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...

In this note we prove that for equilibrium states of axiom A systems the time τB(x) needed for a typical point x to enter for the first time in a typical ball B with radius r scales as τB(x) ∼ r d where d is the local dimension of the invariant measure at the center of the ball. A similar relation is proved for a full measure set of interval excanges. Some applications to Birkoff averages of un...

Hušek defines a space X to have a small diagonal if each uncountable subset of X disjoint from the diagonal, has an uncountable subset whose closure is disjoint from the diagonal. Hušek proved that a compact space of weight ω1 which has a small diagonal, will be metrizable, but it remains an open problem to determine if the weight restriction is necessary. It has been shown to be consistent tha...

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...

A supermanifold, Mm/n, can be caracterired by its smooth superfunctions which constitute an algebra A (Leites, Kostant). We associate canonically ~a Ia Gelfand, certain fibred manifolds on which the automorphisms (the Jordan automorphisms) of A actas diffeomorphisms. For example, the kernelsof all homomorphisms from the algebra of superfunctions onto the Grassmann algebra of dimension n form na...

We present equiconsistency results at the level of subcompact cardinals. Assuming SBHδ, a special case of the Strategic Branches Hypothesis, we prove that if δ is a Woodin cardinal and both 2(δ) and 2δ fail, then δ is subcompact in a class inner model. If in addition 2(δ) fails, we prove that δ is Π21 subcompact in a class inner model. These results are optimal, and lead to equiconsistencies. A...

Let f be a diffeomorphism of a closed C∞ manifold M . In this paper, we introduce the notion of the C1-stable specification property for a closed f -invariant set Λ of M , and we prove that f|Λ satisfies a C 1-stable specification property if and only if Λ is a hyperbolic elementary set. As a corollary, the C1-interior of the set of diffeomorphisms of M satisfying the specification property is ...