2.1 Thermodynamic formalism and reparameterizations

Let $\phi =(\phi _t:X \to X)_{t\in \mathbb {R}}$ be a continuous flow without fixed points. The space of $\phi $ -invariant probability measures on X is denoted by $\mathcal M^\phi .$ It is a convex, weakly compact subset of $C^*(X),$ the dual space to the space of continuous functions equipped with the uniform topology. The metric entropy of $m\in \mathcal M^\phi $ will be denoted by $h(\phi ,m).$ Its definition can be found in Aaronson’s book [Reference Aaronson1]. Via the variational principle, we will define the topological pressure (or just pressure) of a function $f:X\to \mathbb {R}$ as the quantity

(2) $$ \begin{align}P(\phi,f)= \sup_{m\in\mathcal M^{\phi}} \bigg(h(\phi,m)+\int_X f\,dm\bigg).\end{align} $$

A probability measure m realizing the least upper bound is called an equilibrium state of $f.$ An equilibrium state for $f\equiv 0$ is called a measure of maximal entropy, and its entropy is called the topological entropy of $\phi ,$ denoted by $h(\phi ).$

Let $f:X\to \mathbb {R}_{>0}$ be continuous. For every $x\in X$ , the function $k_f:X\times \mathbb {R}\to \mathbb {R},$ defined by $k_f(x,t)=\int _0^tf(\phi _sx)\,ds,$ is an increasing homeomorphism of $\mathbb {R}.$ There is thus a continuous function $\alpha _f:X\times \mathbb {R}\to \mathbb {R}$ such that for all $(x,t)\in X\times \mathbb {R},$

$$ \begin{align*}\alpha_f(x,k_f(x,t))=k_f(x,\alpha_f(x,t))=t.\end{align*} $$

The reparameterization of $\phi $ by $f:X\to \mathbb {R}_{>0}$ is the flow $\phi ^f\!=(\phi ^f_t\!:X \to X)_{t\in \mathbb {R}}$ defined, for all $(x,t)\in X\times \mathbb {R}$ , by

$$ \begin{align*}\phi^f_t\!(x)=\phi_{\alpha_f(x,t)}(x).\end{align*} $$

The Abramov transform of $m\in \mathcal M^\phi $ is the probability measure $m^\#\in \mathcal M^{\phi ^f\!}$ defined by

(3) $$ \begin{align}m^\#=\frac{f\cdot m}{\int f\,dm}.\end{align} $$

One has the following lemma.

Lemma 2.1.1. (Sambarino [Reference Sambarino65, Lemma 2.4])

Let $f:X\to \mathbb {R}_{>0}$ be a continuous function. Assume the equation

$$ \begin{align*}P(\phi,-sf)=0\quad s\in\mathbb{R}\end{align*} $$

has a finite positive solution $h,$ then h is the topological entropy of $\phi ^f\!.$ Conversely, if $h(\phi ^f\!)$ is finite, then it is a solution to the last equation. In this situation, the Abramov transform induces a bijection between the set of equilibrium states of $-hf$ and the set of probability measures maximizing entropy for $\phi ^f\!.$

Two continuous maps $f,g:X\to V$ are Livšic-cohomologous if there exists a $U:X\to ~V$ of class $\operatorname {\mathrm {C}}^1$ in the direction of the flow (that is, such that if for every $x\in X,$ the map $t\mapsto U(\phi _tx)$ is of class $\operatorname {\mathrm {C}}^1$ and the map $x\mapsto {\partial }/{\partial t}|_{t=0}U(\phi _tx)$ is continuous), such that for all $x\in X$ , one has

$$ \begin{align*}f(x)-g(x)=\frac{\partial}{\partial t}\bigg|_{t=0} U(\phi_tx).\end{align*} $$

2.2 Metric-Anosov flows I: Livšic-cohomology

Metric-Anosov flows are a metric version of what is commonly known as hyperbolic flows. The former are called Smale flows by Pollicott [Reference Pollicott56], who transferred to this more general setting the classical theory carried out for the latter. We recall here their definition and some well-known facts on their ergodic theory needed in what follows. Throughout this subsection, we will further assume that $\phi $ is Hölder-continuous with an exponent independent of $t,$ that it is transitive, that it has a dense orbit, and that it is metric-Anosov.

For $\varepsilon>0$ , the local stable/unstable set of x are (respectively)

$$ \begin{align*}W^{{\mathrm{s}}}_{\varepsilon}(x) & =\{y\in X:d(\phi_tx,\phi_t y)\leq\varepsilon \text{ for all } t>0\textrm{ and }d(\phi_tx,\phi_t y)\to0\textrm{ as }t\to\infty\},\\ W^{{\mathrm{u}}}_{\varepsilon}(x) & =\{y\in X:d(\phi_{-t}x,\phi_{-t} y)\leq\varepsilon \text{ for all } t>0\textrm{ and }d(\phi_{-t}x,\phi_{-t} y)\to0\textrm{ as }t\to\infty\}.\end{align*} $$

A translation cocycle over $\phi $ is a map $k: X\times \mathbb {R} \to V$ such that for every $x\in X$ and $t,s\in \mathbb {R},$ one has

$$ \begin{align*}k(x,t+s)=k(\phi_sx,t)+k(x,s),\end{align*} $$

and such that the map $k(\cdot ,t)$ is Hölder-continuous with exponent independent of t and with bounded multiplicative constant when t remains on a bounded set. Two translation cocycles $k_1$ and $k_2$ are Livšic-cohomologous if there exists a continuous map $U:X\to V,$ such that for all $x\in X$ and $t\in \mathbb {R}$ , one has

(4) $$ \begin{align}k_1(x,t)-k_2(x,t)=U(\phi_tx)-U(x).\end{align} $$

If k is a translation cocycle, then the period for k of a periodic orbit $\tau $ is

$$ \begin{align*}\ell_\tau(k)=k(x,p(\tau))\end{align*} $$

for any $x\in \tau .$ The marked spectrum $\tau \mapsto \ell _k(\tau )$ is a cohomological invariant that uniquely determines its class.

Observe that if $f:X\to V$ is Hölder-continuous, then the map

$$ \begin{align*}k_f(x,t)=\int_0^tf(\phi_sx)\,ds\end{align*} $$

is a translation cocycle. Two such functions are Livšic-cohomologous if and only if the associated cocycles are, and the period of f on $\tau $ is, for any $x\in \tau ,$

$$ \begin{align*}\ell_\tau(f)=\int_\tau f=k_f(x,p(\tau)).\end{align*} $$

It turns out that every cocycle is Livšic-cohomologous to a cocycle of the form $k_f$ .

Proof. For any $\kappa>0,$ the function $j(x,t)=1/{2\kappa }\int _{-\kappa }^\kappa k(x,t+s)\,ds$ is differentiable on the second variable. Let $f(x)=(\partial /\partial s)|_{s=0}j(x,s).$ Then,

$$ \begin{align*}k_f(x,t)&=\int_0^tf(\phi_ux)\,du =\int_0^t\frac{\partial}{\partial s}\bigg|_{s=0}j(\phi_ux,s)\,du\\ & =\int_0^t\frac{\partial}{\partial s}\bigg|_{s=0}j(x,s+u)\,du = j(x,t)-j(x,0),\end{align*} $$

so the period $k_f(x,p(\tau ))=j(x,p(\tau ))-j(x,0)=k(x,p(\tau )).$ By Theorem 2.2.2, the cocycles k and $k_f$ are thus Livšic-cohomologous.

We record also the following immediate consequence of Livšic’s theorem.

In this context, much more information can be stated about the pressure function. Recall that the space $\operatorname {\mathrm {Holder}}^\alpha (X)$ of real valued $\alpha $ -Hölder functions is naturally a Banach space when equipped with the norm

$$ \begin{align*}\|f\|_\alpha=\|f\|_\infty+\sup_{x\neq y}\frac{|f(x)-f(y)|}{d(x,y)^\alpha}.\end{align*} $$

Proposition 2.2.5. (Bowen and Ruelle [Reference Bowen and Ruelle16] and Parry and Pollicott [Reference Parry and Pollicott53, Proposition 4.10])

The function $P(\phi ,\cdot )$ is analytic on $\operatorname {\mathrm {Holder}}^\alpha (X).$ If $f,g\in \operatorname {\mathrm {Holder}}^\alpha (X),$ then

$$ \begin{align*}\frac{\partial}{\partial t}\bigg|_{t=0}P(\phi,f+tg)=\int gdm_f,\end{align*} $$

where $m_f$ is the equilibrium state of $f,$ and the function $t\mapsto P(\phi ,f+tg)$ is strictly convex unless g is Livšic-cohomologous to a constant. Finally, one also has

(5) $$ \begin{align}P(\phi,f)=\limsup_{t\to\infty}\frac1t\log\sum_{\tau:p(\tau)\leq t}e^{\ell_\tau(f)}.\end{align} $$

Let $f\in \operatorname {\mathrm {Holder}}^\alpha (X)$ have non-negative (and not all vanishing) periods and define its entropy by

$$ \begin{align*}\mathscr h_f=\limsup_{s\to\infty}\frac1s\log\#\bigg\{\tau\textrm{ periodic}:\int_\tau f\leq s\bigg\}\in(0,\infty].\end{align*} $$

One has the following lemma.

