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

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

For a non-generic, yet dense subset of $C^{1}$ 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 a...

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Veröffentlicht in:Ergodic theory and dynamical systems 2020-09, Vol.40 (9), p.2571-2592
Hauptverfasser: SHINODA, MAO, TAKAHASI, HIROKI
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Optimization
Orbits
Original Article
Perturbation
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description For a non-generic, yet dense subset of $C^{1}$ 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 $C^{1}$ perturbation theorem which allows us to interpolate between expanding Markov maps and the shift map on a finite number of symbols.
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subjects Optimization
Orbits
Original Article
Perturbation
title Lyapunov optimization for non-generic one-dimensional expanding Markov maps
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