The spaces of interval exchange maps and measured foliations are considered and an alternative proof that almost all interval exchange maps and measured foliations are uniquely ergodic is given. These spaces are endowed with a refinement process, called a simplicial system, which is studied abstractly and is shown to be normal under a simple assumption. The results follow and thus are a corollary of a more general theorem in a broader setting.

It is proved, using Krieger's theorem, that ITPFI's of bounded type are ITPFI2. This answers a question asked by E. J. Woods.

We define a ‘core-matrix’ of a transitive sofic subshift, which is unique up to similarity for each transitive sofic subshift. We prove that if there exists a bounded-to-one factor map from one transitive sofic subshift to another, the block of the Jordan form of a core-matrix of this second subshift with non-zero eigenvalues is a principal submatrix of the Jordan form of a core-matrix of the first. We also prove that the subshifts that are almost of finite type are ‘spectrally of finite type’.

The class of k-automorphisms is not contained in a certain class of skew products over a Bernoulli base. The non-identity fibre transformation in the skew is allowed to have positive or even infinite entropy. A difficulty presented by positive entropy is handled via an apparently new property of independent processes (lemma 7.24).

In this paper we use Thurston's work on the dynamics of diffeomorphisms on surfaces to show that a diffeomorphism ƒ on a surface is isotopic to a Morse- Smale one if and only if the growth rate of the length of the words representing elements of the fundamental group under iteration by ƒ is one. Morse-Smale isotopy classes are also shown to be the same as Nielsen's algebraically finite type.

We introduce an Anosov action on the bundle of Weyl chambers of a riemannian symmetric space of non-compact type, which for rank one spaces coincides with the geodesic flow.

In this paper we prove that if the entropy of an ergodic measure preserved by a C2 surface diffeomorphism is positive then it is equal to the product of the Hausdorff dimension of the quotient measure defined by the family of stable manifolds and the positive Lyapunov exponent.

Bishop [2]has given a proof of Birkhoff's ergodic theorem by establishing upcrossing inequalities similar to those of Doob. Such inequalities can be considered as quantitative improvements of convergence theorems: while convergence a.e. means that the number of upcrossings of any interval is a.e. finite, they assert integrability and prove bounds for the integrals. The main point of this paper is to prove upcrossing inequalities for the class of subadditive superstationary processes introduced by Abid [1] as a common generalization of Kingman's [5] subadditive stationary processes and Krengel's [6] superstationary processes. We make use of ideas of Smeltzer [7] who handled the subadditive stationary discrete parameter case in his unpublished thesis. In the continuous parameter case our upcrossing inequality requires more restrictive conditions than the corresponding convergence theorem, due to Hachem [3]. We actually show by example that the number of upcrossings need not be integrable under the assumptions of Hachem even for additive stationary processes.

Given a compact (two-sided) flow, an isolated invariant set S and a Morse-decomposition (M1, …, Mn) of S, there is a generalized Morse equation, proved by Conley and Zehnder, which relates the Alexander-Spanier cohomology groups of the Conley indices of the sets Mi and S with each other. Recently, Rybakowski developed the technique of isolating blocks and extended Conley's index theory to a class of one-sided semiflows on non-necessarily compact spaces, including e.g. semiflows generated by parabolic equations. Using these results, we discuss in this paper Morse decompositions and prove the above-mentioned Morse equation not only for arbitrary homology and cohomology groups, but also in this more general semiflow setting.

Two Cr, r ≥ 2, expanding maps of the circle which are absolutely continuously conjugate are Cr conjugate. Here ƒ and g: S1 → S1 are expanding if they stretch tangent vectors in some metric, and a conjugacy is an isomorphism h: S1 → S1 such that fh = hg.

We give examples of maps of the interval with zero entropy for which the continuous invariant measure has no dimension, and we prove a dimension property for maps lying in the stable manifold of Feigenbaum's fixed points.

By employing a regularizing transformation, the problem of bifurcation of relative equilibria in the Newtonian 4-body problem is reduced to a study of an algebraic correspondence between real algebraic varieties. The finiteness theorems of algebraic geometry are used to find an upper bound for the number of affine equivalence classes of relative equilibria which holds for all masses in the complement of a proper, algebraic subset of the space of all masses.

We prove that the dynamical systems generated by first order partial differential equations are K-flows and chaotic in the sense of Auslander & Yorke.

We show that in several cases preservation of cones leads to non-vanishing of (some) Lyapunov exponents. It gives simple and effective criteria for nonvanishing of the exponents, which is demonstrated on the example of the billiards studied by Bunimovich. It is also shown that geodesic flows on manifolds of non-positive sectional curvature can be treated from this point of view.

The aim of this paper is to introduce and study the class of boundedly non-additive processes. The main result is the decomposition in theorem (2.1) and theorem (3.1), which says that a boundedly non-additive process is the sum of a non-positive subadditive, a non-negative superadditive and an additive process. By this decomposition we can extend the mean and local ergodic theorems for superadditive processes of M. A. Akcoglu and U. Krengel to boundedly non-additive processes. At the end of this paper some examples are given.

Symmetric S-unimodal functions with positive Liapunov exponent of the critical value have an invariant measure absolutely continuous with respect to Lebesgue measure.

If is a weakly mixing isometric extension of a finite measure preserving, k-fold mixing map T, then must also be k-fold mixing.

A mixing transformation is constructed which has non-simple spectrum of finite multiplicity. This example is based on a rank 1 mixing transformation and is constructed by cutting and stacking. It can be made to be mixing of all orders.

In this paper we study actions of lattices in semisimple groups preserving a G-structure of finite type.