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The essential coexistence phenomenon in Hamiltonian dynamics

Part of:Smooth dynamical systems: general theoryDynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  08 April 2021

JIANYU CHEN,
HUYI HU,
YAKOV PESIN and
KE ZHANG
JIANYU CHEN
Affiliation:
School of Mathematical Sciences & Center for Dynamical Systems and Differential Equations, Soochow University, Suzhou 215006, Jiangsu, P.R. China (e-mail: jychen@suda.edu.cn)
HUYI HU
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA (e-mail: hhu@math.msu.edu)
YAKOV PESIN*
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, State College, PA 16802, USA
KE ZHANG
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario, Canada (e-mail: kzhang@math.toronto.edu)
*
e-mail: pesin@math.psu.edu
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Abstract

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We construct an example of a Hamiltonian flow $f^t$ on a four-dimensional smooth manifold $\mathcal {M}$ which after being restricted to an energy surface $\mathcal {M}_e$ demonstrates essential coexistence of regular and chaotic dynamics, that is, there is an open and dense $f^t$-invariant subset $U\subset \mathcal {M}_e$ such that the restriction $f^t|U$ has non-zero Lyapunov exponents in all directions (except for the direction of the flow) and is a Bernoulli flow while, on the boundary $\partial U$, which has positive volume, all Lyapunov exponents of the system are zero.

Keywords

essential coexistenceHamiltonian flowsLyapunov exponentsBernoulli maps

MSC classification

Primary:37D35: Thermodynamic formalism, variational principles, equilibrium states37C45: Dimension theory of dynamical systems
Secondary:37C40: Smooth ergodic theory, invariant measures37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 2: Anatole Katok Memorial Issue Part 1: Special Issue of Ergodic Theory and Dynamical Systems , February 2022, pp. 592 - 613
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

Footnotes

Dedicated to the memory of Anatole Katok

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