Denote by G the group of all μ-preserving bijections of a Lebesgue probability space (X, Σ, μ) and by C the conjugacy class of an antiperiodic transformation σ in G. We present several new results concerning the denseness of C in G with respect to various topologies. One of these asserts that given any weakly mixing transformation τ in G and any F with μ(F) < 1, there is a transformation in C which agrees with τ a.e. on F.

We study minimal flows and their extensions by means of the associated maximally highly proximal flows. These, in turn, can be represented by highly proximal generators, which are certain subsets of the universal minimal flow. From this point of view we obtain information on relative disjointness, coalescence, the Bronstein property, and RIC extensions.

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.

Let M be a two-dimensional, compact manifold and g:Μ→ΜM be a diffeomorphism with a hyperbolic chain recurrent set. We find restrictions on the reduced zeta function p(t) of anyzero-dimensional basic set of g. If deg (p(t)) is odd, then p(1) = 0 (in ). Since there are infinitely many subshifts of finite type whose reduced zeta functions do not satisfy these restrictions, there are infinitely many subshifts which cannot be basic sets for any diffeomorphism of any surface.

Let f be a continuous map of the circle to itself. Let P(f) denote the set of periods of the periodic points. In this paper the set P(f) is studied for functions without fixed points, so 1∉P(f). In particular, it is shown that if s, t are the two smallest integers in P(f) and s and t are relatively prime then αs+βt∈P(f) for any positive integers α and β.

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.

In this paper we construct an example of a homeomorphism of the circle onto itself which is C∞, has no periodic points and no dense orbits. Moreover, the homeomorphism will have no more than two points of zero derivative. We alter this example to form a C∞ map of an interval to itself which has homtervals.

For each n≥2, we find the minimum value of the topological entropies of all continuous self-maps of the circle having a fixed point and a point of least period n, and we exhibit a map with this minimal entropy.

On the infinite torus X = , using a category argument, we produce a large family of homeomorphisms such that for every element S in this family the flow (S, X) is weakly mixing and strictly ergodic. Moreover, writing Xn = {n, n + 1…} and letting Πn, m, for n<m, be the projection of Xn on Xm, S induces for every n, a homeomorphism of Xn and the extensions are isometric. We also show that, for every S in this family, (S, X) is disjoint from every purely weakly mixing flow.

There exists a Bernoulli shift with non-identical factor measures for which no invariant σ-finite equivalent measure exists.

Let α be an automorphism of a finite von Neumann algebra and let H(α) be its Connes-Størmer's entropy. We show that H(α) = 0 if α is the induced automorphism on the crossed product of a Lebesgue space by a pure point spectrum transformation. We also show that H is not continuous in α and we compute H(α) for some α.

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.

We prove that for any amenable non-singular countable equivalence relation R⊂X×X, there exists a non-singular transformation T of X such that, up to a null set:

It follows that any two Cartan subalgebras of a hyperfinite factor are conjugate by an automorphism.

Transformations on [0, 1] which are piecewise monotonic and piecewise continuous are considered. Using symbolic dynamics, the structure of their nonwandering set is determined. This is then used to prove results about maximal and absolutely continuous invariant measures.

If p and q are probability vectors of equal entropy each having at least three non-zero components then there exists a finitary homomorphism between the corresponding one-sided Bernoulli shifts.

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 consider a certain analytic function β (t) which is an invariant of finite equivalence between two finite state Markov chains. If two such chains P, Q have the same β-function we wish to prove that they are finitely equivalent. To this end we show that U(t)Pt = QtU (t) has a nontrivial matrix solution U over the ring (exp) of integral combinations of exponential functions. In fact we can force U(t) to be strictly positiveat any specified t0. If U(t) has entries from (exp), the sub-semi-ring of positive integral combinations of exponential functions, then P, Q are finitely equivalent. Many examples reinforce the conjecture that U(t) may always be chosen over (exp) when P, Q have the same β-function. We relate the β-function to topological entropy, measure entropy and information variance.

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.

For an Axiom A diffeomorphism of a surface with an ergodic invariant measure we prove that the entropy is the product of the positive Lyapunov exponent and the Hausdorff dimension of the set of generic points in an unstable manifold.

We examine generalizations of R. Mañé's results on the topological dimension of spaces supporting an expansive homeomorphism to the case of real-expansive flows. We show that a space supporting a real-expansive flow must be finite dimensional, and a minimal real-expansive flow not exhibiting a type of spiral behaviour must be one-dimensional. This latter class includes all known examples and a slight generalization of Axiom A flows. These results are obtained by introducing a new concept of stable and unstable sets for real flows, and examining real-expansive flows in terms of these sets.