Statistical properties of time-reversible triangular maps of the square

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.