On the minimum ergodic average and minimal systems
- Manuel Saavedra firstname.lastname@example.org
- Helmuth Villavicencio email@example.com
We prove some equivalences associated with the case when the average lower time is minimal. In addition, we characterize the minimal systems by means of the positivity of invariant measures on open sets and also the minimum ergodic averages. Finally, we show that a minimal system admits an open set whose measure is minimal with respect to a set of ergodic measures and its value can be chosen in [0, 1].
S. Addas-Zanata and F. A. Tal, “Support of maximizing measures for typical C0 dynamics on compact manifolds”, Discrete Contin. Dyn. Syst., vol. 26, no. 3, pp. 795–804, 2010.
W. Huang, Z. Lian, S. Shao and X. Ye, “Minimal systems with finitely many ergodic mea- sures”, J. Funct. Anal., vol. 280, no. 12, Paper No. 109000, 42 pages, 2021.
O. Jenkinson, “Every ergodic measure is uniquely maximizing”, Discrete Contin. Dyn. Syst., vol. 16, no. 2, pp. 383–392, 2006.
O. Jenkinson,“Ergodic optimization in dynamical systems”,Ergodic Theory Dynam. Systems, vol. 39, no. 10, pp. 2593–2618, 2019.
K. Liu, L. Xu and R. Zhang, “Time-restricted sensitivity and entropy”, J. Differential Equations, vol. 293, pp. 70–85, 2021.
I. Morris, “Lyapunov-maximizing measures for pairs of weighted shift operators”, Ergodic Theory Dynam. Systems, vol. 39, no. 1, pp. 225–247, 2019.
I. Morris, “Prevalent uniqueness in ergodic optimisation”, Proc. Amer. Math. Soc., vol. 149, no. 4, pp. 1631–1639, 2021.
M. Viana and K. Oliveira, Foundations of ergodic theory, Cambridge studies in advanced mathematics 151, Cambridge: Cambridge University Press, 2016.
P. Walters, An introduction to ergodic theory, Graduate Texts in Mathematics 79, New York: Springer New York, 1982.