Let us fix an exponent $\alpha $ and consider the cone $\operatorname {\mathrm {Holder}}^\alpha _+(X,\mathbb {R})$ of Hölder-continuous functions that are Livšic-cohomologous to a strictly positive function. The implicit function theorem for Banach spaces (see [Reference Akerkar2]) and the explicit formula for the derivative of pressure (Proposition 2.2.5) give the following corollary.

Proof. Indeed, Lemma 2.2.7 gives the equation $P(\phi ,-\mathscr h_ff)=0$ and equation (2.2.5) gives the non-vanishing derivative

$$ \begin{align*}d_{-\mathscr h_ff}P(\phi,f)=\int f\,dm_{-h_ff}>0,\end{align*} $$

so the implicit function theorem completes the result.

2.3 Metric-Anosov flows II: ergodic theory

A fundamental tool for studying the ergodic theory of metric-Anosov systems is the existence of a Markov coding.

Let $\Sigma $ be an irreducible sub-shift of finite type equipped with its shift transformation $\sigma :\Sigma \to \Sigma ,$ and $r:\Sigma \to \mathbb {R}_{>0}$ be Hölder-continuous. Let $\hat r:\Sigma \times \mathbb {R}\to \Sigma \times \mathbb {R}$ be defined as

$$ \begin{align*}\hat r(x,s)=(\sigma x, s-r(x)),\end{align*} $$

and consider the quotient space It is equipped with the flow induced on the quotient by the translation flow.

Definition 2.3.1. (Markov coding)

A triplet $(\Sigma ,\pi ,r)$ is a Markov coding for $\phi $ if $\Sigma $ and r are as above, $\pi :\Sigma \to X$ is Hölder-continuous, and the function $\pi _r:\Sigma \times \mathbb {R}\to X$ defined as

$$ \begin{align*}\pi_r(x,t)=\phi_t\pi(x)\end{align*} $$

verifies the following conditions:

1. (i) $\pi _r$ is Hölder-continuous, surjective, and $\hat r$ -invariant, it passes then to the quotient

2. (ii) is bounded-to-one and injective on a residual set which is of full measure for every ergodic $\unicode{x3c3} ^r$ -invariant measure of total support;

3. (iii) for every $t\in \mathbb {R}$ , one has $\pi _r\unicode{x3c3} ^r_t=\phi _t\pi _r.$

The following result has a long history, see for example the works of Bowen [Reference Bowen14, Reference Bowen15], Ratner [Reference Ratner63], Pollicott [Reference Pollicott56], and more recently Constantine, Lafont, and Thompson [Reference Constantine, Lafont and Thompson27].

The above is a fundamental tool to obtain the following, see for example the works of Bowen and Ruelle [Reference Bowen and Ruelle16], Parry and Pollicott [Reference Parry and Pollicott53], and more recently Giulietti et al [Reference Giulietti, Kloeckner, Lopes and Marcon31]. Recall from §2.1 the definition of equilibrium state.

A final fact we will require in this setting (introduced by Margulis [Reference Margulis48]) is the decomposition of the measure of maximal entropy along the stable/central-unstable sets of $\phi .$

The stable/unstable leaf of x is

$$ \begin{align*} W^{\mathrm{s}}(x) & = \bigcup_{t\in\mathbb{R}_+} \phi_{-t}(W^{{\mathrm{s}}}_\varepsilon(\phi_tx)),\\ W^{{\mathrm{u}}}(x) & = \bigcup_{t\in\mathbb{R}_+} \phi_{t}(W^{{\mathrm{u}}}_\varepsilon(\phi_{-t}x)),\end{align*} $$

and the central stable/unstable leaf is (respectively) the $\phi $ -orbit of $W^{\mathrm {s}}(x)$ (respectively $W^{{\mathrm {u}}}(x)$ ). These sets are independent of (any small enough) $\varepsilon $ (that is smaller than the $\varepsilon $ given by Definition 2.2.1).

One has the following, see for example the works of Margulis [Reference Margulis49], Pollicott [Reference Pollicott55] for a construction via Markov codings or Katok and Hasselblatt’s book [Reference Katok and Hasselblatt41, §5 of Ch. 20] for the discrete-time case.

2.4 Skew-products over sub-shifts

Consider now the two-sided subshift $\Sigma $ and let ${\mathsf {K}}:\Sigma \to V$ be Hölder-continuous.

The skew-product system is defined by $f=f^{\mathsf {K}}:\Sigma \times V\to \Sigma \times V$

(7) $$ \begin{align}f(p,v)=(\sigma(p),v-{\mathsf{K}}(p)).\end{align} $$

If $\nu $ is a $\sigma $ -invariant probability measure on $\Sigma $ , then the measure is f-invariant.

The following proposition seems to be well known but we have not been able to find a specific reference in the literature, for completeness, we added a short proof in Appendix A.

We record also the following classical lemma. Let us say that a subset of $\Sigma \times V$ is bounded if it has compact closure, and that it has total mass (with respect to ) if its complement has measure zero. As the space $\Sigma \times V$ is non-compact, it is natural to study the subset of points of $\Sigma \times V$ whose future orbit returns infinitely many times to a fixed open bounded set:

$$ \begin{align*}\mathcal K(f)=\{p\in \Sigma\times V: \text{ there exists } \mathrm{B}\textrm{ open bounded set and }n_k\to\infty\textrm{ with }f^{n_k}(p)\in\mathrm{B}\}.\end{align*} $$

One can be more specific. If $\mathrm {B}_1,\mathrm {B}_2\subset \Sigma \times W$ , we want to understand the measure of

$$ \begin{align*}\mathcal K(\mathrm{B}_1,\mathrm{B}_2)=\{p\in\mathrm{B}_1:\text{ there exists } n_k\to\infty\textrm{ with }f^{n_k}(p)\in\mathrm{B}_2\}.\end{align*} $$

To this end, one considers the sum .

Proof. This is a standard argument valid for any measure-preserving transformation. The first assertion follows by looking at the tail of the series in question:

where, for each $k\in \mathbb {N}, E_k =\{p\in \mathrm {B}_1:\text { there exists } N\geq k\textrm { with }f^N(p)\in \mathrm {B}_2\}.$ The second assertion can be found in, for example, Aaronson’s book [Reference Aaronson1, p. 22].

2.5 An ergodic dichotomy

As in §2.3, let $\unicode{x3c3} ^r$ be the suspension of the shift on $\Sigma $ by the function $r.$ If $\nu $ is a $\sigma $ -invariant probability measure, then $\nu \otimes \,dt/\int rd\nu $ is invariant under the translation flow $\Sigma \times \mathbb {R}$ and induces thus a $\unicode{x3c3} ^r$ -invariant probability measure on denoted by $\hat \nu .$

Let $K:\Sigma \times \mathbb {R} \to V$ be a $\langle\hat r\rangle $ -invariant Hölder-continuous function and consider the flow

Consider the measure on

defined by

.

Theorem 2.5.2. (Sambarino [Reference Sambarino66, Theorem 3.8])

Assume the group generated by the periods of $(r,K)$ is dense in $\mathbb {R}\times V$ and that $\int Kd\hat \nu =0$ for the equilibrium state $\nu $ of $-h(\unicode{x3c3} ^r)r.$ Then there exists $\kappa>0$ such that given two compactly supported continuous functions

one has

as $t\to \infty .$

We include the main outline of its proof in Appendix B. As it is classical, the above result holds for characteristic functions of open bounded sets whose boundary has measure zero.

Proof. The skew-product system $f^{\mathsf {K}}:\Sigma \times V\to \Sigma \times V$ of equation (7), where ${\mathsf {K}}:\Sigma \to V$ is defined by

(8) $$ \begin{align}{\mathsf{K}}(x)=\int_0^{r(x)}K(x,s)\,ds,\end{align} $$

is the first-return map of $\psi $ to its global section $(\Sigma \times \{0\})/_\sim \,\times \, V.$ Consequently, following the flow-lines until reaching the section, one finds a natural (measurable) bijection between $\psi $ -invariant subsets and f-invariant subsets. If $\mu $ is a $\sigma $ -invariant probability measure on $\Sigma ,$ then this bijection preserves the class of invariant-zero-sets between the measures $\hat \mu \otimes \operatorname {\mathrm {Leb}}$ on and $\mu \otimes \operatorname {\mathrm {Leb}}$ on $\Sigma \times V.$ Thus, we can translate ergodicity results from f to the flow $\psi $ and vice versa.

The case $\dim V=1$ is hence settled by Proposition 2.4.2.

For $\dim V\geq 3$ , one considers an open $A\subset \Sigma $ with $\nu (\partial A)=0,$ an open interval $I\subset \mathbb {R}$ with length $<\min r$ , and $B\subset V$ an open ball. Applying Theorem 2.5.2 to

$$ \begin{align*}\bar {\mathrm{B}}=\bar{\mathrm{B}}_1=\bar{\mathrm{B}}_2=A\times I\times B\end{align*} $$

gives a positive C such that for large t, one has

Thus, for a fixed $t_0>0$ , one has that

If $\dim V\geq 3$ , then and Lemma 2.4.3 gives in particular, the system is not ergodic.

