Decay of correlations for certain isometric extensions of Anosov flows

We establish exponential decay of correlations of all orders for locally $G$-accessible isometric extensions of transitive Anosov flows, under the assumption that the strong stable and strong unstable foliations of the base Anosov flow are jointly $C^1$. This is accomplished by translating accessibility properties of the extension into local non-integrability estimates measured by Dolgopyat's infinitesimal transitivity group, from which we obtain contraction properties for a class of 'twisted' symbolic transfer operators.


Introduction
One of the strongest characteristics of chaotic behaviour in dynamical systems is the exponential decay of correlations, or exponential mixing; besides being of intrinsic interest, this is typically accompanied by other strong statistical properties for regular observables. Naturally, there has been substantial interest in understanding and characterizing the dynamical systems with this property.
Hyperbolicity and the joint structure of the strong stable and strong unstable foliations are among the most well-understood mechanisms driving chaotic behaviour in both discrete-time and continuous-time dynamical systems. For continuous-time systems in particular, the extent to which these foliations fail to be integrable is known to be especially important.
Unfortunately, even among Anosov flows, there is no complete characterization of those which are exponentially mixing. There have been many significant advances, however, with perhaps the most notable due to Dolgopyat, who showed in [Dol98] that the uniform local non-integrability of the strong stable and unstable foliations leads to exponential mixing for smooth Anosov flows, for a large class of equilibrium measures. We will not attempt to give an account of the considerable progress that has been achieved since then, but we refer the reader to [BW16,§1] for an excellent narrative.
Our attention will be restricted to the class of Anosov flows studied in [Dol98], which are known to be exponentially mixing. We will consider compact isometric extensions of these flows, and give criteria for these extensions to be exponentially mixing. We prove the following: Theorem A. Let M, N and F be closed Riemannian manifolds, where π: M → N is a fiber bundle with fibers isometric to F . Suppose that g t : N → N is a smooth, transitive Anosov flow preserving an equilibrium measure ν with a Hölder potential ς: N → R. Moreover, suppose that the strong stable and strong unstable foliations are C 1 , and that ν has unstable conditional measures ν u that are diametrically regular.

Preliminaries
We fix some notation that we will use throughout this paper: N will be a closed Riemannian manifold equipped with a probability measure ν and g t : N → N a C ∞ Anosov flow preserving ν. Let M be a compact Riemannian fiber bundle π: M → N whose fibers π −1 (x) are each isometric to a fixed compact, Riemannian manifold F , and let f t : M → M be an extension of g t satisfying π • f t = g t • π. We equip M with the product measure µ of ν and a probability measure on F .
The motivating example for everything that follows is when N is the unit tangent bundle for a closed n-manifold of quarter-pinched negative curvature, M is the oriented orthonormal frame bundle, g t is the geodesic flow and f t is the frame flow. The natural choice for ν in this case is the Liouville measure on N , and µ is then locally a product of the Liouville measure and the normalized Haar measure on SO(n − 1, R).
which is independent of . Once again, we note that the base of the exponential term is at most 1 and does not depend on ϕ or ξ, completing our proof.
Of course, whether a system is exponentially mixing depends on the measure under consideration. We will be interested in equilibrium measures for Hölder potentials. Definition 1.3. For a continuous function ς: N → R, we call a measure ν an equilibrium state for g t with potential ς if ν maximizes N ς dν + h ν (g 1 ) among all g t -invariant probability measures on N . To emphasize the potential, we will write ν ς for the equilibrium state corresponding to ς when it exists and is unique.
Of particular importance to us is the fact that equilibrium states admit a local product structure with respect to the strong stable and unstable foliations, and that they are invariant under the appropriate transfer operators; we will expand on both of these properties in due course.
When g t is an Anosov flow on a compact manifold, it is a classical result of Bowen and Ruelle [BR75] that equilibrium states for Hölder potentials exist and are unique. Of course, the measure of maximal entropy is always an equilibrium state for the trivial potential ς = 0. In the case of the geodesic flow in negative curvature, the Liouville measure is the equilibrium state for the geometric potential on the unit tangent bundle. We are interested in extensions of Anosov flows that act fiberwise by isometries.
Definition 1.4. We call a smooth flow f t : (M, µ) → (M, µ) on a closed Riemannian manifold M a G-extension of g t : (N, ν) → (N, ν) if π: M → N is a smooth fiber bundle where • the fibers π −1 (x), with the induced metric, are all isometric to a closed Riemannian manifold F , • G is a closed, connected normal subgroup of the isometry group Isom(F ), • G acts transitively on F and has no proper transitive normal subgroups, • π • f t = g t • π, • there is an atlas of trivializations of π: M → N for which all transition functions lie in G, • with respect to these trivializations, the isometries of F induced by the flow f t all lie in G, • f t preserves a measure µ satisfying π * (µ) = ν, and • the fiberwise disintegration of µ along the fibers of π: M → N is the pushforward of the normalized Haar measure on G to each fiber.
The primary driver of exponential mixing in our case will be a stronger variant of local accessibility.
Definition 1.5. Let f t : M → M be a G-extension of g t : N → N . We call f t locally G-accessible if, for every > 0, any trivialization φ: π −1 (B (x)) → B (x) × F defined near x ∈ N and any isometry h ∈ G, there is a sequence of points x 0 , . . . , x k ∈ N for which • we either have x i+1 ∈ W su gt (x i ) or x i+1 ∈ W ss gt (x i ) for each i, and : F → F is given (via φ) by the isometry π −1 (x i ) → π −1 (x i+1 ) induced by leaves of the strong stable or strong unstable foliation of f t .
1.2. Symbolic dynamics. In this section, we will build a discrete, symbolic model for f t -we follow [Win16], and accomplish this by artificially extending a standard Markov partition for the base flow. The results of [Bow73;Rat73] on the existence of Markov partitions for hyperbolic dynamical systems are classical and well-understood; as such, we will recall some of the important points but refrain from delving into the details.
Theorem 1.6 (Bowen, Ratner). If g t is Anosov, then g t has a Markov partition of size for any sufficiently small > 0.
Specifically, Bowen and Ratner construct a Markov partition by taking local strong stable and unstable segments and forming a Markov rectangle.
Definition 1.7. Given x ∈ N and > 0, consider the local strong and weak stable and unstable manifolds of size through x given by where in each case we have taken the connected component through x. For u ∈ W su (x) and s ∈ W ss (x), we define the bracket of u and s to be the point of intersection which is necessarily unique when > 0 is sufficiently small. We define the Markov rectangle [W su (x), W ss (x)] to be the set [W su (x), W ss (x)] := {[u, s] | u ∈ W su (x) and s ∈ W ss (x)} assuming > 0 is sufficiently small.
For now, fix > 0 chosen to be small enough that a Markov partition exists; we will likely need to adjust our choice of as we proceed. We let R = {R 1 , . . . , R k } be a Markov partition of size for g t , where each rectangle R i := [U i , S i ] is generated by local strong stable and unstable manifolds S i := W ss (z i ) and U i := W su (z i ) through points z i ∈ N . Set R := R i , U := U i and S := S i , and let P: R * → R * be the Poincaré return map defined on the appropriate full-measure residual subset R * ⊂ R with return time τ : R * → R.
Remark 1.8. Since we assumed that the strong stable and unstable foliations for g t were C 1 , each R i is naturally an open C 1 submanifold of N . It is important to note that both the Poincaré return map P and the return time map τ defined above are the restrictions of locally C 1 functions on R, but neither P nor τ is even continuous on R. Indeed, the points of discontinuity for P, τ and their iterates are exactly what is removed in passing from R to R * .
