An elementary proof is given of Krengel's stochastic ergodic theorem in the setting of multiparameter superadditive processes.

Let f:T3→T3 be a hyperbolic toral automorphism lifting to a linear automorphism with real eigenvalues. We prove that there is a Hölder continuous path in T3 whose orbit-closure is 1-dimensional. This strengthens results of Hancock and Przytycki concerning continuous paths, and contrasts with results of Franks and Mañé concerning rectifiable paths.

In this paper the existence of a unique measure of maximal entropy for rational endomorphisms of the Riemann sphere is established. The equidistribution of pre-images and periodic points with respect to this measure is proved.

The global maps of homomorphisms of directed graphs are very closely related to homomorphisms of a class of symbolic dynamical systems called subshifts of finite type. In this paper, we introduce the concepts of ‘induced regular homomorphism’ and ‘induced backward regular homomorphism’ which are associated with every homomorphism between strongly connected graphs whose global map is finite-to-one and onto, and using them we study the structure of constant-to-one and onto global maps of homorphisms between strongly connected graphs and that of constant-to-one and onto homomorphisms of irreducible subshifts of finite type. We determine constructively, up to topological conjugacy, the subshifts of finite type which are constant-to-one extensions of a given irreducible subshift of finite type. We give an invariant for constant-to-one and onto homomorphisms of irreducible subshifts of finite type.

If (M, g) is a riemannian nilmanifold, the homothetic metrics εg˜ on the universal cover M converge in the sense of Gromov for small ε. In this convergence the volume of balls and the number of closed geodesies go to a limit, and precise asymptotic estimates are given for these numbers.

Real analytic actions of connected Lie groups locally isomorphic to SL (2, ℝ) on compact surfaces, possibly with boundary, are classified up to topological conjugacy and up to real analytic conjugacy. Finite dimensional universal unfoldings of the real analytic conjugacy relation are also constructed. These are local transversals to the conjugacy classes in the space of actions. Sometimes the unfolding is a variety but not a manifold, and thus the space of actions is not naturally modelled on a vector space. We find many rigid actions and some unexpected bifurcation.