Conformal measures for multidimensional piecewise invertible maps

Given a piecewise invertible map T:X\to X and a weight g:X\rightarrow\ ]0,\infty[ , a conformal measure \nu is a probability measure on X such that, for all measurable A\subset X with T:A\to TA invertible, \nu(TA)= \lambda \int_{A}\frac{1}{g}\ d\nu with a constant \lambda>0 . Such a measure is an essential tool for the study of equilibrium states. Assuming that the topological pressure of the boundary is small, that \log g has bounded distortion and an irreducibility condition, we build such a conformal measure.

Erratum to “Number of equilibrium states of piecewise monotonic maps of the interval”

Markov extensions for multi-dimensional dynamical systems

By a result of F. Hofbauer [11], piecewise monotonic maps of the interval can be identified with topological Markov chains with respect to measures with large entropy. We generalize this to arbitrary piecewise invertible dynamical systems under the following assumption: the total entropy of the system should be greater than the topological entropy of the boundary of some reasonable partition separating almost all orbits. We get a sufficient condition for these maps to have a finite number of invariant and ergodic probability measures with maximal entropy. We illustrate our results by quoting an application to a class of multi-dimensional, non-linear, non-expansive smooth dynamical systems.

Specification on the interval

We study the consequences of discontinuities on the specification property for interval maps. After giving a necessary and sufficient condition for a piecewise monotonic, piecewise continuous map to have this property, we show that for a large and natural class of families of such maps (including the β \beta -transformations), the set of parameters for which the specification property holds, though dense, has zero Lebesgue measure. Thus, regarding the specification property, the general case is at the opposite of the continuous case solved by A.M. Blokh (Russian Math. Surveys 38 (1983), 133–134) (for which we give a proof).