A more extensive discussion of this can be found in [Che02,. We will write (Σ, P) for the Markov shift corresponding to the partition R. Definition 1.9. The suspension of (Σ, P) with roof function τ is the flow g t : Σ × R/∼→ Σ × R/∼ defined by g t (x, s) = (x, s + t), where we have declared (x, τ (x)) ∼ (P(x), 0). Theorem 1.6 says exactly that the natural inclusion of R into N induces a Hölder-continuous semi-conjugacy between g t and g t -this is precisely the result we wish to extend to f t . Fix a finite cover {V i } that trivializes the fiber bundle π: M → N , with isometries φ i : π −1 (V i ) → V i × F . At this point, we may need to revisit our choice of : let us assume that we have taken to be smaller than the Lebesgue number of the cover {V i }. By definition, this means that each R j lies entirely in some V k(j) , and hence φ k(j) induces an isometry π −1 (R j ) → R j × F . We can put these together to form an isometry φ: It is worth noting that we have considerable freedom in our choice of isometries φ i , and this is something we will want to exploit to simplify our arguments in §3. Since the center-stable foliation of f t is C 1 , we can modify a given φ i so that it is constant along the (local) leaves of this foliation. Specifically, we will assume that the projection of each φ i : We will realize f t as a suspension flow on Σ × F with roof function τ . To do this, we will need information on the fiberwise action of g t . Definition 1.10. For x ∈ R j1 with P(x) ∈ R j2 , we define the temporal holonomy Hol(x) at x to be the isometry between φ k(j1) (π −1 (x)) and φ k(j2) (π −1 (P(x))) induced by f t . This defines a function Hol: R * → G.
We will often write Hol (n) (x) for Hol(P n−1 (x)) • . . . • Hol(x). Note that function composition is the multiplication operation in G.
Given the temporal holonomy function Hol: R * → G, we can of course recover f t as a suspension flow on (Σ × F, P × Hol) with roof function τ . However, we will push this a step further.
We can define a projection along the leaves of the strong stable foliation proj S : R → U by setting proj S ([u, s]) = u. This allows us to construct a uniformly expanding model U for g t , where the Poincaré return map descends to a map σ: U * → U * given by σ := proj S •P.
Remark 1.11. By construction, the return time map τ and the temporal holonomy function Hol are both constant along the leaves of the strong stable foliation of g t , and hence descend to functions τ : U → R and Hol: U → G.
Set U τ := U * × R/∼ and U τ,Hol := U * × R × F/∼, where we declare (u, τ (u)) ∼ (P(u), 0) and (u, τ (u), k) ∼ (P(u), 0, (Hol(u))(k)) respectively. We write f t for the suspension flow f t (u, s, k) = (u, s + t, k) on U τ,Hol . It is not too difficult to show that the exponential mixing of f t is equivalent to exponential mixing for f t , though we must first fix a measure on U τ,Hol to make sense of this.
Remark 1.12. Since ν is an equilibrium state for a Hölder potential φ, there are measures ν s i and ν u i on each of the strong stable and unstable segments S i and U i so that ν is absolutely continuous with respect to the product ν u i × ν s i × dt. Moreover, these measures can be chosen so that the Radon-Nikodym derivative of ν with respect to the product ν u i × ν s i × dt is uniformly bounded above by a constant K > 1 and below by K −1 < 1. We will write ν u and ν s for the corresponding measures on U and S respectively, and suppose that they have been normalized so that ν U (U ) = ν S (S) = 1.
Remark 1.13. Up to replacing φ with a cohomologous function on (Σ, P), we can assume that φ is the extension of a Hölder potential φ U on U . As a consequence, we may as well assume that ν u is in fact an equilibrium state for a Hölder potential φ U on (U, σ).
These are classical results in thermodynamic formalism -we refer the reader to [Lep00] and [Mar04, respectively for more details. We will in addition require the conditional measure ν u to have a doubling property later on, in order to control the spectrum of our transfer operators using Dolgopyat's methods.
Definition 1.14. We say that a measure ν u has the doubling or Federer property, or is diametrically regular, if for any k > 1 there is a uniform constant C > 0 so that for all x ∈ U and r > 0.
With the following lemma, we are reduced to establishing exponential mixing for the suspension flow f t on the expanding model U τ,Hol . Lemma 1.15. If f t is exponentially mixing of order k for functions in C 1 (U τ,Hol , C), then f t is exponentially mixing of order k for functions in C 1 (M, C).
Proof. Once again, we will perform the argument in the case k = 1. The general case can be obtained by repeating this inductively. Given ϕ, ψ ∈ C 1 (M, C) with M ϕ dµ = M ψ dµ = 0 and a fixed k ∈ F , we consider corresponding functions ϕ t , ψ t ∈ C 1 (U τ,Hol , C) given by after simply integrating both sides with respect to s. This can be rewritten as since s 0 is fixed. We then see quickly that the difference in the integrals is at most C 1 q t ϕ C 1 ψ C 0 . But now, the same argument shows that and 17). From the local product structure of ν, we see that this is within a constant multiplicative factor of K of the integral for any ϕ and ψ. To conclude, we simply observe that if M ϕ dµ = M ψ dµ = 0, then we must have for any t ∈ R. Moreover, the regularity of ϕ t and ψ t is determined by the regularity of the bracket operation [·, ·] -since we assumed that the foliations were C 1 , we see that ϕ t and ψ t must also be C 1 . Hence, if f t is exponentially mixing for functions in C 1 (U τ,Hol , C), (1.16) must decay exponentially in t, from which we conclude that (1.18) must also decay exponentially.
1.3. Representation theory. In this section, we recall some classical results from the representation theory and harmonic analysis of compact Lie groups; our primary references are [Sug71] and [App14]. Following [Win16, §3.6], our strategy is to decompose functions on U τ,Hol into components corresponding to irreducible representations of G. A function ϕ ∈ C 1 (U τ,Hol , C) can be viewed as a C 1 functionφ: U τ → L 2 (G) by settingφ(u, r) := ϕ(u, ·, r).
Since G is a compact, connected Lie group, we can decomposeφ(u, r) ∈ L 2 (G) into isotypic components corresponding to irreducible representations of G -this is, of course, the classical Peter-Weyl theorem.
Theorem 1.19 (Peter-Weyl). If G is a compact, connected Lie group, then there is a decomposition where the sum is taken over pairwise non-isomorphic irreducible representations of G, and the (V ρ )⊕ dim ρ associated to non-isomorphic irreducible representations are pairwise orthogonal with respect to the standard inner product on L 2 (G).
We fix such a decomposition and writeφ(u, r) = ρφ ρ (u, r) for the decomposition ofφ(u, r) obtained by projecting onto each (V ρ )⊕ dim ρ. Abusing notation, we will use ϕ to refer interchangeably to a function U τ → L 2 (G) or to the function U τ,Hol → C -there should be little ambiguity in either case.
For our later analysis, it will be helpful to consider the derived representation dρ of the Lie algebra g of G acting on L 2 (G), induced by the representation ρ of G on L 2 (G) -see [App14, §2.5.1] for details. We will always assume that we have a fixed Ad-invariant norm · g on g.
Definition 1.20. Given an irreducible representation ρ: G → GL(V ρ ) of G (where we view V ρ ⊂ L 2 (G)), we define the norm ρ of ρ to be the supremum where dρ(X) L 2 (G) is the operator norm of dρ(X) viewed as an automorphism of L 2 (G).
It is a classical fact that ρ is finite, and can be bounded in terms of the highest weight associated to ρ.
Proposition 1.21. Let ρ be a nontrivial irreducible representation, and let λ be its highest weight. There are uniform constants C > 0 and m > 0 so that ρ ≤ Cλ m for all X ∈ g with X g = 1.
