Ruelle Transfer Operator Research Articles

Spectral Properties of Ruelle Transfer Operators for Regular Gibbs Measures and Decay of Correlations for Contact Anosov Flows

In this work we study strong spectral properties of Ruelle transfer operators related to a large family of Gibbs measures for contact Anosov flows. The ultimate aim is to establish exponential decay of correlations for Hölder observables with respect to a very general class of Gibbs measures. The approach invented in 1997 by Dolgopyat in “On decay of correlations in Anosov flows” and further developed in Stoyanov (2011) is substantially refined here, allowing to deal with much more general situations than before, although we still restrict ourselves to the uniformly hyperbolic case. A rather general procedure is established which produces the desired estimates whenever the Gibbs measure admits a Pesin set with exponentially small tails, that is a Pesin set whose preimages along the flow have measures decaying exponentially fast. We call such Gibbs measures regular. Recent results in Gouëzel and Stoyanov (2019) prove existence of such Pesin sets for hyperbolic diffeomorphisms and flows for a large variety of Gibbs measures determined by Hölder continuous potentials. The strong spectral estimates for Ruelle operators and well-established techniques lead to exponential decay of correlations for Hölder continuous observables, as well as to some other consequences such as: (a) existence of a non-zero analytic continuation of the Ruelle zeta function with a pole at the entropy in a vertical strip containing the entropy in its interior; (b) a Prime Orbit Theorem with an exponentially small error.

The double transpose of the Ruelle operator

In this paper we study the double transpose of the $$L^1(X,\mathscr {B}(X),\nu )$$ -extensions of the Ruelle transfer operator $$\mathscr {L}_{f}$$ associated to a general real continuous potential $$f\in C(X)$$ , where $$X=E^{\mathbb {N}}$$ , the alphabet E is any compact metric space and $$\nu $$ is a maximal eigenmeasure. For this operator, denoted by $$\mathbb {L}^{**}_{f}$$ , we prove the existence of some non-negative eigenfunction, in the Banach lattice sense, associated to $$\rho (\mathscr {L}_{f})$$ , the spectral radius of the Ruelle operator acting on C(X). As an application, we obtain a sufficient condition ensuring that the extension of the Ruelle operator to $$L^1(X,\mathscr {B}(X),\nu )$$ has an eigenfunction associated to $$\rho (\mathscr {L}_{f})$$ . These eigenfunctions agree with the usual maximal eigenfunctions, when the potential f belongs to the Hölder, Walters or Bowen class. We also construct solutions to the classical and generalized variational problem, using the eigenvector constructed here.

Growth gap in hyperbolic groups and amenability

We prove a general version of the amenability conjecture in the unified setting of a Gromov hyperbolic group G acting properly cocompactly either on its Cayley graph, or on a CAT(-1)-space. Namely, for any subgroup H of G, we show that H is co-amenable in G if and only if their exponential growth rates (with respect to the prescribed action) coincide. For this, we prove a quantified, representation-theoretical version of Stadlbauer's amenability criterion for group extensions of a topologically transitive subshift of finite type, in terms of the spectral radii of the classical Ruelle transfer operator and its corresponding extension. As a consequence, we are able to show that, in our enlarged context, there is a gap between the exponential growth rate of a group with Kazhdan's property (T) and the ones of its infinite index subgroups. This also generalizes a well-known theorem of Corlette for lattices of the quaternionic hyperbolic space or the Cayley hyperbolic plane.

On Gibbs Measures and Spectra of Ruelle Transfer Operators

AbstractWe prove a comprehensive version of the Ruelle–Perron–Frobenius Theorem with explicit estimates of the spectral radius of the Ruelle transfer operator and various other quantities related to spectral properties of this operator. The novelty here is that the Hölder constant of the function generating the operator appears only polynomially, not exponentially as in previously known estimates.

