# Number of visits in arbitrary sets for $\phi$-mixing dynamics

@inproceedings{Gallo2021NumberOV, title={Number of visits in arbitrary sets for \$\phi\$-mixing dynamics}, author={Sandro Gallo and Nicolai T. A. Haydn and Sandro Vaienti}, year={2021} }

It is well-known that, for sufficiently mixing dynamical systems, the number of visits to balls and cylinders of vanishing measure is approximately Poisson compound distributed in the Kac scaling. Here we extend this kind of results when the target set is an arbitrary set with vanishing measure in the case of φ-mixing systems. The error of approximation in total variation is derived using Stein-Chen method. An important part of the paper is dedicated to examples to illustrate the assumptions… Expand

#### Figures from this paper

#### References

SHOWING 1-10 OF 55 REFERENCES

Limiting distribution and error terms for the number of visits to balls in non-uniformly hyperbolic dynamical systems

- Mathematics
- 2014

We show that for systems that allow a Young tower construction with polynomially decaying correlations the return times to metric balls are in the limit Poisson distributed. We also provide error… Expand

Poisson approximation for the number of visits to balls in non-uniformly hyperbolic dynamical systems

- Mathematics, Physics
- Ergodic Theory and Dynamical Systems
- 2012

Abstract We study the number of visits to balls Br(x), up to time t/μ(Br(x)), for a class of non-uniformly hyperbolic dynamical systems, where μ is the Sinai–Ruelle–Bowen measure. Outside a set of… Expand

Spatio-temporal Poisson processes for visits to small sets

- Mathematics
- 2018

For many measure preserving dynamical systems $(\Omega,T,m)$ the successive hitting times to a small set is well approximated by a Poisson process on the real line. In this work we define a new… Expand

Return times distribution for Markov towers with decay of correlations

- Mathematics
- 2010

In this paper we prove two results. First we show that dynamical systems with a $\phi$-mixing measure have in the limit Poisson distributed return times almost everywhere. We use the Chen-Stein… Expand

Poisson law for some non-uniformly hyperbolic dynamical systems with polynomial rate of mixing

- Mathematics
- Ergodic Theory and Dynamical Systems
- 2015

We consider some non-uniformly hyperbolic invertible dynamical systems which are modeled by a Gibbs–Markov–Young tower. We assume a polynomial tail for the inducing time and a polynomial control of… Expand

Compound Poisson Approximation for Nonnegative Random Variables Via Stein's Method

- Mathematics
- 1992

The aim of this paper is to extend Stein's method to a compound Poisson distribution setting. The compound Poisson distributions of concern here are those of the form POIS$(\nu)$, where $\nu$ is a… Expand

Limiting Entry and Return Times Distribution for Arbitrary Null Sets

- Mathematics
- 2020

We describe an approach that allows us to deduce the limiting return times distribution for arbitrary sets to be compound Poisson distributed. We establish a relation between the limiting return… Expand

Rare event process and entry times distribution for arbitrary null sets on compact manifolds

- Mathematics
- 2019

We establish the general equivalence between rare event process for arbitrary continuous functions whose maximal values are achieved on non-trivial sets, and the entry times distribution for… Expand

Decay of Correlations for Non-Hölderian Dynamics. A Coupling Approach

- Mathematics, Physics
- 1998

We present an upper bound on the mixing rate of the equilibrium state of a dynamical system defined by the one-sided shift and a non Holder potential of summable variations. The bound follows from an… Expand

Convergence of Marked Point Processes of Excesses for Dynamical Systems

- Mathematics, Physics
- 2015

We consider stochastic processes arising from dynamical systems simply by evaluating an observable function along the orbits of the system and study marked point processes associated to extremal… Expand