Proof. See [App14, Theorem 3.4.1]; note that the Hilbert-Schmidt norm is an upper bound for the operator norm.
We will also require some particular results on the growth rate of ρ .
Proposition 1.22. There is a constant N > 0 so that ρ ρ −n converges for any n ≥ N .
Proof. This is a combination of [Sug71, Lemma 1.3] and Proposition 1.21.
And more generally, we can obtain decay estimates for the Fourier coefficients ϕ ρ associated to irreducible representations ρ.
Proof. This is contained in the proof of [Sug71, Theorem 1].

Twisted transfer operators
In this section, we will define Dolgopyat's 'twisted' transfer operators, and show how the spectral bounds we intend to obtain for these operators lead to correlation decay estimates for the expanding suspension semi-flow f t constructed in the previous section.
Recall that we can view a smooth function ψ ∈ C 1 (U τ,Hol , C) as a function ψ ∈ C 1 (U τ , L 2 (G)). We can integrate out the time variable to getψ (u) := τ (u) 0 ψ(u, r) dr in C 1 (U, L 2 (G)); this is the space on which we would like to define our operators. The advantages of working with smooth (as opposed to Hölder) functions will become clear in §4, but we will need to assume that ς is C 1 for most of our arguments. We will explain how to modify our proof to deal with the general case where ς is Hölder in Corollary 4.36, using a standard approximation argument.
Let ρ be an irreducible representation of G acting on an isotypic component V ρ ⊂ L 2 (G), and fix z ∈ C. We define the transfer operatorL z,ρ : where ς z is the potential on the one-sided Markov model (Σ + , σ) obtained as the restriction of the potential defined on (Σ, P). Note that ς z is well-defined in light of Remark 1.13, where we assumed that both α and τ are constant in s. It is also worth remarking that both τ and p are C 1 , since we assumed that the strong stable and unstable foliations of g t were C 1 . As a consequence, ς z and henceL z,ρ ϕ are both C 1 , since we have restricted ourselves to the case where ς is smooth.
Let us recall some classical results of thermodynamic formalism; for a slightly more detailed treatment, we refer the reader to [Mar04,. Let P (ς) be the topological pressure of ς for the map g 1 . By the Ruelle-Perron-Frobenius theorem, the operatorL P (ς),0 associated to the trivial representation ρ = 0 has a unique positive eigenvector ϕ ς ∈ C 1 (U, R) with eigenvalue e P (ς) .
Recall that ν u is an equilibrium measure for the restriction of the potential ς to (Σ + , σ), which by construction has the same topological pressure P (ς). By the Lanford-Ruelle variational principle, this means that e P (ς) U ϕ dν u = UL P (ς),0 ϕ dν u for all ϕ ∈ C 1 (U, V ρ ). It will be convenient to renormalizeL P (ς),0 ϕ so that it preserves the measure ν u : let L z,ρ be the operator defined by Remark 2.1. With the renormalizations above, we have We can alternatively write for all u ∈ U -note that the positivity of ϕ ς is required to ensure that log(ϕ ς (u)) is well-defined. It will be helpful to have a similar formulation for the iterates for all u ∈ U . Here, we write for the n th ergodic product along σ.
Our overarching goal is to establish spectral bounds for these operators; in §4, we will ultimately prove: Theorem 2.2. There are constants C > 0 and r < 1 so that The remainder of this section will be devoted to showing how we obtain Theorem A from Theorem 2.2.
In order to show that β k decays exponentially in t 1 , . . . , t k , we will show that the integral defining its Laplace transformβ(ξ 1 , . . . , ξ k ) converges absolutely for some fixed values of ξ 1 , . . . , ξ k . The following lemma expresses the Laplace transform in terms to the transfer operators we have just defined, the proof of which consists almost entirely of elementary integral manipulations.
We will be somewhat cavalier in interchanging sums and integrals, though this is eventually justified as the final expression we obtain is absolutely convergent.
Lemma 2.3. Given ϕ 0 , . . . , ϕ k ∈ C 1 (U τ,Hol , C) as above with U τ,Hol ϕ i dν u dω dr = 0, we can bound the Laplace transform of the k th -order correlation by Here, we useφ i,ξi to denote the function Proof. By definition, the Laplace transformβ k (ξ 1 , . . . , ξ k ) of β k (t 1 , . . . , t k ) is given by for ξ 1 , . . . , ξ k ∈ C. Since the systems of inequalities are obviously equivalent, we can rewrite (2.4) as by reparametrizing the domain of integration. Let us focus on the k innermost integrals for now, where we can break up by changing variables, replacing t i with t i + τ (ni) (u). Now, note that we can rewrite the integrand using the identifications we made in constructing U τ,Hol . If we integrate (2.5) with respect to t 1 through t k , it becomes and so (2.4) becomes ∞ n1,...,n k =1 U Gφ after interchanging the order of integration and summation.
Since ω is bi-invariant, we can replace h with Hol (max nj ) (u) • h. Of course, we have the identity by simply reversing the order of integration. Let us focus on the innermost integral for the time being. Applying L P (ς),0 to the integrand in (2.8) a total of max n j times yields for any given values of n 1 , . . . , n k ∈ N. Now, observe that e − (ξi)τ (n i ) (u) is at most e − (ξi)τ (max n j ) (u) so long as each (ξ i ) is negative. Hence, we can bound the magnitude of the integrand in (2.8) above by using the triangle inequality. Rearranging this expression, we see that the magnitude of the integrand in (2.8) is bounded above by which should be reminiscent of the expression defining the transfer operator. To make this concrete, recall that we have an L 2 (G)-invariant decomposition of the functionφ 0,−ξ ∈ C 1 (U, L 2 (G)) in terms of its isotypic componentsφ where the sum is taken over irreducible representations ρ of G -including the trivial representation. Once again, by the triangle inequality, we can bound (2.9) above by where we quickly recognize the transfer operator L max nj ρ,P (ς)+ (ξ) applied to φ ρ 0,−ξ , noting of course that for each ρ. Putting this back together, we see that (2.4) is bounded above by We now simply use the bounds in Theorem 2.2 to conclude that the Laplace transformβ in Lemma 2.3 converges.
Theorem 2.10. With conditions as above, there are uniform constants C > 0 and r < 1 so that Proof. We assume that Theorem 2.2 holds, and so we have for all non-trivial irreducible ρ, all ϕ ∈ C 1 (U, V ρ ) and for each z ∈ C with | (z) − P (ς)|< 1. Up to choosing larger values of C and r, we can also assume that the same inequality holds when ρ is trivial -this is precisely the main result in [Dol98]. We will show that the expression boundingβ(ξ 1 , . . . , ξ k ) in Lemma 2.3 converges whenever the real parts of ξ i simultaneously lie in the interval − 1 k < (ξ i ) < 0. The decay desired will follow immediately from applying the inverse Laplace transform to the specific bounds we obtain.
Fix ξ 1 , . . . , ξ k with − 1 k < (ξ i ) < 0. As before, we consider the function . . + ξ k as before, noting that the decomposition of L 2 (G) into irreducible subspaces commutes with the Laplace transform. By (2.11), whenever − 1 Combining this with Lemma 2.3, we are reduced to ensuring that converges. While this seems promising, note that we can only bound for some constant D > 0. To salvage this, we will assume for the moment that we have chosen ϕ 0 ∈ C 3 (U τ,Hol , R) so that we can invoke Theorem 1.23 to bound pointwise with a fixed constant D > 0, where N > 0 is the constant guaranteed by Proposition 1.22. With this, it is clear that the expression on the right side of (2.12) converges absolutely, which says immediately that β k must decay exponentially fast in t 1 , . . . , t k ; this is almost what we wanted to show, but we need an explicit bound onβ k in order to obtain uniform estimates with the desired constants.
