On the joint continuity of topological pressures for sub-additive singular-valued potentials

Given a non-conformal repeller [Formula: see text] of a [Formula: see text] map [Formula: see text], we give a variational principle of a dimensional upper bound of the non-conformal repellers. If [Formula: see text] is [Formula: see text] then we prove the joint continuity of topological pressure for sub-additive singular-valued potentials on maps and parameters.

Nonadditive measure-theoretic pressure and applications to dimensions of an ergodic measure

Without any additional conditions on subadditive potentials, this paper defines subadditive measure-theoretic pressure, and shows that the subadditive measure-theoretic pressure for ergodic measures can be described in terms of measure-theoretic entropy and a constant associated with the ergodic measure. Based on the definition of topological pressure on non-compact sets, we give another equivalent definition of subadditive measure-theoretic pressure, and obtain an inverse variational principle. This paper also studies the superadditive measure-theoretic pressure which has similar formalism to the subadditive measure-theoretic pressure. As an application of the main results, we prove that an average conformal repeller admits an ergodic measure of maximal Hausdorff dimension. Furthermore, for each ergodic measure supported on an average conformal repeller, we construct a set whose dimension is equal to the dimension of the measure.

Escape rates for Gibbs measures

In this paper we study the asymptotic behaviour of the escape rate of a Gibbs measure supported on a conformal repeller through a small hole. There are additional applications to the convergence of the Hausdorff dimension of the survivor set.