Multiple returns for some regular and mixing maps

Area of Science:

• * Dynamical systems theory
• * Statistical mechanics
• * Chaos theory

Background:

• * Limit distributions of visit counts are known to follow Poissonian behavior in highly mixing systems.
• * Understanding these distributions is crucial for characterizing system dynamics, especially in regions with mixed ergodic and regular motions.

Purpose of the Study:

• * To investigate limit laws for the number of visits in various dynamical systems, particularly non-mixing ones.
• * To analyze visit distributions in skew integrable maps, irrational rotations, coupled systems, and the Henon map.
• * To explain observed deviations from Poissonian behavior and explore finite-size effects.

Main Methods:

• * Numerical analysis of visit distributions for shrinking domains around points in phase space.
• * Theoretical explanation for observed power-law decay in skew maps.
• * Superposition principle for coupled systems to analyze boundary effects.
• * Statistical analysis of first return times for periodic points in the Henon map.

Main Results:

• * Irrational rotations on a circle suggest limit laws under specific domain choices.
• * Skew integrable maps exhibit power-law decay in visit distributions, with a theoretical explanation provided.
• * Coupled systems show linear superposition of distributions at boundaries; finite-size effects can mask true limit laws.
• * The Henon map shows Poissonian behavior for generic points but a different law for periodic points.

Conclusions:

• * Deviations from Poissonian behavior are observed in non-uniformly hyperbolic and integrable systems.
• * Power-law distributions arise in specific dynamical systems and coupled systems, offering insights into complex dynamics.
• * The study provides a framework for understanding visit distributions in diverse dynamical systems, including those with mixed regular and chaotic behaviors.