Integrating the defining expression forφ i,ξi (u, h) by parts, we havê for each u and h. Assuming that we choose ϕ i ∈ C 2 (U τ,Hol , R), we can crudely bound for some constant D that depends only on τ C 1 -note that it is essential here that (ξ i ) is confined to a bounded interval. We can similarly bound by choosing a larger value for D if necessary. Putting this all together, we have . Now, we simply take the inverse Laplace transform in the variables ξ 1 through ξ k in succession to get where s i is to be interpreted as the imaginary part of ξ i . To complete our decay estimate, we simply need to show that the integral converges. Unfortunately, this is somewhat involved, so we postpone our remarks on the proof for the moment.
Once the convergence of this integral has been established, we will have shown that for all ϕ 0 ∈ C N +2 (U τ,Hol , R) and ϕ i ∈ C 2 (U τ,Hol , R). An identical argument to the one given in Lemma 1.2 extends this to C 1 functions, using [GS14, Lemma 2.4] once again.
We now indicate how to establish the convergence of the integral encountered in the preceding proof; we have included this for the sake of completeness, but the details are not relevant to the rest of our argument and can be safely skipped.
Proof. We will in fact show that the function defined by the integral decays at a rate of for all sufficiently small > 0; the statement of the lemma follows immediately by direct successive integration. We will work in the case when x > 0, and split the domain of integration in (2.14) into regions where x + y, y < 0, where x + y > 0 but y < 0 and where x + y, y > 0. In the first case, where x + y and y are both negative, we evaluate for a fixed x > 0. One can verify that the antiderivative of this expression is given by where 2 F 1 (·, ·; ·; ·) is (the principal branch of the analytic continuation of) the Gaussian hypergeometric function. Hence, the integral evaluates to which can be bounded above by C (1+x) 0.5− for an appropriate choice of constant C > 0, since by [AS64, 15.1.1] the defining series for 2 F 1 (a, b; c; z) converges on the unit disk in the complex plane when c − (a + b) > 0. Similarly, on the region where x + y is positive and y is negative, we evaluate which can once again be bounded above by C (1+x) 0.5− for the same reasons, though with a possibly larger choice of C > 0. Finally, when both x + y and y are positive, we evaluate at which point we note that the expression in parentheses can be bounded above by C(1 + log x). Hence, (2.15) can be bounded above by C (1+x) 0.5−2 with a possibly larger choice of C, as desired. The proof in the case when x < 0 is identical.

Uniform local non-integrability from local G-accessibility
In this section, we will use the local accessibility of f t to establish the uniform local non-integrability estimates necessary to prove Theorem 2.2, drawing on techniques introduced by Dolgopyat in [Dol02] for group extensions of expanding maps. These arguments require some additional care to adapt to our setting, with the principal difficulties stemming from the nontriviality of the fiber bundle π: M → N and the non-integrability of the strong stable and unstable foliations of g t .
We want to translate the local accessibility of f t into an infinitesimal statement on the Markov model we constructed in §2; we will accomplish this in two main steps. The first step is to define a subalgebra of the Lie algebra g of G that measures the 'non-integrability' of the fiber bundle over the weak stable and strong unstable foliations; this will be accomplished before making any reference to our symbolic model. The second step is to translate this into the symbolic model.
For most of what follows, we will need to be careful to specify which chart V * of the trivialization we are working with at any given point. This is a necessary complication to many of our arguments, since many of the objects we are working with are highly sensitive to the choice of trivialization. Fortunately, however, this will also afford us the flexibility later on to work with trivializations that are specially adapted to our needs.
To start, we want to measure and relate three different holonomies associated to f t : namely, the holonomies induced by the leaves of the strong stable foliation, the leaves of the strong unstable foliation and the flow.
Definition 3.1. Fix x, y ∈ N with y ∈ W su gt (x), along with trivializations φ x , φ y of π: M → N at x and y corresponding to subsets V x , V y ⊂ N respectively. We define the unstable holonomy Θ + Vx,Vy (x, y): F → F between x and y to be the isometry induced by the map π −1 (x) → π −1 (y) that takes a ∈ π −1 (x) to the (necessarily unique) point b ∈ π −1 (y) ∩ W su ft (a). The identifications of π −1 (x) and π −1 (y) with F are obtained via the trivializations φ x , φ y . The stable holonomy Θ − Vx,Vy (x, y) is defined analogously for y ∈ W ss gt (x).
Definition 3.2. Fix x, y ∈ N with g t (x) = y, along with trivializations φ x , φ y of π: M → N at x and y corresponding to subsets V x , V y ⊂ N respectively. We define the temporal holonomy Hol φy φx (x, y): F → F between x and y to be the isometry induced by the map π −1 (x) → π −1 (y) that takes a ∈ π −1 (x) to f t (a) ∈ π −1 (y). The identifications of π −1 (x) and π −1 (y) with F are obtained via the trivializations φ x , φ y .
By the end of this section, we will only need to work with a fixed, finite collection of trivializations that cover N . At this stage, however, the flexibility in these definitions will be crucial. Our first observation is that the unstable holonomy can be expressed in terms of the temporal holonomies induced by the flow; this is made precise in the following proposition, whose proof is largely summarized in Figure 1.
Proposition 3.3. Fix x, y ∈ N with y ∈ W su gt (x), along with trivializations φ 0,x and φ 0,y defined at x and y respectively. Let T = (t n ) be a monotonic sequence of times with t n = 0 and t n → −∞, and let I x = (φ n,x ) and I y = (φ n,y ) be sequences of trivializations for which φ k,x = φ k,y for all k ≥ N . Then we can write Proof. The convergence of the limit is simply a consequence of the fact that d(g tn (x), g tn (y)) → 0 as n → ∞. More precisely, we can rewrite Iy,T (y) Hol   and since f t is C 1 , we see that Hol φn,x φn+1,x (g tn+1 (x), g tn (x)) must also be locally C 1 in g tn+1 (x). Since d N (g tn+1 (y), g tn+1 (x)) decay exponentially fast as n → ∞ and the trivializations I x and I y eventually agree, we see that d G Hol φn,y φn+1,y (g tn+1 (y), g tn (y)), Hol φn,x φn+1,x (g tn+1 (x), g tn (x)) must also decay exponentially fast. In particular, for any h ∈ G, this means that d G Hol φn,y φn+1,y (g tn+1 (y), g tn (y)) • h • Hol Hol φ1,y φ2,y (g t2 (y), g t1 (y)) Hol φ0,y φ1,y (g t1 (y), y) Figure 1. Measuring the unstable holonomy between x and y along a sequence of times 0 > t 1 > t 2 > . . . with respect to trivializations (φ n,x ) and (φ n,y ) defined over charts V n , illustrated in the case when the trivializations for x and y coincide. As the unstable leaf through x and y contracts under g tn , the remaining contribution to the unstable holonomy decreases.
decays exponentially fast and hence d G Hol   decays exponentially fast as n → ∞. Since G is complete, the limit must exist. A similar argument shows that this limit is in fact equal to Θ + φ0,x,φ0,y (x, y): since the unstable foliation of f t is invariant under the flow, we can rewrite Θ + φ0,x,φ0,y (x, y) as Hol (n) Iy,T (y) • Θ + φn,x,φn,y (g tn (x), g tn (y)) • Hol for any n > 0 and any sequences I x , I y and T as above. As t n → −∞, d N (g tn (x), g tn (y)) decreases exponentially fast, and so Θ + φn,x,φn,y (g tn (x), g tn (y)) converges to the identity in G. Of course, this means that, as n → ∞, (3.5) converges to the limit in (3.4). Since the expression in (3.5) is constant at Θ + φ0,x,φ0,y (x, y), this proves the proposition.
