Infinite Ergodicity that Preserves the Lebesgue Measure

Abstract

We proved that for the countably infinite number of one-parameterized one dimensional dynamical systems, they preserve the Lebesgue measure and they are ergodic for the measure (infinite ergodicity). Considered systems connect the parameter region in which dynamical systems are exact and the parameter region in which systems are dissipative, and correspond to the critical points of the parameter in which weak chaos occurs (the Lyapunov exponent converges to zero). These results are the generalization of the work by R. Adler and B. Weiss. We show that the distributions of normalized Lyapunov exponent for these systems obey the Mittag-Leffler distribution of order $1/2$ by numerical simulation.

References in corpus (14)

• Route to chaos in optomechanics
• Pesin-Type Identity for Weak Chaos
• Routes to spatiotemporal chaos in Kerr optical frequency combs
• Solution of the Fokker-Planck equation with a logarithmic potential
• Generalized Arcsine Law and Stable Law in an Infinite Measure Dynamical System
• Subexponential instability implies infinite invariant measure
• Infinite Invariant Density Determines Statistics of Time Averages for Weak Chaos
• Infinite invariant density in a semi-Markov process with continuous state variables
• Separation of trajectories and its Relation to Entropy for Intermittent Systems with a Zero Lyapunov exponent
• Distributional behavior of time averages of non-$L^1$ observables in one-dimensional intermittent maps with infinite invariant measures
• Infinite density for cold atoms in shallow optical lattices
• Exact Lyapunov exponents of the generalized Boole transformations
• Generalized Lyapunov exponent as a unified characterization of dynamical instabilities
• Numerical estimate of infinite invariant densities: application to Pesin-type identity