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Yakov Pesin

Publications and source records attributed to Yakov Pesin.

17 recordsLinked to original sources
arXiv2024

Complete regularity of linear cocycles and the Baire category of the set of Lyapunov-Perron regular points

Given a continuous linear cocycle A over a homeomorphism f of a compact metric space X, we investigate its set R of Lyapunov-Perron regular points, that is, the collection of trajectories of f that obey the conclusions of the Multiplicative Ergodic Theorem. We obtain results roughly saying that the set R is of first Baire category (i.e., meager) in X, unless some rigid structure is present. In some settings, this rigid structure forces the Lyapunov exponents to be defined everywhere and to be independent of the point; that is what we call complete regularity.

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arXiv2021

SRB measures and Young towers for surface diffeomorphisms

We give geometric conditions that are necessary and sufficient for the existence of Sinai-Ruelle-Bowen (SRB) measures for $C^{1+α}$ surface diffeomorphisms, thus proving a version of the Viana conjecture. As part of our argument we give an original method for constructing first return Young towers, proving that every hyperbolic measure, and in particular every SRB measure, can be lifted to such a tower. This method relies on a new general result on hyperbolic branches and shadowing for pseudo-orbits in nonuniformly hyperbolic sets which is of independent interest.

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arXiv2020

Equilibrium measures for some partially hyperbolic systems

We study thermodynamic formalism for topologically transitive partially hyperbolic systems in which the center-stable bundle satisfies a bounded expansion property, and show that every potential function satisfying the Bowen property has a unique equilibrium measure. Our method is to use tools from geometric measure theory to construct a suitable family of reference measures on unstable leaves as a dynamical analogue of Hausdorff measure, and then show that the averaged pushforwards of these measures converge to a measure that has the Gibbs property and is the unique equilibrium measure.

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arXiv2020

Area preserving surface diffeomorphisms with polynomial decay of correlations are ubiquitous

We show that any surface admits an area preserving $C^{1+β}$ diffeomorphism with non-zero Lyapunov exponents which is Bernoulli and has polynomial decay of correlations. We establish both upper and lower polynomial bounds on correlations. In addition, we show that this diffeomorphism satisfies the Central Limit Theorem and has the Large Deviation Property. Finally, we show that the diffeomorphism we constructed possesses a unique hyperbolic Bernoulli measure of maximal entropy with respect to which it has exponential decay of correlations.

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arXiv2019

Dimension Estimates for Non-conformal Repellers and Continuity of Sub-additive Topological Pressure

Given a non-conformal repeller $Λ$ of a $C^{1+γ}$ map, we study the Hausdorff dimension of the repeller and continuity of the sub-additive topological pressure for the sub-additive singular valued potentials. Such a potential always possesses an equilibrium state. We then use a substantially modified version of Katok's approximating argument, to construct a compact invariant set on which the corresponding dynamical quantities (such as Lyapunov exponents and metric entropy) are close to that of the equilibrium measure. This allows us to establish continuity of the sub-additive topological pressure and obtain a sharp lower bound of the Hausdorff dimension of the repeller. The latter is given by the zero of the super-additive topological pressure

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arXiv2019

The essential coexistence phenomenon in Hamiltonian dynamics

We construct an example of a Hamiltonian flow $f^t$ on a $4$-dimensional smooth manifold $\mathcal{M}$ which after being restricted to an energy surface $\mathcal{M}_e$ demonstrates essential coexistence of regular and chaotic dynamics that is there is an open and dense $f^t$-invariant subset $U\subset\mathcal{M}_e$ such that the restriction $f^t|U$ has non-zero Lyapunov exponents in all directions (except the direction of the flow) and is a Bernoulli flow while on the boundary $\partial U$, which has positive volume all Lyapunov exponents of the system are zero.

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arXiv2018

Equilibrium states in dynamical systems via geometric measure theory

Given a dynamical system with a uniformly hyperbolic (`chaotic') attractor, the physically relevant Sinai-Ruelle-Bowen (SRB) measure can be obtained as the limit of the dynamical evolution of the leaf volume along local unstable manifolds. We extend this geometric construction to the substantially broader class of equilibrium states corresponding to Hölder continuous potentials; these states arise naturally in statistical physics and play a crucial role in studying stochastic behavior of dynamical systems. The key step in our construction is to replace leaf volume with a reference measure that is obtained from a Carathéodory dimension structure via an analogue of the construction of Hausdorff measure. In particular, we give a new proof of existence and uniqueness of equilibrium states that does not use standard techniques based on Markov partitions or the specification property; our approach can be applied to systems that do not have Markov partitions and do not satisfy the specification property.

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arXiv2017

Thermodynamics of the Katok Map

We effect thermodynamical formalism for the non-uniformly hyperbolic $C^\infty$ map of the two dimensional torus known as the Katok map. It is a slowdown of a linear Anosov map near the origin and it is a local (but not small) perturbation. We prove the existence of equilibrium measures for any continuous potential function and obtain uniqueness of equilibrium measures associated to the geometric $t$-potential $φ_t=-t\log |df|_{E^u(x)}|$ for any $t\in(t_0,\infty)$, $t\neq 1$ where $E^u(x)$ denotes the unstable direction. We show that $t_0$ tends to $-\infty$ as the size of the perturbation tends to zero. Finally, we establish exponential decay of correlations as well as the Central Limit Theorem for the equilibrium measures associated to $φ_t$ for all values of $t\in (t_0, 1)$.