By means of Markov partitions (Theorem 2.3.2), the above corollary immediately translates to the following. As in the previous section, let X be a compact metric space equipped with a topologically transitive, Hölder-continuous, metric-Anosov flow $\phi .$ Let $F:X\to V$ be Hölder-continuous and consider the flow $\Phi =(\Phi _t:X\times V\to X\times V)_{t\in \mathbb {R}}$

$$ \begin{align*}\Phi_t(p,v)=\bigg(\phi_tp,v-\int_0^s F(\phi_sp)\,ds\bigg).\end{align*} $$

It is convenient to call the flow $\Phi $ by the skew product of $\phi $ by F.

Corollary 2.5.5. (Dichotomy)

Assume the group spanned by the periods of $(1,F)$ is dense in $\mathbb {R}\times V$ and that $\int Fdm=0$ for the measure of maximal entropy m of $\phi .$ Then, $\Phi $ is mixing as in Theorem 2.5.2, moreover

$$ \begin{align*}\dim V\leq1 \Rightarrow \Phi\textrm{ is ergodic with respect to }m\otimes\operatorname{\mathrm{Leb}}\Rightarrow \dim V\leq2.\end{align*} $$

If $\dim V\geq 3$ , then $\mathcal K(\Phi )$ has zero measure.

2.6 The critical hypersurface

We recall here two results of Babillot and Ledrappier [Reference Babillot and Ledrappier5]. Their paper concerns differentiable Anosov flows but, as one checks the proofs, only the existence of a Markov coding is required for both their results below. We take the liberty to state them in our broader generality and refer the reader to loc. cit. whose proofs work verbatim.

As before, let $F:X\to V$ be Hölder-continuous.

Assumption A. We will assume throughout the remainder of §2 that the closed group $\Delta $ spanned by the periods of F has rank $\dim V,$ (that is, $\Delta \simeq \mathbb {R}^k\times \mathbb {Z}^{\dim V-k}$ for some ) and that, moreover, the group spanned by

$$ \begin{align*}\bigg\{(p(\tau),\int_\tau F):\tau\textrm{ periodic}\bigg\}\end{align*} $$

is isomorphic to $\mathbb {R}\times \Delta .$

The compact convex subset of V,

$$ \begin{align*} \mathcal M^\phi(F)=\bigg\{\int_XFd\mu:\mu\in\mathcal M^\phi\bigg\},\end{align*} $$

has hence non-empty interior. Also, for each $\varphi \in V^*$ , one can consider the pressure of the function $\varphi (F):X\to \mathbb {R}$ :

$$ \begin{align*}\mathbf{P}(\varphi)=P(\phi,-\varphi\circ F).\end{align*} $$

By Assumption A, the function $\mathbf {P}:V^*\to \mathbb {R}$ is analytic and strictly convex (Proposition 2.2.5). Using the formula for the derivative of pressure (Proposition 2.2.5), and the natural identification $\operatorname {\mathrm {Gr}}_{\dim V-1}(V^*)\to \mathbb {P}(V),$ one has, for $\varphi \in V^*$ , that

(9) $$ \begin{align}d_\varphi\mathbf{P}=\int Fdm_{-\varphi(F)},\end{align} $$

where $m_{-\varphi (F)}$ is the equilibrium state of $-\varphi (F).$ One has the following proposition.

Let us denote by $\mathscr L_F=\mathbb {R}_+\cdot \mathcal M^\varphi (F)$ the closed cone generated by the periods of $F.$ If $0$ does not belong to $\mathcal M^\phi (F)$ , then $\mathscr L_F$ is a sharp cone (that is, does not contain a hyperplane of V) and its interior is

$$ \begin{align*}\operatorname{\mathrm{int}}\mathscr L_F=\mathbb{R}_+\cdot\operatorname{\mathrm{int}}(\mathcal M^\phi(F)).\end{align*} $$

One has moreover the following proposition.

Remark 2.6.3. Observe that if $\varphi \in V^*$ is such that $\sum _{\tau \textrm { periodic}}e^{-\ell _\tau (\varphi \circ F)}<\infty $ , then necessarily $\varphi $ is strictly positive on the cone $\mathscr L_F,$ that is, $\varphi \in \operatorname {\mathrm {int}}(\mathscr L_F^*).$ Indeed, if $\sum _{\tau }e^{-\ell _\tau (\varphi \circ F)}$ is convergent, then the formula for pressure on Proposition 2.2.5 gives that $\mathbf {P}(\varphi )\leq 0.$ Since $\mathbf {P}(0)>0$ , there exists $s\in (0,1]$ such that $\mathbf {P}(s\varphi )=0.$ The variational principle (equation (2)) implies that

$$ \begin{align*}\int_\tau\varphi(F)\geq0\end{align*} $$

for every periodic orbit $\tau ,$ and thus Lemma 2.2.7 applies to give that $\varphi (F)$ is Livšic-cohomologous to a strictly positive function and $\mathscr h_{\varphi (F)}\in (0,1].$

We are thus interested in the convergence domain of F

$$ \begin{align*}\mathcal D_F & =\{\varphi\in\operatorname{\mathrm{int}}(\mathscr L_F^*): \mathscr h_{\varphi(F)}\in(0,1)\}\\ & \subset \bigg\{\varphi\in V^*: \sum_{\tau\textrm{ periodic}} e^{-\ell_\tau(\varphi\circ F)}<\infty\bigg\}, \end{align*} $$

and the critical hypersurface (also usually called the entropy-one set or the Manhattan curve)

$$ \begin{align*}\mathcal Q_F & =\{\varphi\in\operatorname{\mathrm{int}}(\mathscr L_F^*): \mathscr h_{\varphi(F)}=1\}\end{align*} $$

whose name is justified in the next corollary.

Corollary 2.6.4. Assume $0\notin \mathcal M^\phi (F).$ Then $\mathcal D_F=\mathbf {P}^{-1}(-\infty ,0)$ and $\mathcal Q_F=\mathbf {P}^{-1}(0)$ . Consequently, $\mathcal D_F$ is a strictly convex set whose boundary coincides with $\mathcal Q_F.$ The latter is a closed analytic co-dimension-one submanifold of $V^*.$ The map

$$ \begin{align*}\varphi\in\mathcal Q_F\mapsto {\mathsf{T}}_\varphi\mathcal Q_F\end{align*} $$

induces a diffeomorphism between $\mathcal Q_F$ and directions in the interior of the cone $\mathscr L_F.$

We now focus on the variation of the critical hypersurface when F varies. To this end, consider the Banach space $\operatorname {\mathrm {Holder}}^\alpha (X,V)$ of V-valued Hölder continuous functions with exponent $\alpha .$ The pressure can be considered as an analytic map $\mathbf {P}:\operatorname {\mathrm {Holder}}^\alpha (X,V)\times V^*\to \mathbb {R}$ defined as $\mathbf {P}(G,\psi ):=P(\phi ,-\psi (G)).$ Its differential at the point $(F,\varphi )$ on the vector $(G,\psi )$ is

$$ \begin{align*}d_{(F,\varphi)}\mathbf{P}(G,\psi)=-\int(\psi(F)+\varphi(G))\,dm_{-\varphi(F)},\end{align*} $$

and vanishes identically only if $(F,\varphi )=(0,0).$ The pre-image $\mathbf {P}^{-1}(0)$ is thus a Banach-manifold.

If $F\in \operatorname {\mathrm {Holder}}^\alpha (X,V)$ is such that $0\notin \mathcal M^\phi (F),$ then its critical hypersurface

$$ \begin{align*}\mathcal Q_F=\{\varphi\in V^*:(F,\varphi)\in\mathbf{P}^{-1}(0)\}\end{align*} $$

is the intersection of $\{F\}\times V^*$ with the level set $\mathbf {P}^{-1}(0).$ This intersection will vary analytically on compact sets with F as long as the tangent space $\ker d_{(F,\varphi )}\mathbf {P}$ (for fixed $(F,\varphi )$ with $\mathbf {P}(F,\varphi )=0$ ) is transverse to the vector space $\{0\}\times V^*.$ Since $\ker d_{(F,\varphi )}\mathbf {P}$ has co-dimension 1, transversality is implied by $\ker d_{(F,\varphi )}\mathbf {P}\cap \{0\}\times V^*$ being co-dimension $1$ on $V^*.$ However, by Corollary 2.6.4, this latter intersection is, as long as F verifies assumption A and $0\notin \mathcal M^\phi (F),$ the tangent space ${\mathsf {T}}_\varphi \mathcal Q_F,$ which has co-dimension $1.$ We have thus established the following corollary.

2.7 Dynamical intersection and the critical hypersurface

We recall here the concept of dynamical intersection of Bridgeman et al [Reference Bridgeman, Canary, Labourie and Sambarino17]. Similar concepts have been previously studied by Bonahon [Reference Bonahon11], Burger [Reference Burger19], Croke and Fathi [Reference Croke and Fathi28], and Knieper [Reference Knieper42], among others.

Let $f:X\to \mathbb {R}_+$ be a positive Hölder-continuous function and, for $t>0,$ consider the finite set ${\mathsf {R}}_t(f)=\{\tau \textrm { periodic}:\ell _\tau (f)\leq t\}.$ Let $g:X\to \mathbb {R}$ be Hölder-continuous (but not necessarily positive). Then the dynamical intersection between f and g is defined by

$$ \begin{align*}\mathbf{I}(f,g)=\lim_{t\to\infty}\frac1{\#{\mathsf{R}}_t(f)}\sum_{\tau\in {\mathsf{R}}_t(f)}\frac{\ell_\tau(g)}{\ell_\tau(f)}.\end{align*} $$

Then one has the following proposition.

