For every manifold of dimension n ≥ 5 a diffeomorphism f which has n − 1 non-zero characteristic exponents almost everywhere is constructed. The diffeomorphism preserves the Lebesgue measure and is Bernoulli with respect to this measure. To produce this example a diffeomorphism of the 2-disk is extended by means of an Anosov flow, and this skew product is embedded in ℝn.

We show that, for maps of the interval, the non-wandering set of the map coincides with the non-wandering set of each of its odd powers, while the nonwandering set of any of its even powers can be strictly smaller.

The authors investigate which results of the classical mean ergodic theory for bounded linear operators in Banach spaces have analogues for subadditive sequences (Fn) in a Banach lattice B. A sequence (Fn) is subadditive for a positive contraction T in B if Fn+k ≤ Fn + TnFk (n, k ≥ 1). For example, von Neumann's mean ergodic theorem fails to extend to the general subadditive case, but it extends to the non-negative subadditive case. It is shown that the existence of a weak cluster point f = Tf for (n−1Fn) implies In Lp (1 ≤ p < ∞) the existence of a weak cluster point for non-negative (n−1Fn) is equivalent with norm convergence. If T is an isometry in Lp (1 < p < ∞) and sup then n−1Fn converges weakly. If T in L1 has a strictly positive fixed point and sup then n−1Fn converges strongly. Most results are proved even in the d-parameter case.

Let ℬ and be minimal flows which are disjoint over their intersection. A dynamical interpretation is given of the groups H1(B/S, C/S) and H2(B/S, C/S), where B and C are the groups of ℬ and respectively, and S is the group of the flow ℬ ⋁ .

We construct a real analytic diffeomorphism Fα on a compact connected 4-dimensional manifold M, such that Fα preserves a probability measure μ defined by a smooth volume form, Fα is a minimal diffeomorphism of M and furthermore the metrical entropy of Fα with respect to the measure μ is strictly positive. By a theorem of Goodwyn the topological entropy is also strictly positive. We write down the explicit formula of Fα that depends on a parameter α ∈ T1. This parameter is chosen by Baire category.

We are interested in ergodic properties of absolutely continuous invariant measures of positive entropy for a map of an interval. We prove a Bernoulli property and a characterization by some variational principle.

Let f: M → M be a diffeomorphism of a compact manifold M and let χ:M → R be defined by putting χ(x) equal to the sum of the non-negative characteristic exponents of f at x, each being counted with its multiplicity. If μ is an f-invariant probability of M which is absolutely continuous relative to Lebesgue measure, then Pesin has proved the entropy, hμ(f), is given by We prove this formula without using the theory of stable manifolds.

It is proved that for the map f: ℂ → ℂ given by f(z) = ez, the family is not normal at any point. In particular, f is topologically transitive.

This paper concerns a problem which arose from a paper of Sullivan. Let Γ be a discrete group of isometries of hyperbolic space Hd+1. We study the question of when the geodesic flow on the unit tangent bundle UT (Hd+1/Γ) of Hd+1/Γ is ergodic with respect to certain natural measures. As a consequence, we study the question of when Γ is of divergence type. Ergodicity when the non-wandering set of UT (Hd+1/Γ) is compact is already known from the theory of symbolic dynamics, due to Bowen, or from Sullivan's work. For such a Γ, we consider a subgroup Γ1 of Γ with Γ/Γ1 ≅ℤυ and prove the geodesic flow on UT (Hd+1/Γ1) is ergodic (with respect to one of these natural measures) if and only if υ ≤ 2.