f:X_{\varrho }\rightarrow \mathbb {R}$ is a coboundary if there exists an $f:X_{\varrho }\rightarrow \mathbb {R}$ such that $f = g\circ \sigma - g$ . For $\alpha>0$ , it is an $\alpha $ -coboundary if f is $\alpha $ -Hölder and g is $(\alpha -2)$ -Hölder. The $\alpha $ -Hölder cohomology $H_{\alpha }^0(X_{\varrho })$ of $X_{\varrho }$ is the quotient of $H_{\alpha }(X_{\varrho })$ by the equivalence relation $f_1\sim f_2$ if and only if $f_1-f_2$ is an $\alpha $ -coboundary.

In what follows, X is the differential operator in the leaf direction.

Lemma 5.4. For $\alpha>2$ and an $\alpha $ -Hölder function h on $X_{\varrho }$ , $h = g\circ \sigma - g$ for some $g\in H_{\alpha -2}(X_{\varrho })$ if and only if $h_{\varepsilon } = X \Theta $ for some $\Theta \in \mathcal {S}^2_{\alpha -2}$ .

Proof. Suppose $h_{\varepsilon } = X \Theta $ . By Lemma 5.1, $h_{\varepsilon }\in \mathcal {S}^1_{\alpha -1}$ , and so $\Theta \in \mathcal {S}^2_{\alpha -2}$ by Proposition 2.3. Since $u_{\varepsilon }$ has integral one,

$$ \begin{align*}h(c) = \int_{-\varepsilon}^\varepsilon h_{\varepsilon}(t,c)\, dt = \int_{-\varepsilon}^\varepsilon X\Theta(t,c)\, dt = \Theta(\varepsilon,c) - \Theta(-\varepsilon,c).\end{align*} $$

Now, since $h_{\varepsilon }$ is compactly supported in the $\varepsilon $ -neighborhood of $i(X_{\varrho })$ and $0 = h_{\varepsilon }(\pm \varepsilon ,c) = X\Theta (\pm \varepsilon ,c)$ , $\Theta $ is leafwise constant on the complement of the $\varepsilon $ -neighborhood of $i(X_{\varrho })$ , which implies that $\Theta (\varepsilon ,c) = \Theta (-\varepsilon ,\sigma (c))$ . Defining $g(c) = \Theta (-\varepsilon ,c)$ , it follows that $h = g\circ \sigma - g$ .

Write $\Theta = \sum _{k\geq 0} \pi ^{*}_k\Theta _k$ in the canonical way described in §2.1, and, in the local coordinates around $i(X_{\varrho })$ , let $(-\varepsilon ,c_1)\in C_k^{\perp }((-\varepsilon ,c_2))\setminus C_{k+1}^{\perp }((-\varepsilon ,c_2))$ . For $\beta>0$ ,

$$ \begin{align*} \begin{aligned} \frac{|g(c_1)-g(c_2)|}{\unicode{x3bb}^{-\beta k}} &= \unicode{x3bb}^{\beta k}|\Theta(-\varepsilon, c_1) - \Theta(-\varepsilon, c_2)|\! \leq \unicode{x3bb}^{\beta k}\!\sum_{n\geq k} |\pi^{*}_n\Theta_n (-\varepsilon, c_1) -\! \pi^{*}_n\Theta_n (-\varepsilon, c_2)| \\ &\leq \unicode{x3bb}^{\beta n} \sum_{n\geq k} \|\Theta_n\|_{C^2}\leq \unicode{x3bb}^{\beta k} C_\Theta \frac{\unicode{x3bb}^{-k(\alpha-2)}}{1-\unicode{x3bb}^{-(\alpha-2)}}. \end{aligned} \end{align*} $$

Picking $\beta = \alpha -2$ , it follows that g has finite $\alpha -2$ Hölder norm, and so $g\in H_{\alpha -2}(X_{\varrho })$ .

Now suppose $h = g\circ \sigma - g$ and consider $h_{\varepsilon }$ as constructed above. In the $\varepsilon $ -neighborhood of $i(X_{\varrho })$ , define $\Theta (t,c) = g(c)+\int _{-\varepsilon }^t u_{\varepsilon }(s)h(c)\,ds $ and so close to $i(X_{\varrho })$ , it holds that $X\Theta = h_{\varepsilon }$ . By construction, and by the fact that $h = g\circ \sigma - g$ , this function satisfies $h_{\varepsilon } = X\Theta $ globally. The details are left to the reader.

Lemmas 5.1 and 5.4 imply that the induced map $j:H_{\alpha }^0(X_{\varrho })\rightarrow H^1_{1,\alpha -1}(\Omega _{\varrho })\cong \check H^1(\Omega _{\varrho };\mathbb {R})$ defined by $j([f]) = [\star f_{\varepsilon }]$ is injective whenever $\alpha>2$ . This implies that $H^0_{\alpha }(X_{\varrho })$ is finite dimensional and concludes the proof of Theorem 1.5.