Ergodic optimization for some dynamical systems beyond uniform hyperbolicity

ABSTRACT

In Ergodic optimization, one wants to find ergodic measures to maximize or minimize the integral of given continuous functions. This has been succefully studied for uniformly hyperbolic systems for generic continuous functions by Bousch and Brémon. In this paper, we show that for several interesting systems beyond uniform hyperbolicity, any generic continuous function has a unique maximizing measure with zero entropy. In some cases, we also know that the maximizing measure has full support. These interesting systems include singular hyperbolic attractors, $C^{\mathrm{\infty }}$ surface diffeomorphisms and diffeomorphisms away from homoclinic tangencies.We try to give a uniform mechanism for these non-hyperbolic systems.

KEYWORDS:

• Maximizing measure
• entropy
• singular hyperbolicity
• homoclinic tangency

MATHEMATICS SUBJECT CLASSIFICATIONS (2010):

• 37A25
• 37A35
• 37C05
• 37D25

Acknowledgments

We would like to thank P. Varandas for his nice comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 A dense $G_{\delta }$ set is also said to be residual.

2 Brémont [7] gave a deep argument, but without precious statements that we will need.

3 A set is chain transitive if for any $ϵ>0$ and any two points in this set, there exists an $ϵ$-chain connecting these two points. A finite set $x_1,\dots ,x_k$ is called an $ϵ$-chain, if for $i=1,\dots ,k-1$ there exists $t_i\ge 1$ such that $\mathrm{d}\left({\varphi }_{t_i}\right(x_i),x_{i+1})<ϵ$.

Additional information

Funding

D.Y was partially supported by the NSFC [grant numbers 11822109, 11671288, 11790274]. J.Z was partially supported by the National Key R&D Program of China [grant number 2021YFA1001900] and the NSFC [grant number 12001027].