Separation of trajectories and its Relation to Entropy for Intermittent Systems with a Zero Lyapunov exponent

Abstract

One dimensional intermittent maps with stretched exponential separation of nearby trajectories are considered. When time goes infinity the standard Lyapunov exponent is zero. We investigate the distribution of ${\lambda }_{\alpha }={\sum }_{i=0}^{t-1}\ln \left|M^{\prime }(x_i)\right|/t^{\alpha }$, where $\alpha$ is determined by the nonlinearity of the map in the vicinity of marginally unstable fixed points. The mean of ${\lambda }_{\alpha }$ is determined by the infinite invariant density. Using semi analytical arguments we calculate the infinite invariant density for the Pomeau-Manneville map, and with it obtain excellent agreement between numerical simulation and theory. We show that $\alpha ⟨{\lambda }_{\alpha }⟩$ is equal to Krengel's entropy and to the complexity calculated by the Lempel-Ziv compression algorithm. This generalized Pesin's identity shows that $⟨{\lambda }_{\alpha }⟩$ and Krengel's entropy are the natural generalizations of usual Lyapunov exponent and entropy for these systems.

12 pages, 10 figures

References in corpus (5)

• Random Time-Scale Invariant Diffusion and Transport Coefficients
• Distribution of Time-Averaged Observables for Weak Ergodicity Breaking
• Pesin-Type Identity for Weak Chaos
• Weakly non-ergodic Statistical Physics
• Ergodicity Breaking in a Deterministic Dynamical System

Cited by in corpus (12)

• Ergodic properties of continuous-time random walks: finite-size effects and ensemble dependences
• Aging generates regular motions in weakly chaotic systems
• Infinite Invariant Density Determines Statistics of Time Averages for Weak Chaos
• Alternative numerical computation of one-sided Levy and Mittag-Leffler distributions
• Pesin-type relation for subexponential instability
• Lyapunov statistics and mixing rates for intermittent systems
• Numerical estimate of infinite invariant densities: application to Pesin-type identity
• Number of first-passage times as a measurement of information for weakly chaotic systems
• Infinite Ergodicity that Preserves the Lebesgue Measure
• Singularity of Levy walks in the lifted Pomeau-Manneville map
• Transition to anomalous dynamics in a simple random map
• Does an intermittent dynamical system remain (weakly) chaotic after drilling in a hole?