Many problems in mathematics and science require the use of infinite-dimensional spaces. Consequently, there is need for an analogue of the finite-dimensional notions of ‘Lebesgue almost every’ and ‘Lebesgue measure zero’ in the infinite-dimensional setting. The theory of prevalence addresses this need and provides a powerful framework for describing generic behavior in a probabilistic way. We ...

We introduce the Markov extension, represented schematically as a tower, to the study of dynamical systems with holes. For tower maps with small holes, we prove the existence of conditionally invariant probability measures which are absolutely continuous with respect to Lebesgue measure (abbreviated a.c.c.i.m.). We develop restrictions on the Lebesgue measure of the holes and simple conditions ...

Given a Fuchsian group with at least one cusp, Deroin, Kleptsyn and Navas define a Lyapunov expansion exponent for a point on the boundary, and ask if it vanishes for almost all points with respect to Lebesgue measure. We give an affirmative answer to this question, by considering the behavior of the word metric along typical geodesic rays and their excursions into cusps. We also consider the b...

We prove completeness of the propositional modal logic S4 for the measure algebra based on the Lebesgue-measurable subsets of the unit interval, [0, 1]. In recent talks, Dana Scott introduced a new measure-based semantics for the standard propositional modal language with Boolean connectives and necessity and possibility operators, and ♦. Propositional modal formulae are assigned to Lebesgue-me...

It is shown that some Schrödinger operators, with nonprimitive substitution potentials, have pure singular continuous Cantor spectrum with null Lebesgue measure for all elements in the respective hulls.

We construct, for each irrational number α, a minimal C1-diffeomorphism of the circle with rotation number α which is not ergodic with respect to the Lebesgue measure.