Proposition 2.7.1. [Reference Bridgeman, Canary, Labourie and Sambarino17, §3.4]

One has

$$ \begin{align*}\mathbf{I}(f,g)=\frac{\int gdm_{-\mathscr h_ff}}{\int f\,dm_{-\mathscr h_ff}},\end{align*} $$

in particular, $\mathbf {I}$ is well defined and varies analytically with f and g among Hölder- continuous functions with fixed Hölder exponent. If g is moreover positive, then one has $\mathbf {I}(f,g)\geq \mathscr h_f/\mathscr h_g.$

We now place ourselves in the context of the previous subsection, that is, we consider a Hölder-continuous $F:X\to V$ and we assume moreover that $0\notin \mathcal M^\phi (F).$ For $\varphi \in \mathcal Q_F$ , we consider the map $\mathbf {I}_{\varphi }:V^*\to \mathbb {R}$ defined by

(10) $$ \begin{align}\mathbf{I}_{\varphi}(\psi):= \mathbf{I}(\varphi(F),\psi(F))=\frac{\int \psi(F)\,dm_{-\varphi(F)}}{\int\varphi(F)\,dm_{-\varphi(F)}},\end{align} $$

where the last equality comes from Proposition 2.7.1 and the fact that $\mathscr h_{\varphi (F)}=1$ . Observe that it is a linear map. We then have the following explicit interpretation of the tangent space to the critical hypersurface purely in terms of periods.

3.1 The Ledrappier potential of a Hölder cocycle

Let V be a finite dimensional real vector space. A Hölder cocycle is a function such that:

• • for all , one has $c(\gamma h,(x,y))=c(h,(x,y))+c(\gamma ,h(x,y));$

• • there exists $\alpha \in (0,1]$ such that for every , the map $c(\gamma ,\cdot )$ is $\alpha $ -Hölder continuous.

The period of a Hölder cocycle for a hyperbolic is $\ell _c(\gamma ):=c(\gamma ,(\gamma ^-,\gamma ^+)).$ Two cocycles $c,c'$ are cohomologous if there exists a Hölder-continuous function such that for all and , one has

$$ \begin{align*}c(\gamma,(x,y))-c'(\gamma,(x,y))=U(\gamma(x,y))-U(x,y).\end{align*} $$

Two cohomologous cocycles have the same marked spectrum $\gamma \mapsto \ell _c(\gamma ).$ The following should be compared with [Reference Ledrappier44, Théorème 3].

Proposition 3.1.1. For every Hölder cocycle c, there exists a Hölder-continuous function such that for every hyperbolic , one has

$$ \begin{align*}\int_{[\gamma]}\mathcal J_c=\ell_c(\gamma).\end{align*} $$

Cohomologous cocycles induce Livšic-cohomolgous functions.

Proof. The general case follows from the case $V=\mathbb {R}$ by the Riesz representation theorem. Assume thus $V=\mathbb {R}$ and consider the trivial line bundle equipped with the bundle automorphisms

$$ \begin{align*}\gamma\cdot(p,s):=(\gamma p,e^{-c(\gamma,(x,y))}s),\end{align*} $$

where $p=(x,y,t).$ Denote by the quotient line bundle. It is equipped with a flow $(\hat {\mathsf {g}}_t:{\mathsf {F}}\to {\mathsf {F}})_{t\in \mathbb {R}}$ by bundle automorphisms, induced on the quotient by

$$ \begin{align*}t\cdot(p,s)=({\mathsf{g}}_tp,s).\end{align*} $$

Let $|\,|$ be a Euclidean metric on ${\mathsf {F}}$ and define, for $v\in {\mathsf {F}}_p,$

(11) $$ \begin{align}\mathcal T(p,t)=\log\frac{|\hat{\mathsf{ g}}_tv|}{|v|}.\end{align} $$

It is a translation cocycle over ${\mathsf {g}},$ indeed since ${\mathsf {F}}_p$ is one dimensional, the choice of v does not matter, and since $\hat {\mathsf {g}}$ is a flow, one has

$$ \begin{align*}\log\frac{|\hat{\mathsf{ g}}_{t+s}(v)|}{|v|}= \log\frac{|\hat{\mathsf{g}}_{t}(\hat{\mathsf{g}}_sv)|}{|v|}\frac{|\hat{\mathsf{g}}_s(v)|}{|\hat{\mathsf{g}}_s(v)|}=\log\frac{|\hat{\mathsf{g}}_{t}(\hat{\mathsf{g}}_sv)|}{|\hat{\mathsf{g}}_s(v)|}+\log\frac{|\hat{\mathsf{ g}}_s(v)|}{|v|}.\end{align*} $$

By Corollary 2.2.3, there exists a Hölder-continuous function such that $\mathcal T$ and $k_{\mathcal J_c}$ are Livšic-cohomologous. We end the proof by a period computation. For every hyperbolic , one has, for all $s\in \mathbb {R},$ that $\gamma (\gamma _-,\gamma _+,s)=(\gamma _-,\gamma _+,e^{-\ell _c(\gamma )}s),$ or equivalently, for any and $v\in {\mathsf {F}}_x,$

$$ \begin{align*}\ell_c(\gamma)=\log\frac{|\hat{\mathsf{ g}}_{p(\gamma)}v|}{|v|}=\ell_{[\gamma]}(\mathcal T)=\int_{[\gamma]}\mathcal J_c,\end{align*} $$

where $p(\gamma )$ is the period of $[\gamma ]$ for ${\mathsf {g}}.$ Since cohomologous cocycles have the same marked spectrum, the associated functions have the same periods and are thus Livšic-cohomolgous by Theorem 2.2.2.

3.2 Real cocycles and reparameterizations

Assume now $V=\mathbb {R}$ and consider a cocycle c with non-negative (and not all vanishing) periods. Define its entropy by

For such a cocycle, consider the action of on via c:

(12) $$ \begin{align}\gamma\cdot(x,y,t)=(\gamma x,\gamma y,t-c(\gamma,(x,y))).\end{align} $$

Let us denote by ${\unicode{x3c7} ^{c}}$ the quotient space The following is reported by Sambarino [Reference Sambarino65] for fundamental groups of closed negatively curved manifolds and by Carvajales [Reference Carvajales23] for the refraction cocycle of a ${{{\Delta }}}$ -Anosov representation (see Definition 5.3.1).

The topological entropy of $\phi ^c$ is thus $\mathscr h_c.$ (When has torsion elements, this fact requires some work, see the work of Carvajales et al [Reference Carvajales, Dai, Pozzetti and Wienhard24, §5] for details.)

Proof. The first assertion follows at once from Lemma 2.2.7. For the remaining statements, we continue as in the proof of the proposition but for the cocycle $-c.$ Observe first that $-\mathcal J_c=\mathcal J_{-c}.$ Since $\mathcal T$ and $k_{\mathcal J_{-c}}$ are Livšic-cohomologous, there exists such that for all , and $v\in {\mathsf {F}}_p$ , one has (recall equation (11))

$$ \begin{align*}\log\frac{|\hat{\mathsf{ g}}_tv|}{|v|}-\int_0^t\mathcal J_{-c}=U({\mathsf{g}}_tp)-U(x).\end{align*} $$

Since $\mathcal J_{-c}=-\mathcal J_c$ is Livšic-cohomologous to a strictly negative function, the above equation implies that the flow $\hat {\mathsf {g}}$ is contracting on ${\mathsf {F}},$ that is, there exist positive C and $\mu $ such that for all $v\in {\mathsf {F}}$ and $t\in \mathbb {R}$ , one has

$$ \begin{align*}|\hat{\mathsf{g}}_tv|\leq Ce^{-\mu t}|v|.\end{align*} $$

A standard procedure (see for example [Reference Katok and Hasselblatt41] or [Reference Bridgeman, Canary, Labourie and Sambarino17, Lemma 4.3]) provides a Euclidean metric $\|\,\|$ on ${\mathsf {F}}$ such that the constant C equals $1.$ We denote also by $\|\,\|$ the lift of this metric to it is a -invariant family.

Given then , we let $v_{(x,y,t)}\in (\mathbb {R}-\{0\})/\pm $ be such that $\|v_{(x,y,t)}\|=1.$ As in [Reference Bridgeman, Canary, Labourie and Sambarino17, Proposition 4.2], the map

(13)

is -equivariant and an orbit equivalence between the $\mathbb {R}$ -actions.

