As defined in [3], a Hausdorff space is strongly anti-Urysohn (in short: SAU) if it has at least two non-isolated points and any infinite closed subsets of intersect. Our main result answers the questions [3] by providing ZFC construction locally countable SAU cardinality 2c. The hinges on existence 2c weak P-points ω⁎, very deep Ken Kunen. It remains open spaces >2c could exist, while was show...

Abstract The notion of the Urysohn d -width measures to what extent a metric space can be approximated by -dimensional simplicial complex. We investigate how local width bounds on Riemannian manifold affect its global width. bound 1-width in terms first homology and supremal unit balls. Answering question Larry Guth, we give examples n -manifolds considerable <m:math xmlns:m="http://www.w3.org/...

The usual way to study the local structure of Equilibrium State of an Axiom-A diffeomorphism or flow is to use the symbolic dynamic and to push results on the manifold. A new geometrical method is given. It consists in proving that Equilibrium States for Hölder-continuous functions are related to other Equilibrium States of some special sub-systems satisfying a sort of expansiveness. Using diff...

Work on calculations with simplicial and cubical groups in AXIOM was carried out using loan equipment and software from IBM UK and guidance from L A Lambe. We report on the results of this work, and present the AXIOM code written by the second author during this period. This includes an implementation of the monoids which model cubes and simplices, together with a new AXIOM category of near-rin...

In 1969, Hirsch posed the following problem: given a diffeomorphism f : N → N , and a compact invariant hyperbolic set Λ of f , describe the topology of Λ and the dynamics of f restricted to Λ. We solve the problem where Λ = M is a closed 3-manifold: if M is orientable, then it is a connected sum of tori and handles; otherwise it is a connected sum of tori and handles quotiented by involutions....

We investigate the connection between the dynamical Borel-Cantelli and waiting time results. We prove that if a system has the dynamical BorelCantelli property, then the time needed to enter for the first time in a sequence of small balls scales as the inverse of the measure of the balls. Conversely if we know the waiting time behavior of a system we can prove that certain sequences of decreasi...

This paper contributes to the theory of rational choice under sequential criteria. Following the approach initiated by Manzini and Mariotti (2007) for single-valued choice functions, we characterize choice correspondences that are rational by two sequential criteria under a mild consistency axiom. Rationales ensuring the sequential rationalization are explicitly constructed and a uniquely deter...