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arXiv2016

The geometric approach for constructing Sinai-Ruelle-Bowen measures

An important class of `physically relevant' measures for dynamical systems with hyperbolic behavior is given by Sinai-Ruelle-Bowen (SRB) measures. We survey various techniques for constructing SRB measures and studying their properties, paying special attention to the geometric `push-forward' approach. After describing this approach in the uniformly hyperbolic setting, we review recent work that extends it to non-uniformly hyperbolic systems.

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arXiv2016

Non-stationary non-uniform hyperbolicity: SRB measures for dissipative maps

We prove the existence of SRB measures for diffeomorphisms where a positive volume set of initial conditions satisfy an "effective hyperbolicity" condition that guarantees certain recurrence conditions on the iterates of Lebesgue measure. We give examples of systems that do not admit a dominated splitting but can be shown to have SRB measures using our methods.

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arXiv2014

Thermodynamics of Towers of Hyperbolic Type

We introduce a class of continuous maps $f$ of a compact topological space $X$ admitting inducing schemes of hyperbolic type and describe the associated tower constructions. We then establish a thermodynamic formalism, i.e., we describe a class of real-valued potential functions $φ$ on $X$ such that $f$ possess a unique equilibrium measure $μ_φ$, associated to each $φ$, which minimizes the free energy among the measures that are liftable to the tower. We also describe some ergodic properties of equilibrium measures including decay of correlations and the central limit theorem. We then study the liftability problem and show that under some additional assumptions on the inducing scheme every measure that charges the base of the tower and has sufficiently large entropy is liftable. Our results extend those obtained in [PS05, PS08] for inducing schemes of expanding types and apply to certain multidimensional maps. Applications include obtaining the thermodynamic formalism for Young's diffeomorphisms, the Hénon family at the first birfucation and the Katok map. In particular for the Katok map we obtain the exponential decay of correlations for equilibrium measures associated to the geometric potentials with $0\le t<1$.

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arXiv2013

Hadamard-Perron theorems and effective hyperbolicity

We prove several new versions of the Hadamard-Perron Theorem, which relates infinitesimal dynamics to local dynamics for a sequence of local diffeomorphisms, and in particular establishes the existence of local stable and unstable manifolds. Our results imply the classical Hadamard-Perron Theorem in both its uniform and non-uniform versions, but also apply much more generally. We introduce a notion of "effective hyperbolicity" and show that if the rate of effective hyperbolicity is asymptotically positive, then the local manifolds are well-behaved with positive asymptotic frequency. By applying effective hyperbolicity to finite orbit segments, we prove a closing lemma whose conditions can be verified with a finite amount of information.

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arXiv2008

Equilibrium Measures for Maps with Inducing Schemes

We introduce a class of continuous maps f of a compact metric space I admitting inducing schemes and describe the tower constructions associated with them. We then establish a thermodynamical formalism, i.e., describe a class of real-valued potential functions ϕon I which admit unique equilibrium measures μ_ϕminimizing the free energy for a certain class of measures. We also describe ergodic properties of equilibrium measures including decay of correlation and the Central Limit Theorem. Our results apply in particular to some one-dimensional unimodal and multimodal maps as well as to multidimensional nonuniformly hyperbolic maps admitting Young's tower. Examples of potential functions to which our theory applies include ϕ_t=-t\log|df| with t\in(t_0, t_1) for some t_0<1<t_1. In the particular case of S-unimodal maps we show that one can choose t_0<0 and that the class of measures under consideration comprises all invariant Borel probability measures. Thus our results establish existence and uniqueness of both the measure of maximal entropy (by a different method than Hofbauer) and the absolutely continuous invariant measure extending results by Bruin and Keller for the parameters under consideration.

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arXiv2007

Lifting Measures to Inducing Schemes

In this paper we study the liftability property for piecewise continuous maps of compact metric spaces, which admit inducing schemes in the sense of Pesin and Senti [PS05, PS06]. We show that under some natural assumptions on the inducing schemes - which hold for many known examples - any invariant ergodic Borel probability measure of sufficiently large entropy can be lifted to the tower associated with the inducing scheme. The argument uses the construction of connected Markov extensions due to Buzzi [Buz99], his results on the liftability of measures of large entropy, and a generalization of some results by Bruin [Bru95] on relations between inducing schemes and Markov extensions. We apply our results to study the liftability problem for one-dimensional cusp maps (in particular, unimodal and multimodal maps) and for some multidimensional maps.

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arXiv2005

Thermodynamical Formalism Associated with Inducing Schemes for One-dimensional Maps

For a smooth map f of a compact interval I admitting an inducing scheme we establish a thermodynamical formalism, i.e., describe a class of real-valued potential functions $ϕ$ on I which admit a unique equilibrium measure $μ_ϕ$. Our results apply to unimodal maps corresponding to a positive Lebesgue measure set of parameters in a one-parameter transverse family.

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arXiv1999

Dimension and product structure of hyperbolic measures

We prove that every hyperbolic measure invariant under a C^{1+α} diffeomorphism of a smooth Riemannian manifold possesses asymptotically ``almost'' local product structure, i.e., its density can be approximated by the product of the densities on stable and unstable manifolds up to small exponentials. This has not been known even for measures supported on locally maximal hyperbolic sets. Using this property of hyperbolic measures we prove the long-standing Eckmann-Ruelle conjecture in dimension theory of smooth dynamical systems: the pointwise dimension of every hyperbolic measure invariant under a C^{1+α} diffeomorphism exists almost everywhere. This implies the crucial fact that virtually all the characteristics of dimension type of the measure (including the Hausdorff dimension, box dimension, and information dimension) coincide. This provides the rigorous mathematical justification of the concept of fractal dimension for hyperbolic measures.

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