# American Institute of Mathematical Sciences

July  2017, 37(7): 3867-3903. doi: 10.3934/dcds.2017163

## Uniformly expanding Markov maps of the real line: Exactness and infinite mixing

 1 Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5,40126 Bologna, Italy 2 Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Via Irnerio 46,40126 Bologna, Italy

Received  January 2015 Revised  February 2017 Published  April 2017

Fund Project: The author was partially supported by PRIN Grant 2012AZS52J 001 (MIUR, Italy)

We give a fairly complete characterization of the exact components of a large class of uniformly expanding Markov maps of ${\mathbb{R}}$. Using this result, for a class of  $\mathbb{Z}$-invariant maps and finite modifications thereof, we prove certain properties of infinite mixing recently introduced by the author.

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

show all references

##### References:
A uniformly expanding Markov map ${\mathbb{R}} \to {\mathbb{R}}$.
An example of a quasi-lift of an expanding circle map.
A finite modification of a quasi-lift of a circle map, constructed with the procedure given in Section 3.2, for the case $\mu_o = m$.
A map $T$ associated a random walk. The marks on the abscissa indicate the Markov intervals $I_{jk}$, while those on the ordinate represent the intervals $[k, k+1]$.
A rough sketch of the map $T$ of Counterexample 2. The bold segments on the abscissa indicate the set $X$. The bold parts of the graph of $T$ represent $T|_X$, which is invertible.
 [1] Tomás Caraballo, Juan C. Jara, José A. Langa, José Valero. Morse decomposition of global attractors with infinite components. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2845-2861. doi: 10.3934/dcds.2015.35.2845 [2] Mrinal Kanti Roychowdhury. Quantization coefficients for ergodic measures on infinite symbolic space. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2829-2846. doi: 10.3934/dcds.2014.34.2829 [3] Kathryn Lindsey, Rodrigo Treviño. Infinite type flat surface models of ergodic systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5509-5553. doi: 10.3934/dcds.2016043 [4] Cristina Lizana, Vilton Pinheiro, Paulo Varandas. Contribution to the ergodic theory of robustly transitive maps. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 353-365. doi: 10.3934/dcds.2015.35.353 [5] David Ralston, Serge Troubetzkoy. Ergodic infinite group extensions of geodesic flows on translation surfaces. Journal of Modern Dynamics, 2012, 6 (4) : 477-497. doi: 10.3934/jmd.2012.6.477 [6] Ryszard Rudnicki. An ergodic theory approach to chaos. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 757-770. doi: 10.3934/dcds.2015.35.757 [7] Thierry de la Rue. An introduction to joinings in ergodic theory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 121-142. doi: 10.3934/dcds.2006.15.121 [8] Jose F. Alves; Stefano Luzzatto and Vilton Pinheiro. Markov structures for non-uniformly expanding maps on compact manifolds in arbitrary dimension. Electronic Research Announcements, 2003, 9: 26-31. [9] Almut Burchard, Gregory R. Chambers, Anne Dranovski. Ergodic properties of folding maps on spheres. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1183-1200. doi: 10.3934/dcds.2017049 [10] Diogo Gomes, Levon Nurbekyan. An infinite-dimensional weak KAM theory via random variables. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6167-6185. doi: 10.3934/dcds.2016069 [11] Maxim Sølund Kirsebom. Extreme value theory for random walks on homogeneous spaces. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4689-4717. doi: 10.3934/dcds.2014.34.4689 [12] 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 [13] Xiongping Dai, Yu Huang, Mingqing Xiao. Realization of joint spectral radius via Ergodic theory. Electronic Research Announcements, 2011, 18: 22-30. doi: 10.3934/era.2011.18.22 [14] Jean-Pierre Conze, Y. Guivarc'h. Ergodicity of group actions and spectral gap, applications to random walks and Markov shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4239-4269. doi: 10.3934/dcds.2013.33.4239 [15] Marc Kesseböhmer, Sabrina Kombrink. A complex Ruelle-Perron-Frobenius theorem for infinite Markov shifts with applications to renewal theory. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 335-352. doi: 10.3934/dcdss.2017016 [16] Fernando J. Sánchez-Salas. Dimension of Markov towers for non uniformly expanding one-dimensional systems. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1447-1464. doi: 10.3934/dcds.2003.9.1447 [17] Guizhen Cui, Wenjuan Peng, Lei Tan. On the topology of wandering Julia components. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 929-952. doi: 10.3934/dcds.2011.29.929 [18] Nigel P. Byott, Mark Holland, Yiwei Zhang. On the mixing properties of piecewise expanding maps under composition with permutations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3365-3390. doi: 10.3934/dcds.2013.33.3365 [19] Oliver Jenkinson. Ergodic Optimization. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 197-224. doi: 10.3934/dcds.2006.15.197 [20] Rua Murray. Ulam's method for some non-uniformly expanding maps. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 1007-1018. doi: 10.3934/dcds.2010.26.1007

2018 Impact Factor: 1.143