Ruelle transfer operators with two complex parameters and applications

For a $C^2$ Axiom A flow $\phi_t: M \longrightarrow M$ on a Riemannian manifold $M$ and a basic set$\Lambda$ for $\phi_t$ we consider the Ruelle transfer operator $L_{f - s \tau + z g}$, where $f$ and $g$ are real-valuedHölder functions on $\Lambda$, $\tau$ is the roof function and $s, z \in \mathbb{C}$ are complex parameters. Under some assumptionsabout $\phi_t$ we establish estimates for the iterations of this Ruelle operator in the spirit of the estimates for operatorswith one complex parameter (see [4], [21], [22]). Two cases are covered: (i) for arbitrary Hölder $f,g$when $|Im z| \leq B |Im s|^\mu$ for some constants $B > 0$, $0 0$ . Applying these estimates, we obtain a non zero analyticextension of the zeta function $\zeta(s, z)$ for $P_f - \epsilon < Re (s) < P_f$ and $|z|$ small enough with a simple poleat $s = s(z)$. Two other applications are considered as well: the first concerns the Hannay-Ozorio de Almeida sum formula,while the second deals with the asymptotic of the counting function $\pi_F(T)$ for weighted primitive periods of the flow $\phi_t.$

A rigorous version of R.P. Brent's model for the binary Euclidean algorithm

The binary Euclidean algorithm is a modification of the classical Euclidean algorithm for computation of greatest common divisors which avoids ordinary integer division in favour of division by powers of two only. The expectation of the number of steps taken by the binary Euclidean algorithm when applied to pairs of integers of bounded size was first investigated by R.P. Brent in 1976 via a heuristic model of the algorithm as a random dynamical system. Based on numerical investigations of the expectation of the associated Ruelle transfer operator, Brent obtained a conjectural asymptotic expression for the mean number of steps performed by the algorithm when processing pairs of odd integers whose size is bounded by a large integer. In 1998 B. Vallée modified Brent's model via an induction scheme to rigorously prove an asymptotic formula for the average number of steps performed by the algorithm; however, the relationship of this result with Brent's heuristics remains conjectural. In this article we establish previously conjectural properties of Brent's transfer operator, showing directly that it possesses a spectral gap and preserves a unique continuous density. This density is shown to extend holomorphically to the complex right half-plane and to have a logarithmic singularity at zero. By combining these results with methods from classical analytic number theory we prove the correctness of three conjectured formulae for the expected number of steps, resolving several open questions promoted by D.E. Knuth in The Art of Computer Programming.

The Milnor–Thurston Determinant and the Ruelle Transfer Operator

The topological entropy $h_{\rm top}$ of a continuous piecewise monotone interval map measures the exponential growth in the number of monotonicity intervals for iterates of the map. Milnor and Thurston showed that $\exp(-h_{\rm top})$ is the smallest zero of an analytic function, now coined the Milnor-Thurston determinant, that keeps track of relative positions of forward orbits of critical points. On the other hand $\exp(h_{\rm top})$ equals the spectral radius of a Ruelle transfer operator $L$, associated with the map. Iterates of $L$ keep track of inverse orbits of the map. For no obvious reason, a Fredholm determinant for the transfer operator has not only the same leading zero as the M-T determinant but all peripheral (those lying in the unit disk) zeros are the same. The purpose of this note is to show that on a suitable function space, the dual of the Ruelle transfer operator has a regularized determinant, identical to the Milnor-Thurston determinant, hereby providing a natural explanation for the above puzzle.

Ruelle operators with two complex parameters and applications

For a C2 weak-mixing Axiom-A flow ϕt:M⟶M on a Riemannian manifold M and a basic set Λ for ϕt, we consider the Ruelle transfer operator Lf−sτ+zg, where f and g are real-valued Hölder functions on Λ, τ is the roof function and s,z are complex parameters. Under some assumptions about ϕt for arbitrary Hölder f,g, we establish estimates for the iterations of this Ruelle operator when |Imz|≤B|Ims|ν for some constants B>0, 0<ν<1 (ν=1 for Lipschitz f,g), in the spirit of the estimates for operators with one complex parameter (see [2,11,12]). Applying these estimates, we obtain a non-zero analytic extension of the zeta function ζ(s,z) for Pf−ϵ<Re(s)≤Pf and |z| small enough with a simple pole at s=s(z). Two other applications are considered as well: the first concerns the Hannay–Ozorio de Almeida sum formula, while the second deals with the asymptotic of the counting function πF(T) for weighted primitive periods of the flow ϕt.