We need to understand the infinitesimal behaviour of the stable and unstable foliations -rather than working with the unstable holonomy as defined, we will instead consider its derivative along a leaf of the unstable foliation.
Proposition 3.6. The unstable holonomy Θ + φ1,φ2 (x, y) is simultaneously C 1 in x and y, as x and y vary in a fixed leaf of the strong unstable foliation of g t , and within charts associated to fixed C 1 trivializations φ 1 and φ 2 .
Proof. This follows immediately from the fact that the leaves of the strong unstable foliation of f t are C 1 .
Definition 3.7. Fix x ∈ N , a trivialization φ defined near x and a vector w ∈ T 1 x W su gt (x). We define the infinitesimal holonomy at x in the direction of w to be the element of the Lie algebra g of G. Let > 0 be small enough that φ is defined over B (x). The -infinitesimal transitivity group at x is defined to be the linear span taken over all y ∈ W ss (x) and w ∈ T 1 x W su gt (x). Here, w denotes the pushforward of w to T 1 y W su gt (y) along the leaves of the center stable foliation of g t .
We will soon verify that h φ (x) is largely independent of the choice of trivialization φ, but it is worth making a few comments first.
Remark 3.8. Under our hypotheses, the foliation W ws gt is C 1 , and so the holonomy it induces between the leaves of the foliaton W su gt is also C 1 . This is necessary for the pushforward of w ∈ T 1 y W su gt (y) in Definition 3.7 to make sense.
Remark 3.9. It is necessary to consider the relative infinitesimal holonomy, as we did in Definition 3.7. As we will see in the course of proving Proposition 3.11, the vector X φ w (x) in Definition 3.7 is extremely sensitive to the choice of trivialization φ. For instance, it is certainly possible for X φ w (x) to be 0 for all w ∈ T 1 x W su gt (x) if the trivialization φ is built to be constant along the leaves of the strong unstable foliation, and the existence of such trivializations will be extremely helpful in the course of proving Theorem 3.19.
Remark 3.10. The vectors X φ w (x) and X φ w (y) vary continuously in x and w, by Proposition 3.6. However, since h (x) is defined as the linear span of continuously varying vectors, it is only lower semi-continuous. In particular, there can be singular sets where the dimension of h (x) jumps down.
It turns out that h φ (x) is not particularly sensitive to , though we will not prove this directly. We will show instead that, if f t is locally G-accessible, then h (x) is generically equal to g. For most of what follows, we will treat > 0 as a fixed constant with no particular restrictions. Our first important calculation is that the conjugacy class of h φ (x) does not depend on the trivialization φ, if the trivializations are chosen appropriately.
Proposition 3.11. Fix > 0, x ∈ N and trivializations φ i : (x) h φ1 (x) so long as φ 1 and φ 2 have constant projection to F along each leaf of the strong stable foliation of f t and each flowline of f t .
Here, id φ2 φ1 (x): F → F is used to denote the isometry induced by the identity map π −1 (x) → π −1 (x) with the domain and target identified with F via φ 1 and φ 2 respectively. Proof. We can relate the unstable holonomies between x and u ∈ W su gt (x) with respect to φ 1 and φ 2 by by definition. We now simply take the derivative of each side of (3.12) with respect to u at u = x; in the notation of Definition 3.7, this becomes for any w ∈ T 1 x W su gt (x), where dR denotes the derivative of right multiplication in G. Given any y ∈ W ss (x) and w corresponding to w as in Definition 3.7, exactly the same calculation yields assuming, of course, that y is sufficiently close to x that we are able to use the same trivializations φ 1 , φ 2 . Since both trivializations are constant along the strong stable foliation of f t and we chose y ∈ W ss (x), we clearly have id φ2 φ1 (x) = id φ2 φ1 (y) and hence (dR) as functions T 1 G → T 1 G. Moreover, since the trivializations are also constant along the flowlines of f t , we see that id φ2 φ1 must be constant along the leaves of the center stable foliation of g t . Hence, we must have for all w ∈ T 1 x W su gt (x). Subtracting (3.13) from (3.14) and using the fact that id φ2 φ1 (x) = id φ2 φ1 (y), we then get There is an analogous relation between the -infinitesimal transitivity groups at points along a flowline of g t , though the expansion of the unstable leaves prevent us from obtain an equality in this case.
Proposition 3.15. Fix > 0, x ∈ N and t > 0. Let φ x and φ gt(x) be trivializations near x and g t (x) for which B (x) ⊂ V x and B (g t (x)) ⊂ V gt(x) , and write h for the temporal holonomy h(x) := Hol measured with respect to φ x and φ gt (x). We then have so long as φ x and φ gt(x) have constant projection to F along each leaf of the strong stable foliation of f t and each flowline of f t .
Proof. By Proposition 3.3, we have so long as u is sufficiently close to x. Noting the resemblance to (3.12), simply repeating our calculations in Proposition 3.11 yields for all y ∈ W ss (x) and all w ∈ T 1 x W su gt (x). This completes the proof; note that we do not obtain equality this time since the strong stable leaves for g t contract, and there will be y ∈ W ss (g t (x)) that are not of the form g t (y) for y ∈ W ss (x).
Note that Proposition 3.15 only yields an inclusion of the -infinitesimal transitivty groups, and only in forward time. Our goal is to show that h φ (x) is exactly g at every x ∈ N ; unfortunately, the proof of Proposition 3.15 suggests that even the dimension of h (x) may fail to be constant in general. Fortunately, given the topological transitivity of g t , what we have proven so far is enough to show that the dimension is constant on a large set.
In light of Proposition 3.11, we can be somewhat cavalier in specifying the trivialization φ used in defining h φ (x), if we are solely concerned with the dimension and restrict our attention to trivializations that satisfy the hypotheses of the proposition. We will henceforth always assume that every trivialization we work with has constant projection to F along the strong stable leaves and flowlines of f t . Proof. Let x ∈ N be a point at which h * (x) has maximal dimension. Since h * (·) is lower semi-continuous, it has maximal dimension on an open neighborhood W containing x. By Proposition 3.15, h * (·) therefore has maximal dimension on an open set containing the forward orbit of g t . This is a dense set if g t is topologically transitive.
Since g t is ergodic and dim h * (·) is measurable, it must be constant almost everywhere. The measure ν is an equilibrium measure with a Hölder potential and therefore has full support; hence, the open and dense set on which dim h * (·) has maximal dimension must also have full measure.
Our next objective is to relate the -infinitesimal transitivity groups between points along a leaf of the strong unstable foliation of g t . If we indeed had equality in Proposition 3.15, this would be a relatively straightforward application of Proposition 3.3. The lack of equality makes such an approach impossible, but we can still argue as in Corollary 3.16.
Lemma 3.17. Fix > 0, x ∈ N and y ∈ W su gt (x), along with trivializations φ x and φ y for which we have B 2 (x) ⊂ V x and B (y) ⊂ V y . If x is backwards-recurrent under g t and dim h * (x) is maximal, then h φy (y) = Ad Θ + φx ,φy (x,y) h φx (x) and, in particular, dim h * (·) is constant on W su gt (x).
