On the rate of mixing of Circle extensions of Anosov maps.
Abstract.
Let be an Anosov diffeomorphism. Circle extensions are a rich family of nonuniformly hyperbolic diffeomorphisms living on for which the rate of mixing is conjectured to be generically exponential. In this paper, we investigate the possible rates of exponential mixing by exhibiting some explicit lower bounds on the decay rate by Fourier analytic and probabilistic techniques. The rates obtained are related to the topological pressure of two times the unstable jacobian.
Key words and phrases:
Rates of mixing, Transfer operators, Topological pressure, Anisotropic function spaces.1. Introduction
Let be the usual flat torus. And consider an Anosov diffeomorphism, which will assumed to be topologically mixing in the sequel. Let be a smooth map and let be the circle. Then one can define an extension of , denoted by by setting
all coordinates being understood mod . These maps are the simplest prototype of partially hyperbolic systems, for which the neutral direction forms a trivial bundle in the tangent bundle. The qualitative ergodic theory of these maps is well established, and most questions of ergodic stability are settled in the work of Brin [5] and BurnsWilkinson [6]. However, when it comes to quantitative ergodic theory, very few results are known. Let be the SinaiRuelleBowen invariant probability measure, which can be characterized as the unique physical measure, for which Birkhoff averages converge Lebesguealmost surely to the spatial average. A natural invariant extension of to can be defined by
From the pioneering work of Dolgopyat [8], it follows that for generic , the map has rapid decay of correlations for all observables, i.e. for all on , we have as and all ,
It is natural to expect that exponential mixing is also typical, but it is still an open question in the context of extensions. On the other hand, for Anosov Flows, a recent preprint of Tsujii [17] shows that generic volume preserving dimensional Anosov flows are exponentially mixing. See also [7] for new results in higher dimensions. We don’t know if Tsujii’s recent technique can be used to prove exponential mixing in our context, and this should be pursued elsewhere.
One natural question raised by our current knowledge is what’s the typical rate of mixing when observables are very regular ? Could we get super exponential mixing as in the case of linear Anosov diffeomorphisms of ? How does the instability of the system (Lyapunov exponents) affect this rate of mixing ? To formulate our main result, we recall that the unstable jacobian is defined by
where is the unstable direction at of , which is invariant. This is at least a Hölder continuous function on . Given a Hölder function on one can define the topological pressure by taking the supremum
where the sup is taken over all invariant probability measures, and is the measure theoretic entropy of . Our main result is the following.
Theorem 1.
Assume that is a small enough, volume preserving real analytic perturbation of a linear Anosov map.

Then for all real analytic, for all , one can find real analytic observables with such that

For all large enough, almost surely for all random trigonometric polynomial, the extended map is rapidly mixing for the SRBmeasure on .

For all , with positive probability for , one can find real analytic observables with and
The set of random trigonometric polynomials of degree , denoted by , is defined in and is just the obvious guess: independent Gaussian combinations of Laplace eigenfunctions on . The above theorem shows in particular that the rate of mixing, unlike in the uniformly hyperbolic case, can never be super exponential. This fact was already pointed out for suspensions of analytic expanding maps by the author in [12], with a less precise lower bound involving entropy rather than pressure. Note that the first statement is unconditional and holds for all choice of but there is a loss in the lower bound. However, we are also able to show that for ”many” choices of among the set of random polynomials , the rate of mixing is bigger than
which we believe is the optimal lower bound. One of the motivations for this type of quantitative lower bounds is that it shows that when unstable Lyapunov exponents are ”small”, i.e. close to , then the rate of mixing is arbitrarily (exponentially) slow.
An example: Arnold’s cat map. This is the standard Anosov diffeomorphism on induced by the action of the matrix
The eigenvalues are which implies that the topological entropy of the cat map is exactly
The topological pressure is easily computed as
which yields
Theorem 1 tells us that while the map itself mixes at super exponential decay rate for all analytic observables (for an elementary proof of that fact, see [2], chapter 4), there exists (at least rapidly mixing) extensions and analytic observables whose rate of mixing is not faster than .
The paper is organized as follows. In the next section, we show how the lower bounds on correlation functions can be derived from a statement on the spectrum of certain twisted transfer operators that depend on a frequency parameter . These operators act naturally on an anisotropic function space defined by FaureRoy in [9]. All the material and a priori estimates regarding these spaces is gathered in the last section . In , we prove the first part of the main spectral estimate via a technique based on ”frequency averaging”, i.e. we prove certain bounds by summing smoothly over and eventually recover some pointwise bounds. In , we use a different averaging technique with a more probabilistic flavour: we consider some random ”roof functions” (given by a random ensemble of trigonometric polynomials of degree ) and show that one can prove a lower bound on the expectation of the spectral radius . This in turn shows the existence of a set of functions with an improved lower bound on the spectral radius. We also prove, using mostly the old technology of subshifts of finite type, that provided the degree is taken large enough, exponential mixing occurs with probability in . The last section is devoted to properties of the Anisotropic function space and we rely on the existing work [1, 9] and provide proofs of some spectral upper bounds that are necessary for our purpose.
2. Function space and reduction to a spectral problem
The main result (Theorem 1) follows from a statement on the spectrum of certain ”twisted” transfer operators. First, observe that given an observable defined on of the form ()
then we have
this leads naturally to study the following ”twisted” koopman operators acting by
The analysis of will depend crucially on a good choice of function space. We will be working in the realanalytic category and we describe below the functional analytic setup. Let be a hyperbolic matrix so that its induced action on is an Anosov map. Let be a parameter. We recall that a trigonometric polynomial on is simply an expression of the type
where , and . Trigonometric polynomials are obviously realanalytic on and extend holomorphically to .
Theorem 2.
There exists a family of Hilbert spaces which contain densely all trigonometric polynomials on , such that we have:

