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

Part of: Smooth dynamical systems: general theory Dynamical systems with hyperbolic behavior Low-dimensional dynamical systems Ergodic theory

Published online by Cambridge University Press:  25 March 2019

MAO SHINODA  and
HIROKI TAKAHASI
MAO SHINODA
Affiliation:
Department of Mathematics, Keio University, Yokohama, 223-8522, Japan email shinoda-mao@z3.keio.jp, hiroki@math.keio.ac.jp
HIROKI TAKAHASI
Affiliation:
Department of Mathematics, Keio University, Yokohama, 223-8522, Japan email shinoda-mao@z3.keio.jp, hiroki@math.keio.ac.jp
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Abstract

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.

Keywords

expanding Markov mapLyapunov optimizing measurenon-generic property

MSC classification

Primary: 37C40: Smooth ergodic theory, invariant measures 37D35: Thermodynamic formalism, variational principles, equilibrium states 37E05: Maps of the interval (piecewise continuous, continuous, smooth)
Secondary: 37A40: Nonsingular (and infinite-measure preserving) transformations 37D05: Hyperbolic orbits and sets
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 9 , September 2020 , pp. 2571 - 2592
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Copyright
© Cambridge University Press, 2019

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