Statistical properties of time-reversible triangular maps of the square

TL;DR

This paper investigates the statistical properties of time-reversible triangular maps of the square, focusing on conditions for product structure of equilibrium measures and analyzing invariant densities, especially in the presence of intermittent fixed points.

Contribution

It introduces a class of time-reversal symmetric triangular maps, characterizes when their equilibrium measures factorize, and analyzes the invariant densities in cases with intermittent fixed points.

Findings

Equilibrium measures can have a product structure under certain conditions.

Non-product measures reveal additional statistical properties.

Invariant densities can exhibit singularities at intermittent fixed points.

Abstract

Time reversal symmetric triangular maps of the unit square are introduced with the property that the time evolution of one of their two variables is determined by a piecewise expanding map of the unit interval. We study their statistical properties and establish the conditions under which their equilibrium measures have a product structure, i.e. factorises in a symmetric form. When these conditions are not verified, the equilibrium measure does not have a product form and therefore provides additional information on the statistical properties of theses maps. This is the case of anti-symmetric cusp maps, which have an intermittent fixed point and yet have uniform invariant measures on the unit interval. We construct the invariant density of the corresponding two-dimensional triangular map and prove that it exhibits a singularity at the intermittent fixed point.