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Register a new userSymbolic dynamics for nonuniformly hyperbolic systems
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Authors
Yuri Lima
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This survey describes the recent advances in the construction of Markov partitions for nonuniformly hyperbolic systems. One important feature of this development comes from a finer theory of nonuniformly hyperbolic systems, which we also describe. The Markov partition defines a symbolic extension that is finite-to-one and onto a non-uniformly hyperbolic locus, and this provides dynamical and statistical consequences such as estimates on the number of closed orbits and properties of equilibrium measures. The class of systems includes diffeomorphisms, flows, and maps with singularities.
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We construct Markov partitions for non-invertible and/or singular nonuniformly hyperbolic systems defined on higher dimensional Riemannian manifolds. The generality of the setup covers classical examples not treated so far, such as geodesic flows in closed manifolds, multidimensional billiard maps, and Viana maps, and includes all the recent results of the literature. We also provide a wealth of applications.
This work constructs symbolic dynamics for non-uniformly hyperbolic surface maps with a set of discontinuities $D$. We allow the derivative of points nearby $D$ to be unbounded, of the order of a negative power of the distance to $D$. Under natural geometrical assumptions on the underlying space $M$, we code a set of non-uniformly hyperbolic orbits that do not converge exponentially fast to $D$. The results apply to non-uniformly hyperbolic planar billiards, e.g. Bunimovich stadia.
A major motivation for the development of semigroup theory was, and still is, its applications to the study of formal languages. Therefore, it is not surprising that the correspondence $mathcal Xmapsto B(mathcal X)$, associating to each symbolic dynamical system $mathcal X$ the formal language $B(mathcal X)$ of its blocks, entails a connection between symbolic dynamics and semigroup theory. In this article we survey some developments on this connection, since when it was noticed in an article by Almeida, published in the CIM bulletin, in 2003.
Given a piecewise $C^{1+beta}$ map of the interval, possibly with critical points and discontinuities, we construct a symbolic model for invariant probability measures with nonuniform expansion that do not approach the critical points and discontinuities exponentially fast almost surely. More specifically, we code the lift of these measures in the natural extension of the map.
Let $DeltasubsetneqV$ be a proper subset of the vertices $V$ of the defining graph of an irreducible and aperiodic shift of finite type $(Sigma_{A}^{+},S)$. Let $Sigma_{Delta}$ be the subshift of allowable paths in the graph of $Sigma_{A}^{+}$ which only passes through the vertices of $Delta$. For a random point $x$ chosen with respect to an equilibrium state $mu$ of a Holder potential $phi$ on $Sigma_{A}^{+}$, let $tau_{n}$ be the point process defined as the sum of Dirac point masses at the times $k>0$, suitably rescaled, for which the first $n$-symbols of $S^k x$ belong to $Delta$. We prove that this point process converges in law to a marked Poisson point process of constant parameter measure. The scale is related to the pressure of the restriction of $phi$ to $Sigma_{Delta}$ and the parameters of the limit law are explicitly computed.
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