On ergodic and mixing properties of the triangle map

TL;DR

This paper analyzes the ergodic and mixing properties of the triangle map, a piecewise parabolic automorphism of the torus, using analytical, numerical, and stochastic approaches to understand its dynamics.

Contribution

It provides a detailed comparison of two symbolic encoding schemes and introduces a stochastic version of the triangle map to study its statistical properties.

Findings

Encoding schemes are compatible but not equivalent

Ergodic properties are characterized via Markov matrices and Koopman operator

Stochastic triangle map reproduces statistical behaviors of the original

Abstract

In this paper we study in detail, both analytically and numerically, the dynamical properties of the triangle map, a piecewise parabolic automorphism of the two-dimensional torus, for different values of the two independent parameters defining the map. The dynamics is studied numerically by means of two different symbolic encoding schemes, both relying on the fact that it maps polygons to polygons: in the first scheme we consider dynamically generated partitions made out of suitable sets of disjoint polygons, in the second we consider the standard binary partition of the torus induced by the discontinuity set. These encoding schemes are studied in detail and shown to be compatible, although not equivalent. The ergodic properties of the triangle map are then investigated in terms of the Markov transition matrices associated to the above schemes and furthermore compared to the spectral properties of the Koopman operator in L2(T2). Finally, a stochastic version of the triangle map is introduced and studied. A simple heuristic analysis of the latter yields the correct statistical and scaling behaviours of the correlation functions of the original map.