3.3 Patterson–Sullivan measures

Let us consider now Hölder cocycles with $V=\mathbb {R}$ and only depending on one variable, that is, Assume moreover that c has non-negative periods and finite entropy. A cocycle is dual to c if for every hyperbolic , one has

$$ \begin{align*}\ell_{\bar c}(\gamma)=\ell_c(\gamma^{-1}).\end{align*} $$

Consider a pair of dual cocycles $(\bar c,c)$ and assume a Patterson–Sullivan measure of exponent $\delta $ exists for each c and $\bar c,$ denoted by $\mu $ and $\bar \mu $ , respectively. Assume moreover that a Gromov product for the pair $(\bar c, c)$ exists. The measure

(15) $$ \begin{align}\widetilde{\mathscr m}= e^{-\delta[\cdot,\cdot]}\bar\mu\otimes \mu\otimes \,dt\end{align} $$

on

is hence

-invariant (the action being via $c,$ as in equation (12) and $\mathbb {R}$ -invariant. Passing to the quotient, one obtains a measure $\mathscr m$ on ${\unicode{x3c7} ^{c}}$ invariant under the flow $\phi ^c.$ Observe that we can write $\widetilde {\mathscr m}$ as

The measure $e^{-\delta t}\,d\mu \,dt$ is

-invariant on

(indeed, if

is continuous, then

$$ \begin{align*}\int f(\gamma(y,t))e^{-\delta t} \,d\mu \,dt,&=\int f(\gamma y,t-c(\gamma,y))e^{-\delta t} \,d\mu \,dt & \\& =\int f(\gamma y,t)e^{-\delta (t-c(\gamma,y))} \,d\mu \,dt \,\quad\qquad\textrm{(by translation invariance)}&\\& =\int f( y,t)e^{-\delta (t-c(\gamma,\gamma^{-1}y))}e^{-\delta c(\gamma^{-1},y)} \,d\mu \,dt \quad\textrm{(by definition of }\mu)&\\& =\int f(y,t)e^{-\delta t}\,d\mu \,dt)& \end{align*} $$

and the family

$$ \begin{align*}m_{(y,t)}=e^{\delta t}e^{-\delta[\cdot,y]}d\bar\mu\end{align*} $$

is

-equivariant.

This decomposition induces then a local decomposition of $\mathscr m$ as a product of measures on the laminations induced on the quotient by and The local product structure induced by permits to transport the measures $m_{(y,t)},$ parallel to the central stable leaf $W^{\mathrm {cu}}[(x,y,t)],$ to the stable leaf of $[(x,y,t)]$ for $\phi ^c$ to obtain a family of measures at each strong stable leaf $\nu ^{{\mathrm {s}}}_p$ such that

(16) $$ \begin{align}\frac{d(\phi^c_{-t})_*\nu^{{\mathrm{s}}}_p}{\,d\nu^{{\mathrm{s}}}_{\phi^c_tp}}=e^{-\delta t}.\end{align} $$

Margulis’s description of the measure maximizing entropy (§2.3.4) then gives that $\delta =\mathscr h_c$ and that $\mathscr m$ maximizes entropy for $\phi ^c.$ Thus, subject to the existence of the Patterson–Sullivan measures and the Gromov product, we summarize the above discussion in the following proposition.

Let us consider again the measure $\widetilde {\mathscr m}$ on from equation (15) but let us instead study the -action on the $\mathbb {R}$ -coordinate by the Gromov–Mineyev cocycle, so that and the induced flow is ${\mathsf {g}}.$ The quotient measure, $\mathscr a,$ is ${\mathsf {g}}$ -invariant and the orbit equivalence $\unicode{x3be} $ from equation (13) preserves, by the way it is defined, zero flow-invariant sets between $\mathscr a$ and $\mathscr m.$ Since $\unicode{x3be} $ is a conjugation between ${\mathsf {g}}^{\mathcal J_c}$ and $\phi ^c,$ and $\mathscr m$ maximizes entropy for $\phi ^c,$ the measure $\unicode{x3be} _*\mathscr m$ is ${\mathsf {g}}^{\mathcal J_c}$ -invariant, maximizes entropy for ${\mathsf {g}}^{\mathcal J_c}$ , and has the same zero sets as $\mathscr a.$

One concludes that the Abramov transform in equation (3) ${\mathscr a}^\#$ (with respect to $\mathcal J_c$ ) is a measure of maximal entropy of the flow ${\mathsf {g}}^{\mathcal J_c}.$ Lemma 2.1.1 implies then that $\mathscr a/|\mathscr a|$ is the equilibrium state of $-\mathscr h_c\mathcal J_c.$ Let us summarize in the following remark.

Since the zero sets of an equilibrium state are uniquely determined by the Livšic-cohomology class of the associated potential up to an additive constant (Theorem 2.3.3), one concludes the following corollary.

Proof. A final argument is required, indeed from §2.3.3, there exists a constant K such that $\mathscr h_c\mathcal J_c$ and $\mathscr h_\kappa \mathcal J_\kappa +K$ are Livšic-cohomologous. However, since $\mathscr h_c$ is the topological entropy of ${\mathsf {g}}^{\mathcal J_c},$ Lemma 2.1.1 gives

$$ \begin{align*}0=P({\mathsf{g}},-\mathscr h_c\mathcal J_c)=P({\mathsf{g}},-\mathscr h_\kappa\mathcal J_\kappa+K)=K,\end{align*} $$

where the second equality holds by Remark 2.1.2, and the third equality by the definition of P (equation (2)).

The above corollary was reported by Ledrappier [Reference Ledrappier44] when is the fundamental group of a negatively curved closed manifold. The proof uses also a disintegration argument. One may also check [Reference Kaimanovich39] and Babillot’s survey [Reference Babillot4] specifically for the Buseman cocycle ( as in Ledrappier’s aforementioned situation), and [Reference Carvajales23, Appendix] for the refraction cocycle $\unicode{x3b2} _{\varphi }$ (see §5.3 for the definition) of a ${{{\Delta }}}$ -Anosov representation of an arbitrary word-hyperbolic group.

3.4 Vector-valued cocycles I: the critical hypersurface

Let now

be a Hölder cocycle and consider the compact convex set $\mathcal M^{\mathsf {g}}(\mathcal J_c)\subset V.$ Since this set depends on the base flow ${\mathsf {g}}$ , it is natural to consider the limit cone of c:

We will work from now on with flow ${\mathsf {g}}^f$ given by the remark and the Ledrappier potential $\mathcal J_c^{\mathsf {g^f}}.$ We will rename these by ${\mathsf {g}}$ and $\mathcal J_c$ though as to not overcharge the paper with notation and keep in mind that when we restrict the image of $\mathcal J_c$ to $V',$ Assumption A is verified.

Let $(\mathscr L_c)^*=\{\psi \in V^*:\psi |\mathscr L_c\geq 0\}$ be the dual cone of $c.$ For $\psi \in V^*$ , denote and $\mathscr h_\psi =\mathscr h_{c_\psi }.$

Since $0\notin \mathcal M^{\phi ^{c_\psi }}(\mathcal J_c)$ , we can apply Corollary 2.6. Denote by

respectively the critical hypersurface and the convergence domain of $c.$

Since we have not required the cone $\mathscr L_c$ to have non-empty interior, consider its annihilation space

$$ \begin{align*}\mathrm{Ann}(\mathscr L_c)=\{\psi\in V^*:\mathscr L_c\subset\ker\psi\}.\end{align*} $$

If $\varphi \in \operatorname {\mathrm {int}}(\mathscr L_c)^*$ and $\psi \in \mathrm {Ann}(\mathscr L_c)$ , then the potentials $\varphi (\mathcal J_c)$ and $(\varphi +\psi )(\mathcal J_c)$ are Livšic-cohomologous. Let $\pi ^c:V^*\to V^*/\mathrm {Ann}(\mathscr L_c)$ be the quotient projection.

We also import the concept of dynamical intersection of §2.7 to this setting using the Ledrappier potential of $c.$ For $\varphi \in \mathcal Q_c$ , define the dynamical intersection map associated to c by $\mathbf {I}_\varphi =\mathbf {I}^c_\varphi :V^*\to \mathbb {R}$ be defined by

$$ \begin{align*}\mathbf{I}_\varphi(\psi)=\mathbf{I}(\varphi(\mathcal J_c),\psi(\mathcal J_c)).\end{align*} $$

By definition, $\mathbf {I}(\varphi (\mathcal J_c),\psi (\mathcal J_c))$ only depends on the Livšic-cohomology classes of $\varphi (\mathcal J_c)$ and $\psi (\mathcal J_c),$ so we may freely consider $\mathbf {I}$ as defined on $\mathcal Q_c\times V^*$ or on $\pi ^c(\mathcal Q_c)\times V^*/\mathrm {Ann}(\mathscr L_c).$ One has the following corollary.

Corollary 3.4.3. Under Assumption C, one has that $\pi ^c(\mathcal D_c)$ is a strictly convex set whose boundary is $\pi ^c(\mathcal Q_c).$ The latter is an analytic co-dimension-one sub-manifold. The map ${\mathsf {u}}:\pi ^c(\mathcal Q_c)\to \mathbb {P}(\operatorname {\mathrm {span}}\{\mathscr L_c\})$ defined by

$$ \begin{align*}\varphi\mapsto{\mathsf{u}}_\varphi:={\mathsf{T}}_\varphi\pi^c(\mathcal Q_c)=\ker\mathbf{I}_\varphi\end{align*} $$

is an analytic diffeomorphism between $\pi ^c(\mathcal Q_c)$ and directions in the relative interior of $\mathscr L_c.$

3.5 Vector-valued cocycles II: skew-product structure

We remain in the situation of §3.4, that is, we keep Assumption C. It follows at once from Theorem 3.2 that the -action

$$ \begin{align*}\gamma(x,y,v)=(\gamma x,\gamma y,v-c(\gamma,y))\end{align*} $$

is properly discontinuous. We aim to give a description of the V-action on the quotient space .

