Weak Gibbs measures and the local product structure

Abstract

We establish a version of the local product structure (weak local product structure) for ergodic measures \overline{\mu} which are the invertible extension of ergodic weak Gibbs measures \mu invariant under piecewise C^0-invertible (infinite to one) Markov maps T. As a special case, \overline{\mu} possesses asymptotically ‘almost’ local product structure in the sense of Barreira, Pesin and Schmeling. Under piecewise conformality of T and the existence of a piecewise smooth representation of the dual map of T, the weak local product structure allows one to show that the pointwise dimension of \overline{\mu} exists almost everywhere and is the sum of the pointwise dimension of \mu and the pointwise dimension of the dual of \mu. Our results can be applicable to a natural extension of piecewise conformal two-dimensional Markov map which is related to a complex continued fraction and admits indifferent periodic points.