Invariant measures for Markov maps of the interval

Invariant Measures for Interval Maps with Critical Points and Singularities

We prove that, under a mild summability condition on the growth of the derivative on critical orbits any piecewise monotone interval map possibly containing discontinuities and singularities with infinite derivative (cusp map) admits an ergodic invariant probability measures which is absolutely continuous with respect to Lebesgue measure.

Invariant Measures for Real Analytic Expanding Maps

Let X be a compact connected subset of Rd with non-empty interior, and T : X → X a real analytic full branch expanding map with countably many branches. Elements of a thermodynamic formalism for such systems are developed, including criteria for compactness of transfer operators acting on spaces of bounded holomorphic functions. In particular a new sufficient condition for the existence of a T ...

INVARIANT MEASURES FOR MARKOV PROCESSES(i)

Invariant Measures for Typical Quadratic Maps

A sufficient geometrical condition for the existence of absolutely continuous invariant probability measures for S−unimodal maps will be discussed. The Lebesgue typical existence of such measures in the quadratic family will be a consequence.

Return Time Statistics for Invariant Measures for Interval Maps with Positive Lyapunov Exponent

We prove that multimodal maps with an absolutely continuous invariant measure have exponential return time statistics around a.e. point. We also show a ‘polynomial Gibbs property’ for these systems, and that the convergence to the entropy in the Ornstein-Weiss formula has normal fluctuations. These results are also proved for equilibrium states of some Hölder potentials.