By Lemma 3.4.2 and Theorem 3.2.2, we can, for every $\varphi \in \operatorname {\mathrm {int}}(\mathscr L_c)^*,$ consider the refraction flow $\phi ^{c_\varphi }=(\phi ^{c_\varphi }_t:{\unicode{x3c7} ^{c_\varphi }}\to {\unicode{x3c7} ^{c_\varphi }})_{t\in \mathbb {R}}.$ Such a $\varphi $ is fixed from now on.

Remark 3.5.1. Let us still denote by $\mathcal J_c$ the Ledrappier potential for c over the flow $\phi ^{c_\varphi },$ that is, for every hyperbolic , one has

$$ \begin{align*}\int_{[\gamma]}\mathcal J_c=\ell_c(\gamma)\in V.\end{align*} $$

Let $\mathscr p$ be the probability measure of maximal entropy of $\phi ^{c_\varphi }.$ The growth direction of $\varphi $ is the line of V

(17) $$ \begin{align}{\mathsf{u}}_\varphi={\mathsf{T}}_\varphi\mathcal Q_c= \mathbb{R}\cdot\int_{{\unicode{x3c7}^{c_\varphi}}}\mathcal J_c d\mathscr p\end{align} $$

(the last equality follows from equation (9) and Corollary 2.6.4). Consider also the projection $\pi ^\varphi :V\to \ker \varphi $ parallel to ${\mathsf {u}}_\varphi $ and denote by $\mathcal J_c^\varphi :{\unicode{x3c7} ^{c_\varphi }}\to \ker \varphi $ the composition $\mathcal J_c^\varphi =\pi ^\varphi \circ \mathcal J_c.$ Observe that

(18) $$ \begin{align} \int_{{\unicode{x3c7}^{c_\varphi}}} \mathcal J_c^\varphi d\mathscr p=0.\end{align} $$

Fix $u_\varphi \in {\mathsf {u}}_\varphi $ with $\varphi (u_\varphi )=1$ and define the directional flow by induction on the quotient of

$$ \begin{align*}t\cdot(x,y,v)=(x,y,v-tu_\varphi).\end{align*} $$

Proposition 3.5.2. (Sambarino [Reference Sambarino66])

There exists a (bi)-Hölder-continuous homeomorphism

commuting with the $\ker \varphi $ action, that conjugates the flow $\unicode{x3c9} ^\varphi $ with $\Phi ^\varphi =(\Phi _t^\varphi :{\unicode{x3c7} ^{c_\varphi }}\times \ker \varphi \to {\unicode{x3c7} ^{c_\varphi }}\times \ker \varphi )_{t\in \mathbb {R}}$

$$ \begin{align*}{\displaystyle\Phi_t^\varphi(p,v):=\bigg(\phi^{c_\varphi}_t(p),v-\int_0^t\mathcal J_c^\varphi(\phi^{c_\varphi}_s p)\,ds\bigg).}\end{align*} $$

Let us moreover place ourselves in the existence assumptions of §3.3 for the cocycle $c_\varphi ,$ that is, assume there exists:

• • a dual cocycle $\bar c_\varphi $ together with a Gromov product $[\cdot ,\cdot ]$ for the pair $(\bar c_\varphi ,c_\varphi );$

• • a Patterson–Sullivan measure for each $c_\varphi $ and $\bar c_\varphi ,$ denoted by $\mu $ and $\bar \mu $ , respectively. Recall from Proposition 3.3.2 that, necessarily, the exponent of both $\mu $ and $\bar \mu $ is $\mathscr h_{c_\varphi },$ the topological entropy of $\phi ^{c_\varphi }.$

By Proposition 3.3.2, the measure $\mathscr m$ maximizes entropy for $\phi ^{c_\varphi }.$ One has then $\mathscr p=\mathscr m/|\mathscr m|.$ Consider the $\varphi $ -Bowen–Margulis measure $\Omega ^\varphi $ on defined as induction on the quotient by

$$ \begin{align*}e^{-\mathscr h_{c_\varphi}[\cdot,\cdot]}\bar\mu\otimes\mu\otimes \operatorname{\mathrm{Leb}}_{V},\end{align*} $$

for a Lebesgue measure on $V.$ One has the following result.

3.6 Vector-valued cocycles III: dynamical consequences

We remain in the situation of §3.4. Let us say that c is non-arithmetic if the periods of c span a dense subgroup in $V.$

One concludes at once the following consequences.

4.1 Restricted roots and parabolic groups

Consider $\vartheta \subset {{{\Delta }}}$ and let ${\mathsf {P}}_\vartheta ,$ respectively $\check {\mathsf {P}}_\vartheta ,$ be the normalizers in ${\mathsf {G}}$ of, respectively the Lie algebras

$$ \begin{align*} & \bigoplus_{\alpha\in<{{{\Delta}}}-\vartheta>}{\mathfrak g}_{-\alpha}\oplus \bigoplus_{\alpha\in{{{\Phi}}}^+}{\mathfrak g}_\alpha,\\ & \bigoplus_{\alpha\in<{{{\Delta}}}-\vartheta>}{\mathfrak g}_{\alpha}\oplus \bigoplus_{\alpha\in{{{\Phi}}}^+}{\mathfrak g}_{-\alpha}. \end{align*} $$

The $\vartheta $ -flag space is $\mathcal F_\vartheta ={\mathsf {G}}/{\mathsf {P}}_\vartheta .$ The orbit ${\mathsf {G}}\cdot ([\check {\mathsf {P}}_\vartheta ],[{\mathsf {P}}_\vartheta ])\subset \mathcal F_{\operatorname {\mathrm {i}}\vartheta }\times \mathcal F_\vartheta $ is the unique open orbit on this product space, we will denote it by $\mathcal F^{(2)}_\vartheta $ and say that $(x,y)\in \mathcal F_{\operatorname {\mathrm {i}}\vartheta }\times \mathcal F_\vartheta $ are transverse if in fact $(x,y)\in \mathcal F^{(2)}_\vartheta .$

Denote by $(\cdot ,\cdot )$ a $\mathcal W$ -invariant inner product on ${\mathsf {E}}, (\cdot ,\cdot )$ the induced inner product on ${\mathsf {E}}^*,$ define $\langle,\rangle $ on ${\mathsf {E}}^*$ by

$$ \begin{align*}\langle\chi,\psi\rangle=\frac{2(\chi,\psi) }{(\psi,\psi)},\end{align*} $$

and let $\{\varpi _\alpha \}_{\alpha \in \Pi }$ be the fundamental weights of ${{{\Phi }}},$ defined by the equations $\langle\varpi _\alpha ,\sigma\rangle=d_\alpha \delta _{\alpha \sigma },$ where $d_\alpha =1$ if $2\alpha \notin {{{\Phi }}}$ and $d_\alpha =2$ otherwise.

4.2 The center of the Levi group

We now consider the vector subspace

$$ \begin{align*}{\mathsf{E}}_\vartheta=\bigcap_{\alpha\in{{{\Delta}}}-\vartheta}\ker\alpha^\omega\end{align*} $$

together with the unique projection $p_\vartheta :{\mathsf {E}}\to {\mathsf {E}}_\vartheta $ invariant under the subgroup of the Weyl group $\mathcal W_\vartheta =\{w\in \mathcal W:w|{\mathsf {E}}_\vartheta =\operatorname {\mathrm {id}}\}.$ The dual space $({\mathsf {E}}_\vartheta )^*$ sits naturally as the subspace of ${\mathsf {E}}^*$ of $p_\vartheta $ -invariant linear forms

$$ \begin{align*}({\mathsf{E}}_\vartheta)^*=\{\varphi\in{\mathsf{E}}^*:\varphi\circ p_\vartheta=\varphi\}.\end{align*} $$

It is spanned by the fundamental weights $\{\varpi _\sigma |{\mathsf {E}}_ \vartheta :\sigma \in \vartheta \}.$

4.3 Cartan decomposition

Let ${\mathsf {K}}\subset {\mathsf {G}}$ be a compact group that contains a representative for every element of the Weyl group $\mathcal W,$ this is to say, such that the normalizer $N_{\mathsf {G}}({\mathsf {A}})$ verifies $N_{\mathsf {G}}({\mathsf {A}})=(N_{\mathsf { G}}({\mathsf {A}})\cap {\mathsf {K}}){\mathsf {A}}.$ One has ${\mathsf {G}}={\mathsf {K}}{\mathsf {A}}^+{\mathsf {K}}$ and if $z,w\in {\mathsf {A}}^+$ are such that $z\in {\mathsf {K}}w{\mathsf {K}}$ , then $\nu (z)=\nu (w).$ There exists thus a function

$$ \begin{align*}a:{\mathsf{G}}\to {\mathsf{E}}^+\end{align*} $$

such that for every $g_1,g_2\in {\mathsf {G}}$ , one has $g_1\in {\mathsf {K}}g_2 {\mathsf {K}}$ if and only if $a(g_1)=a(g_2).$ It is called the Cartan projection of ${\mathsf {G}}.$

4.4 Jordan decomposition

Recall that the Jordan decomposition states that every $g\in {\mathsf {G}}$ has a power $g^k$ ( $k=1$ if $\mathbb {K}$ is Archimedean) that can be written as a commuting product $g=g_eg_hg_n,$ where $g_e$ is elliptic, $g_h$ is semi-simple over $\mathbb {K}$ , and $g_n$ is unipotent. The component $g_h$ is conjugate to an element $z_g\in {\mathsf {A}}^+$ and we let

$$ \begin{align*}\unicode{x3bb}(g)=(1/k)\nu(z_g)\in{\mathsf{E}}^+.\end{align*} $$

