arXiv subjects

Luchezar Stoyanov

Publications and source records attributed to Luchezar Stoyanov.

At least 19 recordsLinked to original sources

arXiv2026

Gibbs measures for contact Anosov flows are all exponentially mixing

In this work we study strong spectral properties of Ruelle transfer operators related to Gibbs measures for contact Anosov flows. As a consequence we establish exponential decay of correlations for Hölder observables with respect to any Gibbs measure. The approach invented in 1997 by Dolgopyat, and further developed in our papers in 2011 and 2023, is substantially enhanced here, allowing to deal with the general case of arbitrary contact Anosov flows and arbitrary Gibbs measures. The results obtained here naturally apply to geodesic flows on compact Riemannian manifolds. As is now well-known, the strong spectral estimates for Ruelle operators and a well-established technique by Dolgopyat lead to exponential decay of correlations for Hölder continuous potentials. Other immediate consequences are: (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.

math.DS↗

arXiv2024

Lyapunov Exponents for Open Billiards in the Exterior of Balls

In this paper we consider the billiard flow in the exterior of several (at least three) balls in $\R^3$ with centres lying on a plane. We assume that the balls satisfy the no eclipse condition (H) and their radii are small compared to the distances between their centres. We prove that with respect to any Gibbs measure on the non-wandering set of the billiard flow the two positive Lyapunov exponents are different: $λ_1 > λ_2 > 0$.

math.DS↗

arXiv2023

Rigidity of Travelling Times for Strictly Convex Obstacles in Riemannian Manifolds

Let $K$ and $L$ be two disjoint unions of strictly convex obstacles contained within a Riemannian manifold with boundary $S$ of dimension $m\geq 2$. The sets of travelling times $\mathcal{T}_K$ and $\mathcal{T}_L$ of $K$ and $L$, respectively, are composed of triples $(x,y,t)\in\partial S\times\partial S\times\mathbb{R}^+$ where $t$ is the length of a billiard trajectory with endpoints $x$ and $y$ that reflects elastically on $K$ (or $L$ for $(x,y,t)\in\mathcal{T}_L$). It has been shown (arXiv:2309.11141) that (under some natural curvature bounds on $S$) if $\mathcal{T}_K=\mathcal{T}_L$ and $K$ and $L$ were equivalent up to tangency then $K = L$. In this paper we remove this requirement for $K$ and $L$, and show that if $\mathcal{T}_K = \mathcal{T}_L$ then $K = L$ whenever $m\geq 3$.

math.DG↗

arXiv2023

Uniqueness of Obstacles in Riemannian Manifolds from Travelling Times

Suppose that $K$ and $L$ are two disjoint unions of strictly convex obstacles with the same set of travelling times, contained in an $n$-dimensional Riemannian manifold $M$ (where $n\geq2$). Under some natural curvature conditions on $M$, and provided that no geodesic intersects more than two components in $K$ or $L$, we show that $K = L$.

math.DG↗

arXiv2023

Recovering Obstacles from their Travelling Times

Noakes and Stoyanov (2021) introduced a method of recovering strictly convex planar obstacles from their set of travelling times. We provide an extension of this construction for obstacles on Riemannian surfaces under some general curvature conditions. It is required that no smooth geodesic intersect more than two obstacles.

math.DG↗

arXiv2023

Semi-continuity of Oseledets flags and Pesin sets with exponentially small tails

Let $f$ be an invertible transitive subshift of finite type over a bilateral symbol space $X$, let $μ$ be a Gibbs measure for $f$ determined by a Hölder continuous potential on $X$, and let $A$ be an invertible continuous linear cocycle over $f$ acting on a continuous $\R^d$-bundle $E$ over $X$ with Lyapunov exponents $λ_k < λ_{k-1} < \ldots < λ_1$ such that $A^{-1}$ is continuous as well. We prove that if the Oseledets flags $F_j(x) = E_j(x) \oplus E_{j-1}(x) \oplus \cdots \oplus E_1(x)$ depend upper semi-continuously on $x \in X$, then there exists a Pesin set with exponentially small tails for $μ$.

math.DS↗

arXiv2022

Linearisability of divergence-free fields along invariant 2-tori

We find conditions under which the restriction of a divergence-free vector field $B$ to an invariant toroidal surface $S$ is linearisable. The main results are similar in conclusion to Arnold's Structure Theorems but require weaker assumptions than the commutation $[B,\nabla\times B] = 0$. Relaxing the need for a first integral of $B$ (also known as a flux function), we assume the existence of a solution $u : S \to \mathbb{R}$ to the cohomological equation $B|_S(u) = \partial_n B$ on a toroidal surface $S$ mutually invariant to $B$ and $\nabla \times B$. The right hand side $\partial_n B$ is a normal surface derivative available to vector fields tangent to $S$. In this situation, we show that the field $B$ on $S$ is either identically zero or nowhere vanishing with $B|_S/\|B\|^2 |_S$ being linearisable. We are calling the latter the semi-linearisability of $B$ (with proportionality $\|B\|^2 |_S$). The non-vanishing property relies on Bers' results in pseudo-analytic function theory about a generalised Laplace-Beltrami equation arising from Witten cohomology deformation. With the use of de Rham cohomology, we also point out a Diophantine integral condition where one can conclude that $B|_S$ itself is linearisable. The linearisability of $B|_S$ is fundamental to the so-called magnetic coordinates, which are central to the theory of magnetically confined plasmas.

