# American Institute of Mathematical Sciences

April  2018, 38(4): 2093-2123. doi: 10.3934/dcds.2018086

## Invariance entropy, quasi-stationary measures and control sets

 Institut für Mathematik, Universität Augsburg, Universitätsstrasse 14,86159 Augsburg, Germany

Received  May 2017 Revised  October 2017 Published  January 2018

Fund Project: Research supported by DFG grant 124/19-2.

For control systems in discrete time, this paper discusses measure-theoretic invariance entropy for a subset Q of the state space with respect to a quasi-stationary measure obtained by endowing the control range with a probability measure. The main results show that this entropy is invariant under measurable transformations and that it is already determined by certain subsets of Q which are characterized by controllability properties.

Citation: Fritz Colonius. Invariance entropy, quasi-stationary measures and control sets. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2093-2123. doi: 10.3934/dcds.2018086
##### References:

show all references

##### References:
Extremal graphs for (24) and the set $[d(\alpha),0.5]$ in $Q = [0.2,0.5\dot{]}$ (here $A = 0.05,\sigma = 0.1$ and $\alpha = 0.08$)
Extremal graphs for (44) and the $W$-control sets $D_1(\alpha) = [a(\alpha),b(\alpha))$ and $D_2(\alpha) = [d(\alpha),0.7)$ in $Q = [0.1,0.7\dot {]}$ (here $A = 0.05,\sigma = 0.1$ and $\alpha = 0.08$)
 [1] Adriano Da Silva, Christoph Kawan. Invariance entropy of hyperbolic control sets. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 97-136. doi: 10.3934/dcds.2016.36.97 [2] Dietmar Szolnoki. Set oriented methods for computing reachable sets and control sets. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 361-382. doi: 10.3934/dcdsb.2003.3.361 [3] Christoph Kawan. Upper and lower estimates for invariance entropy. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 169-186. doi: 10.3934/dcds.2011.30.169 [4] Xueting Tian, Paulo Varandas. Topological entropy of level sets of empirical measures for non-uniformly expanding maps. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5407-5431. doi: 10.3934/dcds.2017235 [5] Peter E. Kloeden. Asymptotic invariance and the discretisation of nonautonomous forward attracting sets. Journal of Computational Dynamics, 2016, 3 (2) : 179-189. doi: 10.3934/jcd.2016009 [6] François Blanchard, Wen Huang. Entropy sets, weakly mixing sets and entropy capacity. Discrete & Continuous Dynamical Systems - A, 2008, 20 (2) : 275-311. doi: 10.3934/dcds.2008.20.275 [7] S. Astels. Thickness measures for Cantor sets. Electronic Research Announcements, 1999, 5: 108-111. [8] Michihiro Hirayama. Periodic probability measures are dense in the set of invariant measures. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1185-1192. doi: 10.3934/dcds.2003.9.1185 [9] Zhihong Xia. Hyperbolic invariant sets with positive measures. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 811-818. doi: 10.3934/dcds.2006.15.811 [10] Mario Roldan. Hyperbolic sets and entropy at the homological level. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3417-3433. doi: 10.3934/dcds.2016.36.3417 [11] Dante Carrasco-Olivera, Roger Metzger Alvan, Carlos Arnoldo Morales Rojas. Topological entropy for set-valued maps. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3461-3474. doi: 10.3934/dcdsb.2015.20.3461 [12] Martin Gugat, Rüdiger Schultz, Michael Schuster. Convexity and starshapedness of feasible sets in stationary flow networks. Networks & Heterogeneous Media, 2020, 15 (2) : 171-195. doi: 10.3934/nhm.2020008 [13] Katja Polotzek, Kathrin Padberg-Gehle, Tobias Jäger. Set-oriented numerical computation of rotation sets. Journal of Computational Dynamics, 2017, 4 (1&2) : 119-141. doi: 10.3934/jcd.2017004 [14] Yaotang Li, Suhua Li. Exclusion sets in the Δ-type eigenvalue inclusion set for tensors. Journal of Industrial & Management Optimization, 2019, 15 (2) : 507-516. doi: 10.3934/jimo.2018054 [15] Yaofeng Su. Almost surely invariance principle for non-stationary and random intermittent dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6585-6597. doi: 10.3934/dcds.2019286 [16] Yujun Ju, Dongkui Ma, Yupan Wang. Topological entropy of free semigroup actions for noncompact sets. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 995-1017. doi: 10.3934/dcds.2019041 [17] Wael Bahsoun, Christopher Bose. Quasi-invariant measures, escape rates and the effect of the hole. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1107-1121. doi: 10.3934/dcds.2010.27.1107 [18] Victor Ayala, Adriano Da Silva, Luiz A. B. San Martin. Control systems on flag manifolds and their chain control sets. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2301-2313. doi: 10.3934/dcds.2017101 [19] Peidong Liu, Kening Lu. A note on partially hyperbolic attractors: Entropy conjecture and SRB measures. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 341-352. doi: 10.3934/dcds.2015.35.341 [20] Masayuki Asaoka, Kenichiro Yamamoto. On the large deviation rates of non-entropy-approachable measures. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4401-4410. doi: 10.3934/dcds.2013.33.4401

2018 Impact Factor: 1.143