Typical Uniqueness in Ergodic Optimization

2025/06/02 by Oliver Jenkinson, Xiaoran Li, Jenkinson, Oliver +5 · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2506.01518

Abstract

For ergodic optimization on any topological dynamical system, with real-valued potential function f belonging to any separable Banach space B of continuous functions, we show that the f-maximizing measure is typically unique, in the strong sense that a countable collection of hypersurfaces contains the exceptional set of those f∈ B with non-unique maximizing measure. This strengthens previous results asserting that the uniqueness set is both residual and prevalent.

Citations

• Ergodic Optimization for Open Expanding Multi-valued Topological Dynamical Systems
• Typicality of periodic optimization over an expanding circle map
• Ground states and periodic orbits for expanding Thurston maps
• Points of continuity of semi-continuous functions
• Prevalent uniqueness in ergodic optimisation
• Ergodic optimization theory for a class of typical maps
• Ergodic optimization in dynamical systems
• Ergodic optimization of Birkhoff averages and Lyapunov exponents
• Haar null and Haar meager sets: a survey and new results
• Ergodic optimization of prevalent super-continuous functions
• Ground States are generically a Periodic Orbit
• Ergodic optimization of super-continuous functions on shift spaces
• Maximizing measures of generic Hölder functions have zero entropy
• Lyapunov minimizing measures for expanding maps of the circle
• Some convexity questions arising in statistical mechanics.

Cited by

• Ergodic Optimization with Linear Constraints

Related

• Prevalent uniqueness in ergodic optimisation
• Ergodic optimization for some dynamical systems beyond uniform hyperbolicity
• Ergodic Maximizing Measures of Non-Generic, Yet Dense Continuous Functions
• Effective uniqueness of Parry measure and exceptional sets in ergodic theory
• Noncommutative Ergodic Optimization and Unique Ergodicity