Proof. Since dim h * (·) is lower semi-continuous, there is an open set W ⊂ B (x) on which it is maximal. Because x is backwards recurrent, we can find a monotonic sequence of times T = {t n } with t n → −∞ for which g tn (x) ∈ W for each n > 0. Moreover, we can suppose that t 1 is large enough that we also have g tn (y) ∈ W for each n > 0. Now, write h (n) (y) := Hol φx φx (g tn (y), g tn−1 (y)) • . . . • Hol φy φx (g t1 (y), y) for the n-step holonomies at g tn (x) and g tn (y). Since we have g tn (x), g tn (y) ∈ W by construction, we have and Ad h (n) (y) h φx (g tn (y)) = h φy (y) by Proposition 3.15 -note that we have implicitly used the fact that B 2 (x) ⊂ V x in writing h φx (g tn (x)) and h φx (g tn (y)), where V x is the chart over which φ x is defined. By rearranging these equations, we then have where distances are measured in the standard metric on the Grassmannian of (dim h * (x))-dimensional subspaces of g. Since h φx (·) is lower semi-continuous and has maximal dimension on W , it must be continuous on W . Up to passage to the interior of a compact subset of W , we can assume that h (·) is uniformly continuous on W . Since d N (g tn (y), g tn (x)) → 0 as n → ∞, we must then have at which point we simply observe that Ad g (·) is continuous in g and that h (n) (y) h (n) (x) −1 converges to Θ + φx,φy (x, y) by Proposition 3.3.
In addition to the preceding lemma, we will require its analogue for the stable holonomies. The proof is identical, and we will not repeat it.
x 0 x 1 x 2 x 3 x 5 Figure 2. A refined stable-unstable sequence x 0 , x 1 , . . . , x k , x k+1 = x 0 . Any stable-unstable sequence can be refined so that for each 0 ≤ n ≤ k, there is a trivialization φ xn over a chart V xn containing both x n and x n+1 . This refinement has the same total holonomy.
Lemma 3.18. Fix > 0, x ∈ N and y ∈ W ss gt (x), along with trivializations φ x and φ y for which we have B 2 (x) ⊂ V x and B (y) ⊂ V y . If x is forwards-recurrent under g t and dim h * (x) is maximal, then h φy (y) = Ad Θ − φx,φy (x,y) h φx (x) and, in particular, dim h * (·) is constant on W ss gt (x). Now that we have Lemma 3.17 and Lemma 3.18 to connect the -infinitesimal transitivity groups to the unstable and stable holonomies respectively, we can achieve the first major goal of this section: translating the local accessibility of f t into a statement about h * (·). To begin, we will show that if f t is G-accessible, then h φ (x) must be Ad G -invariant for any bi-recurrent x ∈ N .
Theorem 3.19. Fix > 0, x ∈ N and a trivialization φ x for which B (x) ⊂ V x . Suppose that 2 is smaller than the Lebesgue number of a finite cover Proof. Suppose that x is bi-reccurent. Fix an isometry g ∈ G and consider a stable-unstable sequence x 0 , x 1 , . . . , x k , x k+1 = x 0 in N with x 0 = x and whose total holonomy is g, where x i+1 is either on the strong stable or strong unstable leaf through x i for g t . For the total holonomy to be g, we want where we can freely assume that each consecutive pair x i , x i+1 has a common trivialization φ xi for which B (x i ), B (x i+1 ) ⊂ V xi -this is true up to refining the sequence. Suppose, moreover, that we have chosen We would like to now invoke Proposition 3.11, Lemma 3.17 and Lemma 3.18 to show that h φx (x) is Ad g -invariant for the g corresponding to the total holonomy along this sequence. The x i we have chosen, however, may fail to be forwards-or backwards-recurrent as necessary. However, note that id (·, ·) and Θ − φx i ,φx i (·, ·) are all locally continuous in all of their arguments. Since bi-recurrent points are dense in N , given any δ > 0, we can find a sequence of bi-recurrent points x 0 , x 1 , . . . , x k , x k+1 = x 0 near Since each x i is bi-recurrent, successive applications of Proposition 3.11, Lemma 3.17 and Lemma 3.18 show that we have for g arbitrarily close to G. Since Ad g is continuous in g , we then obtain It is worth remarking that, under our standing assumption that trivializations must have constant projection to F along strong stable leaves of f t , the stable holonomies Θ − φx i ,φx i (x i , x i+1 ) that appeared in the preceding proof must have all be trivial.
We want to show that h * (·) is typically equal to the full Lie algebra g, though it seems unlikely that this should be true if we merely assume that f t is globally G-accessible. With Theorem 3.19, however, we can show that having h φ (x) = g at any point is (typically) equivalent to f t being locally G-accessible at that point.
Theorem 3.20. Fix > 0, x ∈ N and a trivialization φ x defined over V x ⊂ N for which we have B 3 (x) ⊂ V x . Moreover, suppose that h φx (·) is continuous and has maximal dimension on B 3 (x), and that the forwards orbit of x under g t is dense in B 2 (x). If f t is locally G-accessible at x, then h φx (x) = g.
Proof. Without loss of generality, suppose that we have chosen φ x so that Θ + φx,φx (x, u) is trivial for all u ∈ W su gt (x) ∩ B 2 (x) • , and so that Θ − φx,φx (s 1 , s 2 ) is trivial whenever s 1 and s 2 lie on the same (local) leaf of the strong stable foliation of g t in B 2 (x).
We will write h := h φx (x) and suppose for the sake of contradiction that h g is a proper subalgebra. By Proposition 3.17 and our choice of φ x , we must have h φx (u) = h for all u ∈ B (x) lying on the local leaf through x of the strong unstable foliation of g t . Hence, for any u 1 ∈ B (x), each vector X φx w (u 1 ) used in the definition of h φx (u) must lie in the Lie algebra h. Integrating this, we see that the unstable holonomies Θ + φx,φx (u 1 , u 2 ) are constrained to exp(h) for all u 1 , u 2 ∈ B (x) that lie on the same local leaf of the strong unstable foliation of g t . Now, consider H := exp(h) and consider an element g ∈ G \ H that lies in the complement. By our construction of φ x , all unstable holonomies are constrained to H and all stable holonomies are trivial -hence, no local sequence of stable and unstable holonomies along a sequence of points x, x 1 , x 2 , . . . x k , x lying in B (x) can result in a total holonomy of g. Moreover, since h is an ideal by Theorem 3.19, H is a normal subgroup of G; hence, we cannot obtain a total holonomy of g for any choice of trivialization. Since f t is locally G-accessible at x, this is a contradiction.
Theorem 3.20 gives us the infinitesimal analogue of local accessibility that we sought, and all that we need to do now is verify that this translates properly into the symbolic model we are working with. In principle, the difficulty is that unstable leaves for the discrete dynamical system (R, P) are typically not unstable leaves for g t -fortunately, our choice of trivializations will circumvent almost all of these problems.
Recall that (R, P) is a Markov partition associated to g t , which descends to a C 1 expanding model (U, σ) by projecting along leaves of the strong stable foliation of g t . We want to define unstable holonomies entirely within the symbolic model; a natural candidate for a definition comes from Proposition 3.3. For everything that follows, we will assume that each R i ⊂ N has been assigned a fixed trivialization φ i defined on a neighborhood B (R i ), with the property that φ i has constant projection to F along each leaf of the strong stable foliation of f t and each flowline of f t . By choosing and fixing trivializations at each point in R, we will no longer need to specify which trivialization we are using, at least when dealing with the symbolic model. It is straightforward to verify that this limit exists. Moreover, our choice of trivializations ensures that the symbolic unstable holonomies agree with the appropriate (non-symbolic) unstable holonomies.
Proposition 3.22. Fix x ∈ R and y ∈ W su P (x). There is a (possibly negative) t so that g t (y) ∈ W su gt (x) for which we have Θ + symb (x, y) = Θ + φx,φy (x, g t (y)) assuming x and y are sufficiently close, where φ x , φ y are trivializations corresponding to the respective parts of the Markov partition.
Proof. The existence of such a t satisfying |t|< τ (y), τ P −1 (y) follows immediately from the construction of the Markov partition (R, P). We then clearly have Θ + symb (x, y) = Hol φy φy (g t (y), y) • Θ + φx,φy (x, g t (y)) by Proposition 3.3 and Definition 3.21. By our choice of trivialization and the fact that t does not exceed the return time of y, Hol φy φy (g t (y), y) is the identity in G, as desired.