math.DG↗

arXiv2021

Integrability of normal distributions Part 2: Neat foliations by manifolds with boundary

This paper completes the foundations of neatly integrable normal distribution theory on manifolds with boundary. Normal distributions are those which contain vectors transverse to the boundary along its entirety. The theory is observed to be entirely analogous with the theory of integrable distributions on manifolds due to Stefan and Sussmann. The main result is a one-to-one correspondence between so-called neatly integrable normal distributions and neat foliations by manifolds with boundary. Neat foliations are allowed to have non-constant dimension and the leaves have boundary contained in the ambient boundary. The leaves satisfy a characteristic property formally identical to that of weakly embedded submanifolds except in the category of manifolds with boundary.

math.DG↗

arXiv2021

A Stefan-Sussmann theorem for normal distributions on manifolds with boundary

An analogue of the Stefan-Sussmann Theorem on manifolds with boundary is proven for normal distributions. These distributions contain vectors transverse to the boundary along its entirety. Plain integral manifolds are not enough to "integrate" a normal distribution; the next best "integrals" are so-called neat integral manifolds with boundary. The conditions on the distribution for this integrability is expressed in terms of adapted collars and integrability of a pulled-back distribution on the interior and on the boundary.

math.DG↗

arXiv2021

Metric vs topological receptive entropy of semigroup actions

We study the receptive metric entropy for semigroup actions on probability spaces, inspired by a similar notion of topological entropy introduced by Hofmann and Stoyanov. We analyze its basic properties and its relation with the classical metric entropy. In the case of semigroup actions on compact metric spaces we compare the receptive metric entropy with the receptive topological entropy looking for a Variational Principle. With this aim we propose several characterizations of the receptive topological entropy. Finally we introduce a receptive local metric entropy inspired by a notion by Bowen generalized in the classical setting of amenable group actions by Zheng and Chen, and we prove partial versions of the Brin-Katok Formula and the local Variational Principle.

math.DS↗

arXiv2020

Convex Obstacles from Travelling Times

A construction is given for the recovery of a disjoint union of strictly convex smooth planar obstacles from travelling-time information. The obstacles are required to be such that no Euclidean line meets more than two of them.

math.DS↗

arXiv2020

Sharp large deviations for hyperbolic flows

For hyperbolic flows $φ_t$ we examine the Gibbs measure of points $w$ for which $$\int_0^T G(φ_t w) dt - a T \in (- e^{-εn}, e^{- εn})$$ as $n \to \infty$ and $T \geq n$, provided $ε> 0$ is sufficiently small. This is similar to local central limit theorems. The fact that the interval $(- e^{-εn}, e^{- εn})$ is exponentially shrinking as $n \to \infty$ leads to several difficulties. Under some geometric assumptions we establish a sharp large deviation result with leading term $C(a) ε_n e^{γ(a) T}$ and rate function $γ(a) \leq 0.$ The proof is based on the spectral estimates for the iterations of the Ruelle operators with two complex parameters and on a new Tauberian theorem for sequence of functions $g_n(t)$ having an asymptotic as $ n \to \infty$ and $t \geq n.$

math.DS↗

arXiv2020

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 \cite{D1} and further developed in \cite{St2} 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 \cite{GSt} 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.

math.DS↗

arXiv2020

Travelling Times in Scattering by Obstacles in Curved Space

We consider travelling times of billiard trajectories in the exterior of an obstacle K on a two-dimensional Riemannian manifold M. We prove that given two obstacles with almost the same travelling times, the generalised geodesic flows on the non-trapping parts of their respective phase-spaces will have a time-preserving conjugacy. Moreover, if M has non-positive sectional curvature we prove that if K and L are two obstacles with strictly convex boundaries and almost the same travelling times then K and L are identical.

math.DG↗

arXiv2019

Lens Rigidity in Scattering by Unions of Strictly Convex Bodies in $\R^2$

It was proved in \cite{NS1} that obstacles $K$ in $\R^d$ that are finite disjoint unions of strictly convex domains with $C^3$ boundaries are uniquely determined by the travelling times of billiard trajectories in their exteriors and also by their so called scattering length spectra. However the case $d = 2$ is not properly covered in \cite{NS1}. In the present paper we give a separate different proof of the same result in the case $d = 2$.

math.DS↗

arXiv2018

Spectral estimates for Ruelle operators with two parameters and sharp large deviations

We obtain spectral estimates for the iterations of Ruelle operator $L_{f + (a + ıb)τ+ (c + ıd) g}$ with two complex parameters and Hölder functions $f,\: g$ generalizing the case $\Pr(f) =0$ studied in [PeS2]. As an application we prove a sharp large deviation theorem concerning exponentially shrinking intervals which improves the result in [PeS1].

math.DS↗

arXiv2017

Lens rigidity in scattering by non-trapping obstacles

We prove that if two non-trapping obstacles in $\mathbb{R}^n$ satisfy some rather weak non-degeneracy conditions and the scattering rays in their exteriors have (almost) the same travelling times or (almost) the same scattering length spectrum, then they coincide.

math-ph↗

arXiv2017

On Gibbs measures and spectra of Ruelle transfer operators

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.

math.DS↗