Let (X, S) be a shift of finite type. Let G be the group of automorphisms of (X, S) which are compositions of elements of finite order in the kernel of the dimension representation. We characterize the action of G on finite subsystems of (X, S).

We show that any symplectic Anosov diffeomorphism of a four torus T4 with sufficiently smooth stable and unstable foliations is smoothly conjugate to a linear hyperbolic automorphism of T4.

For certain Cr 3-dimensional cooperative or competitive vector fields F, where r is any positive integer, it is shown that for any nonwandering point p, every neighborhood of F in the Cr topology contains a vector field for which p is periodic, and which agrees with F outside a given neighborhood of p. The proof is based on the existence of invariant planar surfaces through p.

We prove two extensions of Sacksteder's Theorem for the action A: Γ × S1 → S1 of a finitely-generated group Γ on the circle by C1+α-diffeomorphisms. If the action A has an exceptional minimal set K with a gap endpoint of exponential orbit growth rate, or if the action A on K has positive topological entropy, then the exceptional set K is hyperbolic. That is, A has a linearly contracting fixed-point in K. A key point of the paper is to prove a foliation closing lemma using the foliation geodesic flow technique.

We consider small random perturbations of matrix cocycles over Lipschitz homeomorphisms of compact metric spaces. Lyapunov exponents are shown to be stable provided that our perturbations satisfy certain regularity conditions. These results are applicable to dynamical systems, particularly to volume-preserving diffeomorphisms.

In this paper we prove the following theorem.

Theorem 1. For a measure-preserving system (X, β, μ, T) and a positive integer k, if f ∈ L2(X, β, μ), the averages

,

converge μ almost everywhere. Here p runs over the rational primes and πN denotes their number in [1, N].

For a mixing stationary Markov chain we prove a Poisson limit law for the recurrence to small cylindrical sets. Since hyperbolic torus automorphisms are Markov chains, the result carries over to these transformations.

By using ergodic theory new estimates of Weyl sums and new estimates of the remainder term in the law of distribution of fractional parts of the polynomial are obtained.

We consider the problem of when does a positive entropy topological system have a continuous factor with strictly smaller entropy. In many cases it is shown that such small entropy factors exist. On the other hand, classes of examples are given where differentiable factors must preserve some of the original entropy.

Liggett has proved that for spin systems, Markov processes with state space {0,1}ℤn, there is a unique stationary distribution in the M < ε regime and all initial configurations uniformly approach this unique stationary distribution exponentially in the weak topology. Here, M and ε are two parameters of the system. We extend this result to discrete time but strengthen it by proving exponential convergence in the stronger - metric instead of the usual weak topology. This is then used to show that the unique stationary process with state space {0,1}ℤn and index set ℤ is isomorphic (in the sense of ergodic theory) to an independent process indexed by ℤ. In the translation invariant case, we prove the stronger fact that this stationary process viewed as a {0, l}-valued process with index set ℤn × ℤ (spacetime) is isomorphic to an independent process also indexed by ℤn × ℤ. This shows that this process is in some sense the most random possible. An application of this last result to approximating by an infinite number of finite systems concatenated independently together is also presented. Finally, we extend all of these results to continuous time.

Herein the authors show that discrete groups of motions on Hn+1 may be conservative on Sn but have no positive measure ergodic components for this boundary action. An explicit example of such a group is given for H3 using the Apollonian circle packing of R2.

We consider some ways in which regularity of a mapping influences dynamics of its iterations and growth of various complexity-type invariants.