American Institute of Mathematical Sciences

Advanced
July  2004, 10(3): 589-616. doi: 10.3934/dcds.2004.10.589

The limiting distribution and error terms for return times of dynamical systems

Nicolai Haydn 1, and  Sandro Vaienti 2,

1. 

Department of Mathematics, University of Southern California, Los Angeles, 90089-1113, United States
2. 

Phymat, Université de Toulon, Centre de Physique Théorique and FRUMAM, Luminy Case 907, 13288 Marseille Cedex 09, France

Received  October 2002 Revised  July 2003 Published  January 2004

We develop a new framework that allows to prove that the limiting distribution of return times for a large class of mixing dynamical systems are Poisson distributed. We demonstrate our technique in several settings and obtain more general results than previously has been proven. We also obtain error estimates. For $\phi$-mixing maps we obtain a close to exhausting description of return times. For $(\phi,f)$-mixing maps it is shown how the separation function affects error estimates for the limiting distribution. As examples of $(\phi,f)$-mixing we prove that for piecewise invertible maps and for rational maps return times are in the limit Poisson distributed.
Keywords: ö-mixing, rational maps, Poisson distribution, Axiom-A systems, Mixing dynamical systems, return times statistics, piecewise expanding maps..
Mathematics Subject Classification: 37A25, 37A50, 37F3.
Citation: Nicolai Haydn, Sandro Vaienti. The limiting distribution and error terms for return times of dynamical systems. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 589-616. doi: 10.3934/dcds.2004.10.589
[1]

Nigel P. Byott, Mark Holland, Yiwei Zhang. On the mixing properties of piecewise expanding maps under composition with permutations. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3365-3390. doi: 10.3934/dcds.2013.33.3365
[2]

Ralf Spatzier, Lei Yang. Exponential mixing and smooth classification of commuting expanding maps. Journal of Modern Dynamics, 2017, 11: 263-312. doi: 10.3934/jmd.2017012
[3]

Marco Lenci. Uniformly expanding Markov maps of the real line: Exactness and infinite mixing. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3867-3903. doi: 10.3934/dcds.2017163
[4]

Richard Sharp, Anastasios Stylianou. Statistics of multipliers for hyperbolic rational maps. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1225-1241. doi: 10.3934/dcds.2021153
[5]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217
[6]

Rui Kuang, Xiangdong Ye. The return times set and mixing for measure preserving transformations. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 817-827. doi: 10.3934/dcds.2007.18.817
[7]

Benoît Saussol. Recurrence rate in rapidly mixing dynamical systems. Discrete and Continuous Dynamical Systems, 2006, 15 (1) : 259-267. doi: 10.3934/dcds.2006.15.259
[8]

Ian Melbourne, Dalia Terhesiu. Mixing properties for toral extensions of slowly mixing dynamical systems with finite and infinite measure. Journal of Modern Dynamics, 2018, 12: 285-313. doi: 10.3934/jmd.2018011
[9]

Viviane Baladi, Daniel Smania. Smooth deformations of piecewise expanding unimodal maps. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 685-703. doi: 10.3934/dcds.2009.23.685
[10]

Damien Thomine. A spectral gap for transfer operators of piecewise expanding maps. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 917-944. doi: 10.3934/dcds.2011.30.917
[11]

Magnus Aspenberg, Viviane Baladi, Juho Leppänen, Tomas Persson. On the fractional susceptibility function of piecewise expanding maps. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 679-706. doi: 10.3934/dcds.2021133
[12]

Michal Málek, Peter Raith. Stability of the distribution function for piecewise monotonic maps on the interval. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2527-2539. doi: 10.3934/dcds.2018105
[13]

Claudio Bonanno, Marco Lenci. Pomeau-Manneville maps are global-local mixing. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1051-1069. doi: 10.3934/dcds.2020309
[14]

Daniel Schnellmann. Typical points for one-parameter families of piecewise expanding maps of the interval. Discrete and Continuous Dynamical Systems, 2011, 31 (3) : 877-911. doi: 10.3934/dcds.2011.31.877
[15]

Peyman Eslami. Inducing schemes for multi-dimensional piecewise expanding maps. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 353-368. doi: 10.3934/dcds.2021120
[16]

Renaud Leplaideur, Benoît Saussol. Large deviations for return times in non-rectangle sets for axiom a diffeomorphisms. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 327-344. doi: 10.3934/dcds.2008.22.327
[17]

Malo Jézéquel. Parameter regularity of dynamical determinants of expanding maps of the circle and an application to linear response. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 927-958. doi: 10.3934/dcds.2019039
[18]

Guizhen Cui, Yan Gao. Wandering continua for rational maps. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1321-1329. doi: 10.3934/dcds.2016.36.1321
[19]

Carlangelo Liverani. A footnote on expanding maps. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3741-3751. doi: 10.3934/dcds.2013.33.3741
[20]

Richard Miles, Thomas Ward. A directional uniformity of periodic point distribution and mixing. Discrete and Continuous Dynamical Systems, 2011, 30 (4) : 1181-1189. doi: 10.3934/dcds.2011.30.1181

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (63)
  • HTML views (0)
  • Cited by (16)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy