Hitting Time in Regular Sets and Logarithm Law for Rapidly Mixing Dynamical Systems
نویسنده
چکیده
We prove that if a system has superpolynomial (faster than any power law) decay of correlations (with respect to Lipschitz observables), then the time τ(x, Sr) is needed for a typical point x to enter for the first time a set Sr = {x : f(x) ≤ r} which is a sublevel of a Lipschitz function f scales as 1 μ(Sr) i.e., lim r→0 log τ(x, Sr) − log r = lim r→0 log μ(Sr) log r . This generalizes a previous result obtained for balls. We will also consider relations with the return time distributions, an application to observed systems and to the geodesic flow in negatively curved manifolds.
منابع مشابه
Universal Hitting Time Statistics for Integrable Flows
The perceived randomness in the time evolution of “chaotic” dynamical systems can be characterized by universal probabilistic limit laws, which do not depend on the fine features of the individual system. One important example is the Poisson law for the times at which a particle with random initial data hits a small set. This was proved in various settings for dynamical systems with strong mixi...
متن کاملThe Central Limit Theorem for uniformly strong mixing measures
The theorem of Shannon-McMillan-Breiman states that for every generating partition on an ergodic system, the exponential decay rate of the measure of cylinder sets equals the metric entropy almost everywhere (provided the entropy is finite). In this paper we prove that the measure of cylinder sets are lognormally distributed for strongly mixing systems and infinite partitions and show that the ...
متن کامل2 00 8 Dimension and hitting time in rapidly mixing systems
We prove that if a system has superpolynomial (faster than any power law) decay of correlations then the time τ r (x, x 0) needed for a typical point x to enter for the first time a ball B(x 0 , r) centered in x 0 , with small radius r scales as the local dimension at x 0 , i.e. lim r→0 log τ r (x, x 0) − log r = d µ (x 0). This result is obtained by proving a kind of dynamical Borel-Cantelli l...
متن کاملHitting Time Statistics and Extreme Value Theory
We consider discrete time dynamical system and show the link between Hitting Time Statistics (the distribution of the first time points land in asymptotically small sets) and Extreme Value Theory (distribution properties of the partial maximum of stochastic processes). This relation allows to study Hitting Time Statistics with tools from Extreme Value Theory, and vice versa. We apply these resu...
متن کاملPoisson Processes for Susbsystems of Finite Type in Symbolic Dynamical Systems
Let ∆ ( V be a proper subset of the vertices V of the defining graph of an irreducible and aperiodic shift of finite type (Σ A , T ). Let Σ∆ be the subshift of allowable paths in the graph of Σ A which only passes through the vertices of ∆. For a random point x chosen with respect to an equilibrium state μ of a Hölder potential φ on Σ A , let τn be the point process defined as the sum of Dirac ...
متن کامل