Ruelle operators and decay of correlations for contact Anosov flows

We prove strong spectral estimates for Ruelle transfer operators for arbitrary C2 contact Anosov flows. As a consequence of this we obtain: (a) existence of a non-zero analytic continuation of the Ruelle zeta function with a pole at the entropy in a vertical strip containing the entropy in its interior; (b) a Prime Orbit Theorem with an exponentially small error; (c) exponential decay of correlations for Hölder continuous observables with respect to any Gibbs measure.

Semi-classical approach for Anosov diffeomorphisms and Ruelle resonances

In this paper, we show that some spectral properties of Anosov diffeomorphisms can be obtained by semi-classical analysis. In particular the Ruelle resonances which are eigenvalues of the Ruelle transfer operator acting in suitable anisotropic Sobolev spaces and which govern the decay of dynamical correlations, can be treated as the quantum resonances of open quantum systems in the Aguilar-Baslev-Combes theory or the more recent Helffer-Sjostrand phase-space theory.

The Ruelle Transfer Operator in the Context of Orthogonal Polynomials

We redefine the Ruelle transfer operator, a classical tool from dynamical systems theory, in terms of orthogonal polynomial sequences. This transfer operator will be given via the preimages of the Chebyshev polynomials of the first kind and we will show that function spaces determined by the Chebyshev polynomials of the first kind are left invariant while function spaces determined by various other orthogonal polynomial sequences are not.

Natural boundary for the susceptibility function of generic piecewise expanding unimodal maps

AbstractFor a piecewise expanding unimodal interval map$f$with unique absolutely continuous invariant probability measure$\mu $, a perturbation$X$, and an observable$\varphi $, the susceptibility function is$\Psi _\varphi (z)= \sum _{k=0}^\infty z^k \int X(x) \varphi '( f^k)(x) (f^k)'(x) \, d\mu $. Combining previous results [V. Baladi, On the susceptibility function of piecewise expanding interval maps.Comm. Math. Phys.275(2007), 839–859; V. Baladi and D. Smania, Linear response for piecewise expanding unimodal maps.Nonlinearity21(2008), 677–711] (deduced from spectral properties of Ruelle transfer operators) with recent work of Breuer–Simon [Natural boundaries and spectral theory.Adv. Math.226(2011), 4902–4920] (based on techniques from the spectral theory of Jacobi matrices and a classical paper of Agmon [Sur les séries de Dirichlet.Ann. Sci. Éc. Norm. Supér.(3)66(1949), 263–310]), we show that density of the postcritical orbit (a generic condition) implies that$\Psi _\varphi (z)$has a strong natural boundary on the unit circle. The Breuer–Simon method provides uncountably many candidates for the outer functions of$\Psi _\varphi (z)$, associated with precritical orbits. If the perturbation$X$is horizontal, a generic condition (Birkhoff typicality of the postcritical orbit) implies that the non-tangential limit of$\Psi _\varphi (z)$as$z\to 1$exists and coincides with the derivative of the absolutely continuous invariant probability measure with respect to the map (‘linear response formula’). Applying the Wiener–Wintner theorem, we study the singularity type of non-tangential limits of$\Psi _\varphi (z)$as$z\to e^{i\omega }$for real$\omega $. An additional ‘law of the iterated logarithm’ typicality assumption on the postcritical orbit gives stronger results.

Pinching conditions, linearization and regularity of Axiom A flows

In this paper we study a certain regularity property of$C^2$ Axiom A flows $\phi_t$ over basic sets $Λ$ related to diameters of balls in Bowen'smetric, which we call regular distortion along unstable manifolds. The motivation toinvestigate the latter comes from the study of spectral properties of Ruelle transferoperators in [21]. We prove that if the bottom of the spectrum of $d\phi_t$ over $E^u_{|Λ}$ is point-wisely pinched andintegrable, then the flow has regular distortion along unstable manifolds over $Λ$.In the process, under the same conditions, we show that locally the flow is Lipschitz conjugate toits linearization over the `pinched part' of the unstable tangent bundle.

Fractal Weyl law for skew extensions of expanding maps

We consider compact Lie group extensions of expanding maps of the circle, essentially restricting to SU(2) extensions. The main objective of the paper is the associated Ruelle transfer (or pull-back) operator . Harmonic analysis yields a natural decomposition , where j indexes irreducible representation spaces. Using semi-classical techniques we extend a previous result by Faure proving an asymptotic spectral gap for the family when restricted to adapted spaces of distributions. Our main result is a fractal Weyl upper bound for the number of eigenvalues (the Ruelle resonances) of these operators out of some fixed disc centred on 0 in the complex plane.