For all real analytic on , for all small enough real analytic perturbation of , one can find such that acts as a bounded compact trace class operator.

For all , the spectral radius of is smaller than .

Moreover there exist constants , and , independent of such that the eigenvalue sequence satisfies the bound

The lebesgue measure on the torus extends as a continuous linear functional and has ”full support” in the following generalized sense. Given with , one can find a trigonometric polynomial such that ^{1}^{1}1The fact that given , for all trigonometric polynomial the product belongs to will be clarified in .

For all , we have the trace formula
where the sum runs over all periodic points of period of the map , and .
Here the norm refers to the sup norm of in a complexified neighbourhood of the torus . More precisely if extends holomorphically in small complex neighbourhood of the type , we set
The existence of such function spaces ”adapted to the hyperbolic dynamics” follow in the analytic category from the work of FaureRoy [9]. More recently, these function spaces have been used to study the Ruelle spectrum of Anosov maps by Adam [1], and also BandtlowJustSlipantschuk [16]. We will provide more details for the construction of these spaces later on but roughly speaking, they are designed in Fourier coordinates to impose analyticity in the stable direction (exponential decay of Fourier coefficients) while irregularity is allowed in the unstable direction (exponential growth at most of Fourier modes). Part and follow readily from the above mentioned papers. On the other hand parts and of the above theorem will require an extra amount of work which is the purpose of the last section of the paper. However, we can use Theorem 2 as a ”blackbox” to prove our main result which will follow from the spectral statement below.
Proposition 3.
Under the above assumptions and notations, the following holds.

