# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2020335

## Recurrence for measurable semigroup actions

 Institute for Information Transmission Problems RAS (Kharkevich Institute), National Research University "Higher School of Economics", Moscow, Russia

Received  January 2020 Revised  August 2020 Published  October 2020

We study qualitative properties of the set of recurrent points of finitely generated free semigroups of measurable maps. In the case of a single generator the classical Poincare recurrence theorem shows that these properties are closely related to the presence of an invariant measure. Curious, but otherwise it turns out to be possible that almost all points are recurrent, while there is an wandering set of positive (non-invariant) measure. For a general semigroup the assumption about the common invariant measure for all generators looks somewhat unnatural (despite being widely used). Instead we give abstract conditions (of conservativity type) for this problem and propose a weaker version of the recurrent property. Technically, the problem is reduced to the analysis of the recurrence of a specially constructed Markov process. Questions of inheritance of the recurrence property from the semigroup generators to the entire semigroup and vice versa are studied in detail and we demonstrate that this inheritance might be rather unexpected.

Citation: Michael Blank. Recurrence for measurable semigroup actions. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020335
##### References:

show all references

##### References:
Graphs of the maps $T_i$ in the Example 5
Graphs of the maps $T_i$ in the Example 6
Graphs of the maps $T_i$ in the Example 7
 [1] Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002 [2] Yangjian Sun, Changjian Liu. The Poincaré bifurcation of a SD oscillator. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1565-1577. doi: 10.3934/dcdsb.2020173 [3] Ágota P. Horváth. Discrete diffusion semigroups associated with Jacobi-Dunkl and exceptional Jacobi polynomials. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021002 [4] Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296 [5] Indranil Chowdhury, Gyula Csató, Prosenjit Roy, Firoj Sk. Study of fractional Poincaré inequalities on unbounded domains. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020394 [6] Angelica Pachon, Federico Polito, Costantino Ricciuti. On discrete-time semi-Markov processes. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1499-1529. doi: 10.3934/dcdsb.2020170 [7] Zsolt Saffer, Miklós Telek, Gábor Horváth. Analysis of Markov-modulated fluid polling systems with gated discipline. Journal of Industrial & Management Optimization, 2021, 17 (2) : 575-599. doi: 10.3934/jimo.2019124 [8] Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444 [9] Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048 [10] Haodong Yu, Jie Sun. Robust stochastic optimization with convex risk measures: A discretized subgradient scheme. Journal of Industrial & Management Optimization, 2021, 17 (1) : 81-99. doi: 10.3934/jimo.2019100 [11] Paul A. Glendinning, David J. W. Simpson. A constructive approach to robust chaos using invariant manifolds and expanding cones. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020409 [12] Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324 [13] Xin Zhao, Tao Feng, Liang Wang, Zhipeng Qiu. Threshold dynamics and sensitivity analysis of a stochastic semi-Markov switched SIRS epidemic model with nonlinear incidence and vaccination. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021010 [14] Azmy S. Ackleh, Nicolas Saintier. Diffusive limit to a selection-mutation equation with small mutation formulated on the space of measures. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1469-1497. doi: 10.3934/dcdsb.2020169

2019 Impact Factor: 1.338