We show that history-preserving bisimilarity for higher-dimensional automata has a simple characterization directly in terms of higher-dimensional transitions. This implies that it is decidable for finite higher-dimensional automata. To arrive at our characterization, we apply the open-maps framework of Joyal, Nielsen and Winskel in the category of unfoldings of precubical sets.

Global aspects of the motion of passive scalars in time-dependent incompressible fluid flows are well described by volume-preserving (Liouvillian) three-dimensional maps. In this paper the possible invariant structures in Liouvillian maps and the two most interesting nearlyintegrable cases are investigated. In addition, the fundamental role of invariant lines in organizing the dynamics of this ...

My aim in this article is to provide an accessible introduction to the notion of topological entropy and (for context) its measure theoretic analogue, and then to present some recent work applying related ideas to the structure of iterated preimages for a continuous (in general non-invertible) map of a compact metric space to itself. These ideas will be illustrated by two classes of examples, f...

We show that the Gibbs states (known from [9] to be unique) of Hölder continuous potentials and random distance expanding maps coincide with relative equilibrium states of those potentials, proving in particular that the latter exist and are unique. In the realm of conformal expanding random maps we prove that given an ergodic (globally) invariant measure with a given marginal, for almost every...

The three-dimensional motion of an incompressible inviscid uid is classically described by the Euler equations, but can also be seen, following Arnold 1], as a geodesic on a group of volume-preserving maps. Local existence and uniqueness of minimal geodesics have been established by Ebin and Marsden 16]. In the large, for a large class of data, the existence of minimal geodesics may fail, as sh...

We investigate linear parabolic maps on the torus. In a generic case these maps are non-invertible and discontinuous. Although the metric entropy of these systems is equal to zero, their dynamics is non-trivial due to folding of the image of the unit square into the torus. We study the structure of the maximal invariant set, and in a generic case we prove the sensitive dependence on the initial...

In this paper we revisit once again, see [ShSu], a family of expanding circle endomorphisms. We consider a family {Bθ} of Blaschke products acting on the unit circle, T, in the complex plane obtained by composing a given Blashke product B with the rotations about zero given by mulitplication by θ ∈ T. While the initial map B may have a fixed sink on T there is always an open set of θ for which ...

We obtain large deviation results for non-uniformly expanding maps with non-flat singularities or criticalities and for partially hy-perbolic non-uniformly expanding attracting sets. That is, given a continuous function we consider its space average with respect to a physical measure and compare this with the time averages along orbits of the map, showing that the Lebesgue measure of the set of...

We obtain large deviation bounds for non-uniformly expanding maps with non-flat singularities or criticalities and for partially hyperbolic non-uniformly expanding attracting sets. That is, given a continuous function we consider its space average with respect to a physical measure and compare this with the time averages along orbits of the map, showing that the Lebesgue measure of the set of p...