The map $\unicode{x3bb} :{\mathsf {G}}\to {\mathsf {E}}^+$ is the Jordan projection of ${\mathsf {G}}.$ We will also denote by $\unicode{x3bb} _\vartheta :{\mathsf {G}}\to {\mathsf {E}}_\vartheta $ the composition $p_\vartheta \circ \unicode{x3bb} .$ For ${\mathsf {G}}=\operatorname {\mathrm {{\mathsf {PGL}}}}_d(\mathbb {K})$ , we will denote by $\unicode{x3bb} _1(g)$ the logarithm of the spectral radius of $g.$

4.5 Representations of ${\mathsf {G}}$

Let ${\mathsf {V}}$ be a finite-dimensional $\mathbb {K}$ -vector space and $\unicode{x3c6} :{\mathsf {G}}\to \operatorname {\mathrm {{\mathsf {PGL}}}}({\mathsf {V}})$ be an algebraic irreducible representation. Then the weight space associated to $\chi \in \mathbf X({\mathsf {A}})$ is the vector space

$$ \begin{align*}{\mathsf{V}}_\chi=\{v\in {\mathsf{V}}:\unicode{x3c6}(a) v=\chi(a) v \text{ for all } a\in{\mathsf{A}}\}\end{align*} $$

and if $V_\chi \neq 0$ , then we say that $\chi ^\omega \in {\mathsf {E}}^*$ is a restricted weight of $\unicode{x3c6} .$ Theorem 7.2 of Tits [Reference Tits69] states that the set of weights has a unique maximal element with respect to the order $\chi \geq \psi $ if $\chi -\psi $ is a sum of simple roots with non-negative coefficients. This is called the highest weight of $\unicode{x3c6} $ and denoted by $\chi _\unicode{x3c6} .$

We denote by $\|\,\|_\unicode{x3c6} $ a norm on ${\mathsf {V}}$ invariant under $\unicode{x3c6} {\mathsf {K}}$ and such that $\unicode{x3c6} {\mathsf {A}}$ consists on semi-homotheties (that is, diagonal on an orthonormal basis $\mathcal E$ of $V,$ in the classical sense if $\mathbb {K}$ Archimedean, and such that $\|\sum _{e\in \mathcal E}v_ee\|=\max \{|v_e|\}$ if $\mathbb {K}$ is non-Archimedean). If $\mathbb {K}$ is Archimedean, the existence of such a norm is classical (see for example [Reference Benoist and Quint9, Lemma 6.33]), if $\mathbb {K}$ is non-Archimedean, then this is the content of [Reference Quint60, Théorème 6.1] due to Quint.

For every $g\in {\mathsf {G}}$ , one has then

(19)

Denote by $W_{\chi _\unicode{x3c6} }$ the $\unicode{x3c6} {\mathsf {A}}$ -invariant complement of ${\mathsf {V}}_{\chi _\unicode{x3c6} }.$ The stabilizer in ${\mathsf {G}}$ of ${\mathsf {W}}_{\chi _\unicode{x3c6} }$ is $\check {\mathsf {P}}_{\vartheta ,\mathbb {K}},$ and thus one has a map of flag spaces

(20) $$ \begin{align}(\Xi_\unicode{x3c6},\Xi^*_\unicode{x3c6}):\mathcal F_{\vartheta_\unicode{x3c6}}^{(2)}({\mathsf{G}})\to \operatorname{\mathrm{Gr}}_{\dim {\mathsf{V}}_{\chi_\unicode{x3c6}}}^{(2)}({\mathsf{V}}),\end{align} $$

where $\vartheta _\unicode{x3c6} =\{\sigma \in {{{\Delta }}}:\chi _\unicode{x3c6} -\sigma \textrm { is a weight of }\unicode{x3c6} \}$ . This is a proper embedding which is a homeomorphism onto its image. Here, $\mathscr G_{\dim {\mathsf {V}}_{\chi _\unicode{x3c6} }}^{(2)}({\mathsf {V}})$ is the open $\operatorname {\mathrm {{\mathsf {PGL}}}} ({\mathsf {V}})$ -orbit in the product of the Grassmannians of $(\dim {\mathsf {V}}_{\chi _\unicode{x3c6} })$ -dimensional and $(\dim {\mathsf {V}}-\dim {\mathsf {V}}_{\chi _\unicode{x3c6} })$ -dimensional subspaces.

One has the following proposition from Tits [Reference Tits69] that guarantees existence of certain representations of ${\mathsf {G}}.$ We say that $\phi $ is proximal if $\dim {\mathsf {V}}_{\chi _\unicode{x3c6} }=1.$

4.6 Buseman–Iwasawa cocycle

The Iwasawa decomposition of ${\mathsf {G}}$ states that every $g\in {\mathsf {G}}$ can be written as a product $lzu$ with $l\in {\mathsf {K}}, z\in {\mathsf {A}}$ , and $u\in {\mathsf {U}}_{{{\Delta }}},$ where ${\mathsf {U}}_{{{\Delta }}}$ is the unipotent radical of ${\mathsf {P}}_{{{{\Delta }}}}.$ When $\mathbb {K}$ is non-Archimedean, the Iwasawa decomposition is not unique; however, if $z_1,z_2\in {\mathsf {A}}$ are such that $z_1\in {\mathsf {K}} z_2{\mathsf {U}}_{{{{\Delta }}}}$ , then $\nu (z_1)=\nu (z_2).$

The Buseman–Iwasawa cocycle of ${\mathsf {G}}, \beta :{\mathsf {G}}\times \mathcal F\to {\mathsf {E}},$ is defined, for all $g\in {\mathsf {G}}$ and $k[{\mathsf {P}}_{{{\Delta }}}]\in \mathcal F,$ if $gk=lzu$ is an Iwasawa decomposition of $gk$ , by $\beta (g,k[{\mathsf {P}}_{{{\Delta }}}])=\nu (z).$ Quint proved the following lemma.

The Buseman–Iwasawa cocycle can also be read from the representations of ${\mathsf {G}}.$ Indeed, Quint [Reference Quint62, Lemme 6.4] states that for every $g\in {\mathsf {G}}$ and $x\in \mathcal F_\vartheta $ , one has

(21)

where $v\in \Xi _{\unicode{x3c6} _\sigma }(x)\in \mathbb {P}({\mathsf {V}}_\sigma )$ is non-zero.

4.7 Gromov product

As in [Reference Sambarino66], the Gromov product $\mathscr G_\vartheta :\mathcal F^{(2)}_\vartheta \to {\mathsf {E}}_\vartheta $ is defined such that, for every $(x,y)\in \mathcal F^{(2)}_\vartheta $ and $\sigma \in \vartheta ,$ one has

where $\varphi \in \Xi ^*_{\unicode{x3c6} _\sigma }(x)$ and $v\in \Xi _{\unicode{x3c6} _\sigma }(y)$ are the equivariant maps from equation (20).

A straightforward computation ([Reference Sambarino66, Lemma 4.12]) gives, for all $g{\kern-1.5pt}\in{\kern-1.5pt} {\mathsf {G}}$ and $(x,{\kern-1pt}y){\kern-1.5pt}\in{\kern-1.5pt} \mathcal F^{(2)}_\vartheta ,$

(22) $$ \begin{align}\mathscr G_\vartheta(gx,gy)-\mathscr G_\vartheta(x,y)=-(\operatorname{\mathrm{i}}\beta_{\operatorname{\mathrm{i}}\vartheta}(g,x)+\beta_\vartheta(g,y)).\end{align} $$

4.8 Proximality

Recall that $g\in \operatorname {\mathrm {{\mathsf {PGL}}}}_d({\mathsf {V}})$ is proximal if it has a unique eigenvalue with maximal modulus and that the multiplicity of this eigenvalue in the characteristic polynomial of g is $1.$ The associated eigenline is denoted by $g^+\in \mathbb {P}({\mathsf {V}})$ and $g^-$ is its g-invariant complementary subspace.

We say then that $g\in {\mathsf {G}}$ is $\vartheta $ -proximal if for every $\sigma \in \vartheta $ , one has $\unicode{x3c6} _\sigma (g)$ is proximal. In this situation, there exists a pair $(g^-_\vartheta ,g^+_\vartheta )\in \mathcal F^{(2)}_\vartheta ,$ defined by, for every $\sigma \in \vartheta , \Xi _{\unicode{x3c6} _\sigma }(g^+_\vartheta )=\unicode{x3c6} _\sigma (g)^+,$ and every flag $x\in \mathcal F_\vartheta $ in general position with $g^-_\vartheta $ verifies $g^nx\to g^+_\vartheta .$

It is also useful to consider a quantified version of proximality. Given $r,\varepsilon $ positive, we say that g is $(r,\varepsilon )$ -proximal if it is proximal,

$$ \begin{align*}\varpi_\sigma\mathscr G_\vartheta(g_\vartheta^-,g^+_\vartheta)\geq-r\end{align*} $$

for all $\sigma \in \vartheta $ , and for every $x\in \mathcal F_\vartheta $ with $\min _{\sigma _\in \vartheta }\varpi _\sigma \mathscr G_\vartheta (g_\vartheta ^-,x)\geq - \varepsilon ^{-1}$ , one has $d_{\mathcal F_\vartheta }(gx,g^+_\vartheta )\leq \varepsilon .$ More details on the following can be found in [Reference Sambarino65, Lemma 5.6].

