We show that the stationary measure for some random systems of two piecewise affine homeomorphisms of the interval is singular, verifying partially a conjecture by Alsedà and Misiurewicz and contributing to a question of Navas on the absolute continuity of stationary measures, considered in the setup of semigroups of piecewise affine circle homeomorphisms. We focus on the case of resonant boundary derivatives.

Let X ⊂ R N be a Borel set, μ a Borel probability measure on X and T : X → X a locally Lipschitz and injective map. Fix k ∈ N strictly greater than the (Hausdorff) dimension of X and assume that the set of p-periodic points of T has dimension smaller than p for p = 1, …, k − 1. We prove that for a typical polynomial perturbation h ̃ of a given locally Lipschitz function h : X → R , the k-delay coordinate map x ↦ ( h ̃ ( x ) , h ̃ ( T x ) , … , h ̃ ( T k − 1 x ) ) is injective on a set of full μ-measure. This is a probabilistic version of the Takens delay embedding theorem as proven by Sauer, Yorke and Casdagli. We also provide a non-dynamical probabilistic embedding theorem of similar type, which strengthens a previous result by Alberti, Bölcskei, De Lellis, Koliander and Riegler. In both cases, the key improvements compared to the non-probabilistic counterparts are the reduction of the number of required coordinates from 2  dim  X to dim  X and using Hausdorff dimension instead of the box-counting one. We present examples showing how the use of the Hausdorff dimension improves the previously obtained results.

We study variational principles for metric mean dimension. First we prove that in the variational principle of Lindenstrauss and Tsukamoto it suffices to take the supremum over ergodic measures. Second we derive a variational principle for metric mean dimension involving growth rates of measure-theoretic entropy of partitions decreasing in diameter which holds in full generality and in particular does not necessitate the assumption of tame growth of covering numbers. The expressions involved are a dynamical version of Rényi information dimension. Third we derive a new expression for the Geiger-Koch information dimension rate for ergodic shift-invariant measures. Finally we develop a lower bound for metric mean dimension in terms of Brin-Katok local entropy.