A continuous ℤ⊗TG action on a subshift of finite type consists of a subshift of finite type with its shift transformation, together with a group, G, of homeomorphisms of the subshift and a group automorphism T, so that the commutation relation σ ° g = Tg ° ∑A is any positive entropy subshift of finite type, G is any finite group and T is any automorphism of G then there is a non-trivial ℤ⊗TG action on ∑A. We then classify all such actions up to ‘almost topological‘ conjugacy.

A class of vector fields on the 2-torus, which includes Cherry fields, is studied. Natural paths through this class are defined and it is shown that the parameters for which the vector field is unstable is the closure of has irrational rotation number}, where ƒ is a certain map of the circle and Rt is rotation through t. This is shown to be a Cantor set of zero Hausdorff dimension. The Cherry fields are shown to form a family of codimension one submanifolds of the set of vector fields. The natural paths are shown to be stable paths.

The paper contains the proof of the fact that every solvable locally compact separable group is the range of a cocycle of an ergodic automorphism. The proof is based on the theory of representations of canonical anticommutation relations and the orbit theory of dynamical systems. The slight generalization of reasoning shows further that this result holds for amenable Lie groups as well and can be also extended to almost connected amenable locally compact separable groups.

We give an example of a compact metric space X and a strictly ergodic homeomorphism T of X with invariant probability μ such that for every x ∈ X the set is not generic for μ × μ} is countable.

Every rational function ƒ with degree has a unique invariant probability ƒƒ that maximizes entropy. It has been conjectured that the system (ƒ, μƒ) is equivalent to the one sided Bernoulli shift . In this paper we prove that there exists m >0 such that (ƒm, ƒƒ) is equivalent to .

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’.

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.

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.

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.