And now, we can analogously define the symbolic infinitesimal transitivity group: Definition 3.23. Fix x ∈ U i and a vector w ∈ T 1 x U i . We define the symbolic infinitesimal holonomy at x in the direction of w to be the element of the Lie algebra g of G. The symbolic infinitesimal transitivity group at x is defined to be th linear span where we take the span over s, s ∈ S i , w ∈ T 1 x U i and let w be the projection of w to [x, s ] via center-stable leaves followed by the flow.
With very little work, we can now prove Theorem 3.24. Fix a bi-recurrent x ∈ U i at which dim h (x) is maximal, and suppose that f t is G-accessible. Then h symb (x) = g.
Proof. By Proposition 3.22, the unstable holonomies used in Definition 3.7 and Definition 3.23 are the same. Hence, the regular and symbolic transitivity groups agree, and so by Theorem 3.19, we have h symb (x) = g. This is almost the result we want, but we need to use some Lie theory to extract the explicit estimates that we will use in the next section. We will want to phrase this in terms of σ, which means minor notational changes in the preceding theorems. Recall that σ: U → U is not actually invertible; to make sense of the inverse, we must choose branches of σ −n locally.
Remark 3.26. A consistent past for u ∈ U i corresponds exactly to a choice of stable element s ∈ S i -we can recover the maps v (n) by projecting the Poincaré return map P (−n) along leaves of the strong stable foliation.
Finally, we can establish the main estimate.
We begin by recalling the definition of the twisted transfer operator L n z,ρ ϕ (u) := σ n (u )=u e α (n) z (u ) ρ Hol (n) (u ) ϕ(u ) associated to an irreducible representation ρ of G on V ρ ⊂ L 2 (G). In principle, the difficulty in obtaining contraction for L n z,ρ ϕ L 2 (ν u ) as n → ∞ lies in the possibility that the rotation introduced by the action of ρ may cause resonances between the vectors ϕ(u ) ∈ V ρ , and this may happen on a set of large measure.
The local non-integrability estimate provided by Theorem 3.27, however, suggests that we should typically be able to find u 1 , u 2 with σ n (u 1 ) = σ n (u 2 ) ∈ U lni so that ρ Hol (n) (u 1 ) ϕ(u 1 ) and ρ Hol (n) (u 2 ) ϕ(u 2 ) are 'uniformly' non-parallel. The main argument in the section boils down to verifying that this can be accomplished on an adequately large set, with explicit uniformity estimates. Throughout this section, we will work with a fixed isotypic component V ρ of the regular representation of G on L 2 (G). However, it is worth noting that most of the intermediate constants will fundamentally depend on ρ, and keeping track of these dependencies is essential to obtaining a final bound in Theorem 4.35 that is independent of ρ.
Though we need to deal with L n z,ρ ϕ for any ϕ ∈ C 1 (U, V ρ ), it will be helpful to work instead with slightly more regular real-valued functions Φ ∈ C 1 (U, R + ) with bounded logarithmic derivative; in other words,we will require for all u ∈ U . We will use K C to denote the class of such functions for any constant C > 0. Since g t is Anosov, the expansion rates of g t on U are bounded away from 1. Since the return times τ : R → R are bounded away from 0, the slowest expansion rates of dσ n on U are therefore also bounded away from 1. For what follows, let f κ n and bK n with 1 < κ < K be the slowest and fastest expansion rate of any unit vector in T 1 U under dσ n .
We are interested in functions Φ ∈ K C with bounded logarithmic derivative because they can be used to control less regular functions ϕ ∈ C 1 (U, R). The following lemma makes this precise.
Lemma 4.1. Fix C > 0, ϕ ∈ C 1 (U, V ρ ) and Φ ∈ K C . There is a δ > 0 so that, if for all u ∈ U , then for any u 0 ∈ U , for all u ∈ B δ (u 0 ), each n > 0 and each consistent past v = v (n) defined near u 0 . Moreover, for any u 0 ∈ U , for all x, y ∈ B δ (u 0 ). The choice of constant δ > 0 can be made so that we have δC = A for some uniform constant A, which does not depend on ρ, Φ, ϕ, u 0 , v or n.
Proof. Since we chose Φ ∈ K C , log Φ is C-Lipschitz and we therefore have for any x, y ∈ U . Exponentiating both sides, this means for any x, y ∈ U . Now, suppose that d(x, y) ≤ 2δ for some δ > 0, and fix a unit speed path γ: [0, 2δ] → U 0 with γ(0) = y and γ(2δ) = x. We then have by the fundamental theorem of calculus and our bounds on ϕ. Fix δ small enough to ensure and e C f 2δ ≤ 2 hold simultaneously -note that we really only require that Cδ is sufficiently small, and so δ can be chosen inversely proportional to C. We therefore have whenever d(x, y) < 2δ. To conclude, suppose that we had at some y ∈ B δ (u 0 ), for a given u 0 ∈ U and n. Then by (4.3) and (4.2), we must have for any x ∈ B δ (u 0 ), as desired.
Lemma 4.4. Fix C > 0, ϕ ∈ C 1 (U, V ρ ) and Φ ∈ K C . Take δ > 0 as in Lemma 4.1 and suppose that we have for all u ∈ B δ (u 0 ), and some given u 0 ∈ U , then we have Proof. Differentiating the fraction, we see that we need to bound for any u ∈ U . Note that we always have and so (4.5) is at most by the triangle inequality. Cancelling terms, we can reduce (4.6) to by the chain rule. Since f κ n is the slowest expansion rate of any vector in T 1 U under dσ n , we can bound for all u ∈ U . By hypothesis, we also have for all u ∈ U , and so (4.7) can be bounded above by by hypothesis. Cancelling Φ, we obtain the result desired.
Before we get to the main argument in this section, we need an elementary linear algebra result. Proof. We expand using the polarization identity, obtaining We might be tempted to argue that, if ϕ ∈ C 1 (U, V ρ ) is controlled by Φ ∈ K C as in Lemma 4.1, then we can similarly bound L n z,ρ ϕ (u) L 2 (G) by L n (z),0 Φ (u) pointwise -however, while this turns out to be true, it is unhelpful since L n P (ς),0 Φ fails to contract as n → ∞. Indeed, since Φ is strictly positive by definition, L n P (ς),0 Φ will converge to U Φ dν u > 0. The solution is to artificially introduce contraction into the transfer operators, and Theorem 3.27 is precisely what ensures that we can do this while maintaining a pointwise bound. In the next lemma, we will show that we can uniformly and explicitly bound L n z,ρ ϕ (u) L 2 (G) away from L n (z),0 Φ (u) on a measurable portion of any sufficiently small set.
Lemma 4.9. Fix C > 0, ϕ ∈ C 1 (U, V ρ ) and Φ ∈ K C with for all u ∈ U . Let U lni ⊂ U be the open subset given by Theorem 3.27 and let δ > 0 be the constant given by Lemma 4.1. There are constants n 0 > 0, > 0 and s < 1 so that, for any x ∈ U lni with B δ (x) ⊂ U lni , we can find • a point y ∈ U lni with B sδ (y) ⊂ B δ (x) and • a pair of pasts v for which we can bound for all u ∈ B sδ (y) and all z ∈ C with | (z) − P (ς)| < 1. The constant can be chosen independently of ρ and C, while n 0 depends only on C ρ . The constant s can be chosen uniformly in C and ρ.