For all , there exist infinitely many such that has an eigenvalue with

Furthermore, for all , there exist trigonometric polynomials , non cohomologous to constants, such that one can find such that has an eigenvalue with
Let us show how to derive Theorem 1 from Proposition 3. We fix large enough such that either or from the above statement holds. We use observables of the form
where will be analytic functions on the torus specified later on. Notice that we have
Because is compact, we can use holomorphic functional calculus to write for all
the error term being understood in the operator norm topology. Each is a finite rank projector such that
equals the algebraic multiplicity^{2}^{2}2We cannot discard the possible presence of Jordan blocks. of the eigenvalue . For all we have , and
where . Going back to the correlation function (from now on we assume that we are in the volume preserving case i.e. ), we have
where is a polynomial in with degree at most , given by
Now set . We set for all such that , . We can then write
for some . Let us set ^{3}^{3}3Remark that if there is only one eigenvalue with , then the proof is much simpler, but we cannot exclude this case.
To obtain a lower bound on the oscillating sum , we will use Dirichlet box principle, which for us is the following handy fact.
Lemma 4.
Let and . For all , one can find an integer such that
Applying this lemma with , for all we can find a sequence with as goes to , such that for all , we have
Therefore we have for all ,
Now consider the quantity given by
If we assume that are such that the sum of coefficients
then makes sense for all large and exists and is nonvanishing. We can therefore choose such that for all large we have
The proof is then done because we get for all large ,
for some . It remains to check that we can adjust such that
Without loss in generality, we can pick such that and choose with . Then we have
and by property from Proposition 3, we can choose to be a trigonometric polynomial such that is non vanishing. Consider now the functional . It’s now a non trivial continuous linear form on and by density we can choose a trigonometric polynomial such that again . The proof is done.
3. Existence of non trivial spectra for via frequency averaging
In this section, we will prove Proposition 3 and its two statements. The main ideas will revolve around the trace formula
and different ways to estimate (from below and above) this oscillating sum via averaging techniques. We start by a basic a priori bound.
3.1. An upper bound on the trace
For all , let denote the spectral radius of , i.e.
Proposition 5.
For all , there exists depending only on and the map , such that for all and , we have
3.2. Averaging over the frequency
In this section we shall prove part of Proposition 3. First we need an observation on topological pressure of the unstable jacobian and weighted sums over periodic orbits that arise from the trace formulas. More precisely we have the following fact.
Lemma 6.
Let . For all , one can find a constant such that for all large enough
Proof. We recall that the Anosov structure says that at each point , we have a splitting
with , and there exist constants such that for all ,
Whenever , we have two mappings and . Therefore we have
By exponential decay of both and as , we deduce that there exists a constant , such that for all large, we have
so that
It is then a standard fact, that when is a topologically mixing Anosov map, we have for all real valued Hölder potential ,
For references, see the classics [4, 14]. For a more modern treatment, see also [3], Chapter 7, Corollary 7.7. The proof of the lower bound is done. For the upper bound, the exact same ideas work straightforwardly.
We now proceed toward a proof of proposition 3, first part. Let us set
Our goal is to obtain some decent lower bounds on . Pointwise, this is quite a desperate task, but we will rely instead on an averaged estimate by summing carefully over the frequency parameter . We pick a test function on having the following set of properties:^{4}^{4}4The existence of such a test function is a folklore fact. Start with the usual bump function given by . To make sure that the Fourier transform is positive consider then the convolution which obviously has now the desired properties.
where is the Fourier transform defined by
We now set for some ,
and consider the quantity
To compute this average, we use Poisson summation formula which says that given a rapidly decaying test function , we have the celebrated identity
Therefore we have
Because is positive, we can obviously bound from below (by forgetting all the nondiagonal terms) for all
Observe now that because is in the Schwartz class (rapid decay), then we have
which tells us that for large , we can as well drop all non zero terms in the sum , so that we end up with the lower bound (we use Lemma 6)
for some and all large enough. Using Proposition 5, we have obtained
identity valid for all and large. We now fix and choose . We end the proof by contradiction. Assume that there exists such that for all , we have . We get therefore (using the fact that unconditionally )
For notational simplicity, we set . We set in the sequel
where will be adjusted later on. As we have
we get a contradiction whenever
which leads to choose
for all , and the proof is done.
4. An improved lower bound via a probabilistic technique
The goal of this section is to show how to improve the lower bound of Theorem 1, 1) via a different argument. Instead of averaging over the frequency parameter , we will consider some random ”coupling functions” of the form
where belongs to a suitable ensemble of random trigonometric polynomials . The game is to estimate from below the expectation , where is seen as a random variable. This technique will allow us to overcome the ”exponent loss”, artefact of the frequency averaging technique, and will rely also on a positivity argument. We will also prove that rapid mixing occurs with probability in , which will occupy section .
4.1. The set of random trigonometric polynomials
Our goal is to define a set of random real valued trigonometric polynomials. We could use a Fourier basis of real valued trigonometric functions, based on product of sines and cosines, but that would lead to cumbersome notations. Instead, we choose a more conceptual route using a fixed basis of real valued eigenfunctions of the Laplacian. Let be the flat Laplacian on . We choose an basis of real eigenfunctions of the Laplacian such that
where the eigenvalues are repeated according to multiplicity. Each eigenfunction is a trigonometric polynomial of the form
with , and are coefficients subject to the normalization. We will need to use two basic facts on these eigenfunctions:

(Weyl law). As , we have .

(growth). As , we have , for all .
The first fact is easy and follows from a crude upper bound on the number of lattice points in a disc. The other claim is less trivial but standard and follows from estimating the number of lattice points on a circle, which in turn is related to the number of representations of a given integer as a sum of two squares.
Let be a probability space, and assume that there exists a sequence of real valued, independent random variables on , whose common probability law is Gaussian centered with variance , i.e. each obeys the law (for all measurable )
Fix a large integer and consider the random trigonometric polynomial given by
and this random ensemble is denoted by . We start by a basic Lemma.
Lemma 7.
For all , for all polynomial , with , the expectation
Proof. Let us prove that the random variable has finite even moments to all orders. By Schwarz inequality, we have (, ),
where we have set
Consequently, we have
Because the random gaussian variables are assumed to be independent and have finite moments to all orders, we can conclude (without having to compute it) that
Using Schwarz inequality, we deduce now that all the odds moments are also finite, and the proof is done.
4.2. Rapid mixing occurs almost surely in .
In the following, we prove that rapid mixing is almost sure in . This part of the paper will definitely not surprise the experts, but there are nevertheless some technical details that have to be addressed.
Here we recall that the real analytic Anosov map is assumed to be topologically mixing, which definitely occurs if is close to a linear hyperbolic map. The proof is twofold: first we use symbolic dynamics to show that rapid mixing follows from an estimate on transfer operators due to Dolgopyat [8]. This estimate holds under a diophantine condition satisfied by the ”roof” function . We then show that when , this diophantine hypothesis holds with full probability.
According to Bowen [4], using Markov partitions, there exists a topologically mixing subshift of finite type