The ‘backward continued fraction’ map studied by A. Reyni is defined by y = g(x) where g(x) equals the fractional part of 1/(1−x) for 0 < x < 1. We show that it is a factor map of a special cross-section map for the geodesic flow on the unit tangent bundle of the modular surface. This gives an alternative derivation of the fact that this map preserves the infinite measure dx/x on the unit interval.

We prove the rational egodicity of geodesic flows on divergence type surfaces of constant negative curvature, and identify their asymptotic types.

We investigate a variant of the baker's transformation in which the mapping is onto but is not one-to-one. The Bowen-Ruelle measure for this map is investigated.

We discuss weak disjointness of homomorphisms of minimal transformation groups and use the techniques involved to deepen our knowledge of the equicontinuous structure relation.

Let G be a connected Lie group and let Γ be a lattice in G (not necessarily co-compact). We show that if (ut) is a unipotent one-parameter subgroup of G then every ergodic invariant (locally finite) measure of the action of (ut) on G/Γ is finite. For ‘arithmetic lattices’ this was proved in [2]. The present generalization is obtained by studying the ‘frequency of visiting compact subsets’ for unbounded orbits of such flows in the special case where G is a connected semi-simple Lie group of ℝ-rank 1 and Γ is any (not necessarily arithmetic) lattice in G.

The notions of ergodicity, strong mixing and weak mixing are defined and studied for arbitrary sequences of measure-preserving transformations of a probability space. Several results, notably ones connected with mean ergodic theorems, are generalized from the case of the sequence of all powers of a single transformation to this case. The conditions for ergodicity, strong mixing and weak mixing of sequences of affine transformations of compact groups are investigated.

Continuous maps from the real line to itself give, in a natural way, a partial ordering of permutations. This paper studies the structure of simple permutations which have order a power of two, where simple permutations are permutations corresponding to the simple orbits of Block.

The rotation number of a diffeomorphism f: S1 → S1 with lift is defined as . We investigate the case where f is an endomorphism. Then this limit may not exist and may depend on x. We investigate the set of limit points of , as a function of x.

Let Σ be a commutative semigroup of continuous endomorphisms of the r-dimensional torus. Generalizing a result of Furstenberg dealing with the circle group, necessary and sufficient conditions are given here for Σ to possess the following property: Any Σ-minimal set consists of torsion elements. Semigroups not having this property are shown to admit minimal sets of positive Hausdorff dimension.

The property of a one-parameter C*- (or W*-) dynamical system that the spectral subspaces corresponding to the three subsets (−∞, 0), {0}, and (0, +∞) add up to the whole algebra is reformulated. If the C*-algebra is prime (or the W*-algebra is a factor), an equivalent property is that the spectrum is finite.

Only finitely many shift equivalence classes of non-negative aperiodic integral matrices may share a given diagonal Jordan form away from zero. The diagonal assumption is necessary.

We describe the dynamical behaviour of the entire transcendental function exp(z). We use symbolic dynamics to describe the complicated orbit structure of this map whose Julia Set is the entire complex plane. Bifurcations occurring in the family c exp(z) are discussed in the final section.

A complete set of computable invariants is given for deciding whether two irreducible subshifts of finite type have topologically equivalent suspension flows.

An ergodic flow is said to have the weak Pinsker property if it admits a decreasing sequence of factors whose entropies tend to zero and each of which has a Bernoulli complement. We show that this property is preserved under taking factors and d-limits. In addition, we show that a flow has the weak Pinsker property whenever one ergodic transformation in the flow has this property.

We prove that any locally compact, non-compact, second countable group acts minimally on any metrizable connected manifold modelled on the separable Hilbert space.

A specific singularity of a vector field on is considered, of codimension 2 in the dissipative case and of codimension 1 in the conservative case. In both contexts in generic unfoldings the existence is proved of subordinate Šil'nikov bifurcations, which have codimension 1. Special attention is paid to the C∞-flatness of this subordinate phenomenon.

Let f be a continuous map of degree one of the circle onto itself. We prove that for every number a from the rotation interval of f there exists an invariant closed set A consisting of points with rotation number a and such that f restricted to A preserves the order. This result is analogous to the one in the case of a twist map of an annulus.

Poincare's recurrence theorem says that, given a measurable subset of a space on which a finite measure-preserving transformation acts, almost every point of the subset returns to the subset after a finite number of applications of the transformation. Moreover, Kac's recurrence theorem refines this result by showing that the average of the first return times to the subset over the subset is at most one, with equality in the ergodic case. In particular, the first return time function to any measurable set is integrable. By considering the supremum over all p ≥ 1 for which the first return time function is p-integrable for all open sets, we obtain a number for each almost-topological dynamical system, which we call the return time invariant. It is easy to show that this invariant is non-decreasing under finitary homomorphism. We use the invariant to construct a continuum number of countable state Markov shifts with a given entropy (and hence measure-theoretically isomorphic) which are pairwise non-finitarily isomorphic.

We consider Anosov flows on closed 3-manifolds which are circle bundles. Our main result is that, up to a finite covering, these flows are topologically equivalent to the geodesic flow of a suface of constant negative curvature. The same method shows that, if M is a closed hyperbolic manifold of any dimension, all the geodesic flows which correspond to different metrics on M and which are of Anosov type are topologically equivalent.

We formulate sufficient conditions under which, for a finite subset of SL (2, ℝ), the maximal Liapunov exponent is positive. These conditions are based on the notion of compatible hyperbolicity. We then give an analytical formulation of such a condition and we apply this criterion to prove mixing properties of a particular transformation of the two-dimensional torus.