arXiv subjects
Publications and source records attributed to Luchezar Stoyanov.
We 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 previous known estimates.
We prove exponential decay of correlations for Hölder continuous observables with respect to any Gibbs measure for contact Anosov flows admitting Pesin sets with exponentially small tails. This is achieved by establishing strong spectral estimates for certain Ruelle transfer operators for such flows.
Given Hölder continuous functions $f$ and $ψ$ on a sub-shift of finite type $Σ_A^{+}$ such that $ψ$ is not cohomologous to a constant, the classical large deviation principle holds (\cite{OP}, \cite{Kif}, \cite{Y}) with a rate function $I_ψ\geq 0$ such that $I_ψ(p) = 0$ iff $p = \int ψ\, d μ$, where $μ= μ_f$ is the equilibrium state of $f$. In this paper we derive a uniform estimate from below for $I_ψ$ for $p$ outside an interval containing $\tildeψ = \int ψ\, dμ$, which depends only on the sub-shift, the function $f$, the norm $|ψ|_\infty$, the Hölder constant of $ψ$ and the integral $\tildeψ$. Similar results can be derived in the same way e.g. for Axiom A diffeomorphisms on basic sets.
Pesin sets are measurable sets along which the behavior of a matrix cocycle above a measure preserving dynamical system is explicitly controlled. In uniformly hyper-bolic dynamics, we study how often points return to Pesin sets under suitable conditions on the cocycle: if it is locally constant, or if it admits invariant holonomies and is pinching and twisting, we show that the measure of points that do not return a linear number of times to Pesin sets is exponentially small. We discuss applications to the exponential mixing of contact Anosov flows, and counterexamples illustrating the necessity of suitable conditions on the cocycle.
Billiard trajectories (broken generalised geodesics) are considered in the exterior of an obstacle $K$ with smooth boundary on an arbitrary Riemannian manifold. We prove a generalisation of the well-known Santalo's formula. As a consequence, it is established that if the set of trapped points has positive measure, then for all sufficiently small smooth perturbations of the boundary of $K$ the set of trapped points for the new obstacle obtained in this way also has positive measure. More generally the measure of the set of trapped points depends continuously on perturbations of the obstacle $K$. Some consequences of the generalised Santalo's formula are derived in the case of scattering by an obstacle $K$ in an Euclidean space. For example, it is shown that, for a large class of obstacles $K$, the volume of $K$ is uniquely determined by the average travelling times of scattering rays in the exterior of $K$.
For the time-one map $f$ of a contact Anosov flow on a compact Riemann manifold $M$, satisfying a certain regularity condition, we show that given a Gibbs measure on $M$, a sufficiently large Pesin regular set $P_0$ and an arbitrary $δ\in (0,1)$, there exist positive constants $C$ and $c$ such that for any integer $n \geq 1$, the measure of the set of those $x\in M$ with $f^k(x) \notin P_0$ for at least $δn$ values of $k = 0,1, \ldots,n-1$ does not exceed $C e^{-cn}$.
For $C^2$ weak mixing Axiom A flow $ϕ_t: M \longrightarrow M$ on a Riemannian manifold $M$ and a basic set $Λ$ for $ϕ_t$ we consider the Ruelle transfer operator $L_{f - s τ+ z g}$, 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 $ϕ_t$ we establish estimates for the iterations of this Ruelle operator in the spirit of the estimates for operators with one complex parameter (see \cite{D}, \cite{St2}, \cite{St3}). Two cases are covered: (i) for arbitrary Hölder $f,g$ when $|\Im z| \leq B |\Im s|^μ$ for some constants $B > 0$, $0 < μ< 1$ ($μ= 1$ for Lipschitz $f,g$), (ii) for Lipschitz $f,g$ when $|\Im s| \leq B_1 |\Im z|$ for some constant $B > 0$ . Applying these estimates, we obtain a non zero analytic extension of the zeta function $ζ(s, z)$ for $P_f - ε< \Re (s) < P_f$ and $|z|$ small enough with 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 $π_F(T)$ for weighted primitive periods of the flow $ϕ_t.$
The paper deals with some problems related to recovering information about an obstacle in an Euclidean space from certain measurements of lengths of generalized geodesics in the exterior of the obstacle. The main result is that if two obstacles satisfy some generic regularity conditions and have (almost) the same traveling times, then the generalized geodesic flows in their exteriors are conjugate on the non-trapping part of their phase spaces with a time preserving conjugacy. In the case of a union of two strictly convex domains in the plane, a constructive algorithm is described to recover the obstacle from traveling times.
