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Register a new userPoisson processes for subsystems of finite type in symbolic dynamics
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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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Let $DeltasubsetneqV$ be a proper subset of the vertices $V$ of the defining graph of an aperiodic shift of finite type $(Sigma_{A}^{+},S)$. Let $Delta_{n}$ be the union of cylinders in $Sigma_{A}^{+}$ corresponding to the points $x$ for which the first $n$-symbols of $x$ belong to $Delta$ and let $mu$ be an equilibrium state of a Holder potential $phi$ on $Sigma_{A}^{+}$. We know that $mu(Delta_{n})$ converges to zero as $n$ diverges. We study the asymptotic behaviour of $mu(Delta_{n})$ and compare it with the pressure of the restriction of $phi$ to $Sigma_{Delta}$. The present paper extends some results in cite{CCC} to the case when $Sigma_{Delta}$ is irreducible and periodic. We show an explicit example where the asymptotic behaviour differs from the aperiodic case.
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.
In this paper, we use a unified framework to study Poisson stable (including stationary, periodic, quasi-periodic, almost periodic, almost automorphic, Birkhoff recurrent, almost recurrent in the sense of Bebutov, Levitan almost periodic, pseudo-periodic, pseudo-recurrent and Poisson stable) solutions for semilinear stochastic differential equations driven by infinite dimensional Levy noise with large jumps. Under suitable conditions on drift, diffusion and jump coefficients, we prove that there exist solutions which inherit the Poisson stability of coefficients. Further we show that these solutions are globally asymptotically stable in square-mean sense. Finally, we illustrate our theoretical results by several examples.
In this survey we will present the symbolic extension theory in topological dynamics, which was built over the past twenty years.
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.
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