Proposition 4.8.1. (Benoist [Reference Benoist6, Corollaire 6.3])

For every $\delta>0$ , there exist $r,\varepsilon>0$ such that if $g\in {\mathsf {G}}$ is $(r,\varepsilon )$ -proximal, then

4.9 Cartan attractors

Consider $g\in {\mathsf {G}}$ and let $g=k_gz_gl_g$ be a Cartan decomposition. We say that $g\in {\mathsf {G}}$ has a gap at $\vartheta $ if for all $\sigma \in \vartheta $ , one has

$$ \begin{align*}\sigma(a(g))>0.\end{align*} $$

In that case, the Cartan attractor of g in $\mathcal F_\vartheta $

$$ \begin{align*}U_\vartheta(g)=k_g[{\mathsf{P}}_{\vartheta}]\end{align*} $$

is well defined: uniquely defined if $\mathbb {K}$ is Archimedian; defined up to a ball of radius $q^{-\min _{\sigma \in \vartheta }\sigma (g)}$ if $\mathbb {K}$ is non-Archimedean (see [Reference Pozzetti, Sambarino and Wienhard59, Remark 2.4]).

Lemma 4.9.2. (Bochi, Potrie, and Sambarino [Reference Bochi, Potrie and Sambarino10, Lemma A.5])

Consider $g,h\in {\mathsf {G}}$ such that h and $gh$ have gaps at every $\sigma \in \vartheta ,$ then one has

The Cartan basin of g is defined, for $\alpha>0,$ by (recall Remark 4.7.1)

$$ \begin{align*}B_{\vartheta,\alpha}(g)= \{x\in\mathcal F_\vartheta: \varpi_\sigma(\mathscr G_\vartheta(U_{\operatorname{\mathrm{i}}\vartheta}(g^{-1})),x)>-\alpha \text{ for all }\sigma\in\vartheta\}.\end{align*} $$

It is clear from the definition that given $\alpha>0$ , there exists a constant $K_\alpha $ such that if $y\in \mathcal F_{\vartheta }$ belongs to $B_{\vartheta ,\alpha }(g)$ , then one has

(23) $$ \begin{align} \|a_\vartheta(g)-\beta_\vartheta(g,y)\|\leq K_\alpha. \end{align} $$

4.10 General facts on discrete subgroups

We record here some facts related to the title that we will need in the following work.

Lemma 4.10.1. Let $\Delta \subset {\mathsf {G}}$ be a discrete subgroup. Then for every $\varphi \in {\mathsf {E}}^*$ strictly positive on ${\mathsf {E}}^+$ , the exponential rate

$$ \begin{align*}\delta^\varphi_\Delta=\limsup_{t\to\infty}\frac1t\log\#\{g\in\Delta:\varphi(a(g))\leq t\}\end{align*} $$

is finite.

We record also the following theorem from Benoist [Reference Benoist8].

5.1 Real projective-Anosov representations

We begin by recalling Labourie’s original approach. Let be a finitely generated word-hyperbolic group. If is a representation, then we can consider the natural flat bundle automorphism defined as follows. Consider the flat bundle defined by , where $(p,v)\sim (\gamma p,\gamma _\rho v),$ and define $\hat {\mathsf {g}}=(\hat {\mathsf {g}}_t:{\mathsf {E}}_\rho \to {\mathsf {E}}_\rho )_{t\in \mathbb {R}}$ as the induction on the quotient by $t\cdot (p,v)=(\widetilde {\mathsf {g}}_t p, v).$

Definition 5.1.1. The representation $\rho $ is projective-Anosov if there exists a pair of continuous $\rho $ -equivariant maps

such that:

• • for every , one has $\ker \xi ^{d-1}(x)\oplus \xi ^1(y)=\mathbb {R}^d.$ This induces a $\hat {\mathsf {g}}$ -invariant decomposition $\Xi \oplus \Theta ={\mathsf {E}}_\rho $ ;

• • the decomposition ${\mathsf {E}}_\rho =\Xi \oplus \Theta $ is a dominated splitting for $\hat {\mathsf {g}},$ that is, there exist $c,\alpha $ positive such that for every $v\in \Xi _p$ and $w\in \Theta _p$ , one has

  $$ \begin{align*}\frac{\|\hat{\mathsf{g}}_tv\|}{\|\hat{\mathsf{g}}_tw\|}\leq ce^{-\alpha t}\frac{\|v\|}{\|w\|}.\end{align*} $$

One has the following standard consequences, see for example [Reference Guichard and Wienhard35, Lemma 3.1] or [Reference Bridgeman, Canary, Labourie and Sambarino17, Lemma 2.5 and Proposition 2.6]. Recall from §4.8 that $g\in \operatorname {\mathrm {{\mathsf {PGL}}}}_d(\mathbb {R})$ is proximal if the Jordan block associated to the eigenvalues with maximal modulus is one-dimensional.

Lemma 5.1.2. If $\rho $ is projective-Anosov, then for every hyperbolic $\gamma $ , one has $\gamma _\rho $ is proximal with attracting line $\xi ^1(\gamma _+).$ In particular, the entropy

The equivariant maps $\xi ^1$ and $\xi ^{d-1}$ are Hölder-continuous.

Proof. Let us add a word on finiteness of entropy. Recall from Bowditch [Reference Bowditch13] that, since

is hyperbolic, its action on the space of pairwise distinct triples

is properly discontinuous and co-compact. If

is hyperbolic, one can choose then $\eta \in [\gamma ]$ (the conjugacy class of $\gamma $ ) whose fixed points are far apart by a constant independent of $\gamma .$ Since the image $\gamma _\rho $ is proximal and the equivariant maps are continuous, one has that $\eta _\rho $ is $(r,\varepsilon )$ -proximal for constants $r,\varepsilon $ independent of $[\gamma ].$ By Proposition 4.8.1, one has then $|\log \|\eta _\rho \|-\unicode{x3bb} _1(\eta _\rho )|<K$ for some K independent of $\eta .$ If follows then that for every $t\in \mathbb {R}_+$ ,

Finiteness of entropy then follows from Lemma 4.10.1.

We use the equivariant maps to construct a bundle whose fiber at is

$$ \begin{align*}\widetilde{\mathsf{ F}}_{(x,y)}=\{(\varphi,v)\in\xi^{d-1}(x)\times\xi^1(y):\varphi(v)=1\}/\sim,\end{align*} $$

where $(\varphi ,v)\sim (-\varphi ,-v).$ This bundle is equipped with a -action $\gamma (\varphi ,v)= (\varphi \circ \gamma ^{-1}_\rho ,\gamma _\rho v)$ and an $\mathbb {R}$ -action $(\widetilde {\mathsf {g^\rho }}_t:\widetilde {\mathsf {F}}\to \widetilde {\mathsf {F}})_{t\in \mathbb {R}}$ defined by $\widetilde {\mathsf {g^\rho }}_t\cdot (\varphi ,v)=(e^t\varphi ,e^{-t}v).$ Let and denote by ${\mathsf {g}}^\rho =({\mathsf {g}}^\rho _t:{\mathsf {F}}\to {\mathsf {F}})_{t\in \mathbb {R}}$ the induced flow on the quotient. It is usually called the geodesic flow of $\rho $ .

Theorem 5.1.3. (Bridgeman et al [Reference Bridgeman, Canary, Labourie and Sambarino17])

The above -action is properly discontinuous and co-compact. The flow ${\mathsf {g}}^\rho $ is Hölder-continuous and metric-Anosov with stable/unstable laminations (induced on the quotient by)

$$ \begin{align*}\tilde W^{\mathrm{s}}((x,y,(\varphi,v)) & =\{(x,\cdot,(\varphi,\cdot))\in\widetilde{\mathsf{F}}\},\\ W^{{\mathrm{u}}}((x,y,(\varphi,v)) & =\{(\cdot,y,(\cdot,v))\in\widetilde{\mathsf{F}}\}.\end{align*} $$

It is moreover Hölder-conjugated to the Gromov–Mineyev geodesic flow ${\mathsf {g}}$ of consequently, this latter flow is also metric-Anosov.

Therefore, hyperbolic groups admitting a real projective-Anosov representation verify Assumption B and are thus subject of a Ledrappier correspondence (§3). It is established by Carvajales [Reference Carvajales22, Appendix] that ${\mathsf {g}}^\rho $ is topologically mixing (regardless of the Zariski closure of $\rho $ ) and thus mixing for any equilibrium state.

5.2 Arbitrary ${\mathsf {G}},$ coarse geometry viewpoint

Let ${\mathsf {G}}$ be as in §4. We use freely the notation introduced there and fix from now on a subset $\vartheta \subset {{{\Delta }}}$ of simple roots.

Let be a finitely generated group and denote, for by $|\gamma |$ the word length with respect to a fixed finite symmetric generating set of

Definition 5.2.1. A representation is $\vartheta $ -Anosov if there exist $c,\mu $ positive such that for all and $\sigma \in \vartheta $ , one has

(24) $$ \begin{align}\sigma(a(\gamma_\rho))\geq \mu|\gamma|-c.\end{align} $$

The constants c and $\mu $ will be referred to as the domination constants of $\rho .$