Lyapunov optimization for non-generic one-dimensional expanding Markov maps

2017/05/22 by Mao Shinoda, Hiroki Takahasi, Shinoda, Mao +1
Computer Science · Mathematics · #37A40 #37C40 #37D05 #37D35 #37E05 #Cellular Automata and Applications #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1705.07579

openalex publication_date 2017/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a non-generic, yet dense subset of C¹ expanding Markov maps of the interval we prove the existence of uncountably many Lyapunov optimizing measures which are ergodic, fully supported and have positive entropy. These measures are equilibrium states for some Hölder continuous potentials. We also prove the existence of another non-generic dense subset for which the optimizing measure is unique and supported on a periodic orbit. A key ingredient is a new C¹ perturbation lemma which allows us to interpolate between expanding Markov maps and the shift map on a finite number of symbols.

Citations

• Ergodic optimization, zero temperature limits and the max-plus algebra

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