Mixing rates for intermittent maps of high exponent

Abstract

We obtain higher order theory for the long term behavior of the transfer operator associated with the unit interval map \(f(x)=x(1+2^\alpha x^\alpha )\) if \(0 1\), which corresponds to the infinite measure preserving case. Higher order theory for \(\alpha \ge 2\) is more challenging and requires new techniques. Along the way, we provide higher order theory for scalar and operator renewal sequences with infinite measure and regular variation. Although the present work considers the unit interval map mentioned above as a toy model, our interest focuses on finding sufficient conditions under which the asymptotic behavior of the transfer operator associated to dynamical systems preserving an infinite measure is ’almost like’ the asymptotic behavior of scalar renewal sequences associated to null recurrent Markov chains characterized by regular variation.