Regular decay of ball diameters and spectra of Ruelle operators for contact Anosov flows

For Anosov flows on compact Riemann manifolds we study the rate of decay along the flow of diameters of balls $B^s(x,\ep)$ on local stable manifolds at Lyapunov regular points $x$. We prove that this decay rate is similar for all sufficiently small values of $\epsilon > 0$. From this and the main result in \cite{kn:St1}, we derive strong spectral estimates for Ruelle transfer operators for contact Anosov flows with Lipschitz local stable holonomy maps. These apply in particular to geodesic flows on compact locally symmetric manifolds of strictly negative curvature. As is now well known, such spectral estimates have deep implications in some related areas, e.g. in studying analytic properties of Ruelle zeta functions and partial differential operators, asymptotics of closed orbit counting functions, etc.

Distribution of Periods of Closed Trajectories in Exponentially Shrinking Intervals

For hyperbolic flows over basic sets we study the asymptotic of the number of closed trajectories γ with periods T γ lying in exponentially shrinking intervals $${(x - e^{-\delta x}, x + e^{-\delta x}), \; \delta > 0, \; x \to + \infty.}$$ A general result is established which concerns hyperbolic flows admitting symbolic models whose corresponding Ruelle transfer operators satisfy some spectral estimates. This result applies to a variety of hyperbolic flows on basic sets, in particular to geodesic flows on manifolds of constant negative curvature and to open billiard flows.

Non-integrability of open billiard flows and Dolgopyat-type estimates

AbstractWe consider open billiard flows in ℝn and show that the standard symplectic form dα in ℝn satisfies a specific non-integrability condition over their non-wandering sets Λ. This allows one to use the main result in Stoyanov [Spectra of Ruelle transfer operators for Axiom A flows. Preprint, 2010, arXiv:0810.1126v4 [math.DS]] and obtain Dolgopyat-type estimates for the spectra of Ruelle transfer operators under simpler conditions. We also describe a class of open billiard flows in ℝn(n≥3) satisfying a certain pinching condition, which in turn implies that the (un)stable laminations over the non-wandering set are C1.

Semiclassical origin of the spectral gap for transfer operators of a partially expanding map

We consider a specific family of skew product of partially expanding map on the torus. We study the spectrum of the Ruelle transfer operator and show that in the limit of high frequencies in the neutral direction (this is a semiclassical limit), the spectrum develops a spectral gap, for a generic map. This result has already been obtained by Tsujii (2008 Ergodic Theory Dyn. Syst. 28 291–317). The novelty here is that we use semiclassical analysis which provides a different and quite natural description. We show that the transfer operator is a semiclassical operator with a well-defined ‘classical dynamics’ on the cotangent space. This classical dynamics has a ‘trapped set’ which is responsible for the Ruelle resonances spectrum. In particular, we show that the spectral gap is closely related to a specific dynamical property of this trapped set.

Spectra of Ruelle transfer operators for Axiom A flows

For Axiom A flows on basic sets satisfying certain additional conditions we prove strong spectral estimates for Ruelle transfer operators similar to those of Dolgopyat (1998 Ann. Math. 147 357–90) for geodesic flows on compact surfaces (for general potentials) and transitive Anosov flows on compact manifolds with C1 jointly non-integrable horocycle foliations (for the Sinai–Bowen–Ruelle potential). Here we deal with general potentials and on spaces of arbitrary dimension, although under some geometric and regularity conditions. As is now well known, such results have deep implications in some related areas, e.g. in studying analytic properties of Ruelle zeta functions and partial differential operators, closed orbit counting functions, and in other areas.

Semi-classical approach for Anosov diffeomorphisms and Ruelle resonances

In this paper, we show that some spectral properties of Anosov diffeomorphisms can be obtained by semi-classical analysis. In particular the Ruelle resonances which are eigenvalues of the Ruelle transfer operator acting in suitable anisotropic Sobolev spaces and which govern the decay of dynamical correlations, can be treated as the quantum resonances of open quantum systems in the Aguilar-Baslev-Combes theory or the more recent Helffer-Sjostrand phase-space theory.