Obstacles $K$ and $L$ in $R^d$ ($d\geq 2$) are considered that are finite disjoint unions of strictly convex domains with $C^3$ boundaries. We show that if $K$ and $L$ have (almost) the same scattering length spectrum, or (almost) the same traveling times, then $K = L$.
We prove a sharp large deviation principle concerning intervals shrinking with sub-exponential speed for certain models involving the Poincaré map related to a Markov family for an Axiom A flow restricted to a basic set $Λ$ satisfying some additional regularity assumptions.
We examine the asymptotics of the number of the closed trajectories $γ$ of hyperbolic flows $ϕ_t$ whose primitive periods $T_γ$ lie in exponentially shrinking intervals $(x - e^{-δx}, x + e^{-δx}),\:δ> 0,\: x \to + \infty.$ Our results holds for hyperbolic dynamical systems having a symbolic model with a non-lattice roof function $f$ under the assumption that the corresponding Ruelle operator related to $f$ satisfies strong spectral estimates. In particular, our analysis works for open billiard systems and for the geodesics flow on manifolds with constant negative curvature.
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 $ε> 0$. 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.
Let $s_0 < 0$ be the abscissa of absolute convergence of the dynamical zeta function $Z(s)$ for several disjoint strictly convex compact obstacles $K_i \subset \R^N, i = 1,..., κ_0,\: \ka_0 \geq 3,$ and let $R_χ(z) = χ(-Δ_D - z^2)^{-1}χ,\: χ\in C_0^{\infty}(\R^N),$ be the cut-off resolvent of the Dirichlet Laplacian $-Δ_D$ in $Ω= \bar{\R^N \setminus \cup_{i = 1}^{k_0} K_i}$. We prove that there exists $σ_1 < s_0$ such that $Z(s)$ is analytic for $\Re (s) \geq σ_1$ and the cut-off resolvent $R_χ(z)$ has an analytic continuation for $\Im (z) < - σ_1,\: |\Re (z)| \geq C > 0.$
In this paper we study a certain regularity property of Axiom A flows over basic sets related to diameters of balls in Bowen's metric, which we call regular distortion along unstable manifolds. The motivation to investigate the latter comes from the study of spectral properties of Ruelle transfer operators. We prove that if the bottom of the spectrum of the tangent map of the flow over unstable manifolds is point-wisely pinched and integrable, then the flow has regular distortion along unstable manifolds over the basic set. In the process, under the same conditions, we show that locally the flow is Lipschitz conjugate to its linearization over the `pinched part' of the unstable tangent bundle.
In this paper we consider open billiard flows in Euclidean spaces with C^1 (un)stable laminations over their non-wandering sets. We show that for such billiard flows the standard symplectic form satisfies a specific non-degeneracy condition over the non-wandering set. This allows to use some previous general results and obtain Dolgopyat type estimates for spectra of Ruelle transfer operators under simpler conditions. We also describe a class of open billiard flows in Euclidean spaces satisfying a certain pinching condition, which in turn implies that the (un)stable laminations over the non-wandering set are C^1.
For Axiom A flows on basic sets satisfying certain additional conditions we prove strong spectral estimates for Ruelle transfer operators similar to these of Dolgopyat (1998) for geodesic flows on compact surfaces (for general potentials)and transitive Anosov flows on compact manifolds with C^1 jointly non-integrable horocycle foliations (for the Sinai-Bowen-Ruelle potential). Here we deal with general potentials. 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, decay of correlations for Hölder continuous potentials.
Given a locally maximal compact invariant hyperbolic set $Λ$ for a $C^1$ flow or diffeomorphism on a Riemann manifold with $C^1$ unstable laminations, we construct an invariant continuous bundle of tangent vectors to local unstable manifolds that locally approximates $Λ$ in a certain way.
In this paper we prove two asymptotic estimates for pairs of closed trajectories for open billiards similar to those established by Pollicott and Sharp for closed geodesics on negatively curved compact surfaces. The first of these estimates holds for general open billiards in any dimension. The more intricate second estimate is established for open billiards satisfying the so called Dolgopyat type estimates. This class of billiards includes all open billiards in the plane and open billiards in $\R^N, N \geq 3$ satisfying some additional conditions.