Proof. We will deal with a fixed n > 0 throughout the proof, and specify how large n needs to be as we proceed -it is important to note that we cannot deal with arbitrarily large n without foregoing the uniformity of the bounds we wish to obtain. We proceed in cases depending on which alternative of Lemma 4.1 holds. For any pasts v (n) 1 and v for all u ∈ B δ (x), then since ρ is unitary we can clearly bound for all u ∈ B δ (x). In this case, we are done by simply setting y := x. Similarly, if we had for all u ∈ B δ (x), then we are again done by setting y := x, up to interchanging our choice of v 1 and v 2 . So we may as well assume that the second alternative of Lemma 4.1 holds for both v (n) 1 and v (n) 2 , and that we therefore have for all u ∈ B δ (x). We will temporarily abbreviate for the sake of clarity. Note thatφ andψ are well-defined on B δ (x) as an immediate consequence of (4.10). Now, by reverse the triangle inequality, is at least ρ (g 1 (x))ψ 1 (y) − ρ (g 2 (x))ψ 2 (y) for any y ∈ B δ (x). Since group elements act by isometries with respect to the L 2 (G) norm, (4.12) is equal to ρ g 2 (y)g −1 2 (x)g 1 (x) ψ 1 (y) − ρ (g 2 (y))ψ 2 (y) which we can bound below by ρ g 2 (y)g −1 2 (x)g 1 (x) ψ 1 (y) − ρ (g 1 (y))ψ 1 (y) − ρ (g 2 (y))ψ 2 (y) − ρ (g 1 (y))ψ 1 (y) (4.13) using the reverse triangle inequality once again. We can rewrite (4.13) as ρ g 1 (x)g −1 1 (y)g 1 (y) ψ 1 (y) − ρ g 2 (x)g −1 2 (y)g 1 (y) ψ 1 (y) − ρ (g 2 (y))ψ 2 (y) − ρ (g 1 (y))ψ 1 (y) (4.14) by replacingψ 1 (y) with g −1 1 (y)g 1 (y)ψ 1 (y) in the first term of the first line, and multiplying both terms on the first line by g 2 (x)g −1 2 (y). Hence, (4.11) is bounded below by (4.14), and we see that is at least ρ g 1 (x)g −1 1 (y)g 1 (y) ψ 1 (y) − ρ g 2 (x)g −1 2 (y)g 1 (y) ψ 1 (y) and v (n) 2 , and a y with d(x, y) = δ 2 for which we have ρ g 1 (x)g −1 1 (y)g 1 (y) ψ 1 (y) − ρ g 2 (x)g −1 2 (y)g 1 (y) ψ 1 (y) for all n ≥ N -note that we have applied the theorem to which is certainly a smooth function in C 1 (U, V ρ ). On the other hand, by Lemma 4.4,φ (u) is at worst 8C f κ n -Lipschitz in u on B δ (x), and we can estimate since we chose y ∈ B δ (x). Suppose that n and ρ are large enough so that we have for some constant K > 0. Note that we can make this choice of n so that it depends only on C ρ and not ρ directly; moreover, the requirement that ρ be sufficiently large can be made absolute, and in particular is independent of z. This ensures ψ (x) −ψ (y) since d(x, y) ≤ δ. Note that our choice of n here depends only on C ρ . Plugging (4.16) and (4.17) into (4.15), we conclude that ρ(g 1 (x))ψ 1 (x) − ρ (g 2 (x))ψ 2 (x) L 2 (G) + ρ(g 1 (y))ψ 1 (y) − ρ (g 2 (y))ψ 2 (y) or ρ(g 1 (y))ψ 1 (y) − ρ (g 2 (y))ψ 2 (y) must hold. Without loss of generality, suppose (4.19) holds. Using the Lipschitz estimate onφ , we can bound for all u ∈ B δ (x). A straightforward calculation shows that and we have sup by Definition 1.20; note that since ρ is a homomorphism, the operator norm of dρ at g (u) is equivalent to the norm at the identity. Hence, (4.20) is at most by our choice of δ, since ρ acts by L 2 (G)-isometries. We can make a uniform choice of s < 1 so that and we then have ρ (g 1 (y))ψ 1 (y) − ρ (g 1 (u))ψ 1 (u) and ρ (g 2 (y))ψ 2 (y) − ρ (g 2 (u))ψ 2 (u) for all u ∈ B sδ (y). Combining (4.22) and (4.23) with (4.18), we now have ρ (g 1 (u))ψ 1 (u) − ρ (g 2 (u))ψ 2 (u) for all u ∈ B sδ (y). Now, fix u ∈ B sδ (y). By Proposition 4.8, we can then bound and v (n) 2 -this is not immediate since (4.25) could certainly fail to hold on the entire ball B sδ (y). This will take some extra work. Note that we have for all u ∈ U . By making s smaller if necessary, we can ensure that, in addition to (4.21), we also have where A is the uniform constant guaranteed by Lemma 4.1. Note that this can be accomplished by a uniform choice of s, since n is fixed and z is required to satisfy | (z) − P (ς)| < 1. As a consequence, we see by Lemma 4.1 that 1 2 e α (n) for all u, u ∈ B sδ (y). Now, suppose that for some u ∈ B sδ (y) -this is in contrast to the bound by (4.24) that we have at u ∈ B sδ (y). If we had then we can invoke (4.26) twice to see that for all u ∈ B δ (x i ). We can smoothly extend η i to all of U by setting η i = 0 outside v (n0) 1 (u) (B δ (x i )). To define β, we simply set β(u) := 1 − i η i (u) and by Lemma 4.9 we clearly have L n0 z,ρ ϕ (u) L 2 (G) ≤ L n0 (z),0 (βΦ) (u) for all u ∈ U . It simply remains to estimate L n0 P (ς),0 (βΦ) . Note that we have L n0 P (ς),0 (βΦ) (u) ≤ 1 − for all u ∈ B sδ 2 (y i ) by construction. Moreover, by the monotonicity of L n0 P (ς),0 , we can bound L n0 P (ς),0 (βΦ) (u) ≤ L n0 P (ς),0 Φ (u) for all u ∈ U − B sδ 2 (y i ). Taken together, these inequalities mean that we can bound It is important to recognize that many of the estimates so far do in fact depend on ρ, (z) or C -this will be problematic for the spectral bounds we want to obtain. To isolate some of these dependencies, we will restrict our attention to control functions Φ ∈ K C(1+| (z)|) ρ , where we hope to be able to make a uniform choice of an appropriate C. In particular, we want to find a C so that K C(1+| (z)|) ρ is invariant under L n z,ρ , at least for z with (z) sufficently close to P (ς).
Proposition 4.29. There is a uniform choice of constant C > 0 so that, for all ϕ ∈ C 1 (U, V ρ ) and for all u ∈ U , and all n > 0.
for i ≥ 1. Note that we have just shown that Φ 1 ∈ K C0(1+| (z)|) ρ and that we have for all u ∈ U . As before, we get inductively for all i ≥ 1, and moreover by construction. Chaining these inequalities together, we have L i·n0 z,ρ ϕ L 2 (ν u ) ≤ r i 0 ϕ C 1 which is almost what we need. To conclude, observe that we have where L k z,ρ L 2 (G) denotes the operator norm of L k z,ρ . If k < n 0 and | (z) − P (ς)| < 1, then we can find a uniform bound D > 0 so that L k z,ρ L 2 (G) ≤ D, as desired. The differentiability of the potential ς was essential to much of what we have done so far in this section; to extend our results to the case when ς is only Hölder, however, is a relatively straightforward approximation argument, identical to the one given in [Dol98]. We sketch this below.
Corollary 4.36. With notation as in Theorem 4.35, we have L n (z),ρ ϕ L 2 (ν u ) ≤ Dr n 0 ϕ C 1 when the potential ς is only Hölder continuous.