Records and occupation time statistics for area-preserving maps

TL;DR

This paper compares transport, records, and occupation time statistics of two area-preserving maps, revealing standard map's random walk-like behavior and the triangle map's complex, transient-like dynamics.

Contribution

It provides a detailed analysis of transport and record statistics for the standard and triangle maps, highlighting their distinct dynamical behaviors.

Findings

Standard map exhibits random walk-like transport.

Triangle map shows complex, transient-like behavior.

Transport properties differ significantly between the two maps.

Abstract

A relevant problem in dynamics is to characterize how deterministic systems may exhibit features typically associated to stochastic processes. A widely studied example is the study of (normal or anomalous) transport properties for deterministic systems on a non-compact phase space. We consider here two examples of area-preserving maps: the Chirikov-Taylor standard map and the Casati-Prosen triangle map, and we investigate transport properties, records' statistics and occupation time statistics. While the standard map, when a chaotic sea is present, always reproduces results expected for simple random walks, the triangle map -- whose analysis still displays many elusive points -- behaves in a wildly different way, some of the features being compatible with a transient (non conservative) nature of the dynamics.