Joint typical periodic optimization: systems with stable hyperbolicity

TL;DR

This paper extends the joint typical periodic optimization framework to a broader class of hyperbolic systems, establishing new results for various dynamical systems including Axiom A diffeomorphisms and hyperbolic rational maps.

Contribution

It develops an axiomatic perturbation framework and proves new joint optimization results for several important classes of hyperbolic systems.

Findings

Optimization of periodic orbits persists under perturbations in broader hyperbolic systems.

Established joint typical periodic optimization for Axiom A diffeomorphisms, rational maps, quadratic polynomials, and 1D $C^r$ maps.

Broadened the scope of joint optimization theory to include more complex hyperbolic dynamics.

Abstract

The framework of joint typical periodic optimization, in which both the dynamical system and the potential function are allowed to vary simultaneously, was introduced in [HHJL25], in a direction motivated by the work of Yang, Hunt & Ott [YHO00]. For certain classes of hyperbolic systems, it was shown there that optimizing periodic orbits persist under simultaneous perturbation, yielding joint locking sets that contain open dense subsets of the relevant product spaces. In the present article we broaden the scope of this theory, by developing an axiomatic joint perturbation framework that accommodates a wider class of stably hyperbolic systems, and by establishing new joint typical periodic optimization results for several natural and important families: Axiom A diffeomorphisms with the no-cycle property, hyperbolic rational maps on the Riemann sphere, real quadratic polynomials, and $C^r$ maps in one dimension.