Statistical properties of nonuniformly expanding 1D maps with logarithmic singularities

Abstract

We study the statistical properties of piecewise smooth maps on a circle, with a finite number of critical and singular points with an unbounded derivative, such that the derivative goes like the inverse of the distance to the singular points. We write down a simple set of conditions, and show that when these conditions are met, there exist an absolutely continuous invariant probability measure with exponential decay of correlations. We also rule out the existence of nontrivial coboundary, and obtain a positive variance in the central limit theorem for any nonconstant Hölder continuous observable. Our results apply to a positive measure set of nonuniformly expanding maps on the circle considered by Takahasi and Wang (2012 Nonlinearity 25 533).