We consider quasisymmetric reparametrizations of the parameter space of the quadratic family. We prove that the set of quadratic maps which are either regular or Collet-Eckmann with polynomial recurrence of the critical orbit has full Lebesgue measure.

We show that there are no partially hyperbolic horseshoes with positive Lebesgue measure for diffeomorphisms whose class of differentiability is higher than 1. This generalizes a classical result by Bowen for uniformly hyperbolic horseshoes.

We prove that the set of critical values of the distance function from a submanifold of a complete Riemannian manifold is of Lebesgue measure zero. In this way, we extend a result of Itoh and Tanaka.

We give an explicit expression for the invariant measure, absolutely continuous with respect to the Lebesgue measure, of the greedy βtransformation with three deleted digits. We define a version of the natural extension of the transformation to obtain this expression. We get that the transformation is exact and weakly Bernoulli.

Space, as we typically represent it in mathematics and physics, is composed of dimensionless, indivisible points. But philosophers have for a long time doubted whether there are any point-sized regions of space. On an alternative, region-based approach to space, extended regions together with the relations of parthood and contact are taken as primitive; points are represented as mathematical ab...

We study a d-dimensional coupled map lattice consisting of hyperbolic toral automorphisms (Arnold cat maps) that are weakly coupled by an analytic map. We construct the Sinai–Ruelle–Bowen measure for this system and study its marginals on the tori. We prove that they are absolutely continuous with respect to Lebesgue measure if and only if the coupling satisfies a non-degeneracy condition.

An equivalence between the Gurov-Reshetnyak GR(ε) and Muckenhoupt A∞ conditions is established. Our proof is extremely simple and works for arbitrary absolutely continuous measures. Throughout the paper, μ will be a positive measure on R absolutely continuous with respect to Lebesgue measure. Denote Ωμ(f ;Q) = 1 μ(Q) ∫ Q |f(x) − fQ,μ|dμ(x), fQ,μ = 1 μ(Q) ∫

Generalized Riesz products similar to the type which arise as the spectral measure for a rank-one transformation are studied. A condition for the mutual singularity of two such measures is given. As an application, a probability space of transformations is presented in which almost all transformations are singular with respect to Lebesgue measure. AMS Subject Classification: Primary 28D03; Seco...

We study a d-dimensional coupled map lattice consisting of hyperbolic toral automorphisms (Arnold cat maps) that are weakly coupled by an analytic map. We construct the Sinai–Ruelle–Bowen measure for this system and study its marginals on the tori. We prove that they are absolutely continuous with respect to Lebesgue measure if and only if the coupling satisfies a non-degeneracy condition.