This work is concerned with the generic existence of homoclinic points for area preserving diffeomorphisms of compact orientable surfaces. We give a shorter proof of Pixton's theorem that shows that, Cr-generically, an area preserving diffeomorphism of the two sphere has the property that every hyperbolic periodic point has transverse homoclinic points. Then, we extend Pixton's result to the torus and investigate certain generic aspects of the accumulation of the invariant manifolds all over themselves in the case of symplectic diffeomorphisms of compact manifolds.

The study of area preserving maps of manifolds has an extensive history in the theory of dynamical systems. One interest has been in the behaviour of such maps near an isolated fixed point. In 1974 Carl Simon proved the existence of an upper bound for the index of an isolated fixed point for Ck area preserving diffeomorphisms of a surface. We extend his result to homeomorphisms of an orientable two manifold. The proof utilizes the notion of free modification, developed by Morton Brown, and enlarges the scope of the problem to the consideration of ‘nice’ measures, i.e. uniformly equivalent to Lebesgue measure on compact sets. By suitably modifying the homeomorphism and the measure, we obtain the following theorem.

The Riemannian flat metrics on tori T2 are characterized by a weakly ergodic property of the geodesic flows.

This paper discusses a division of one-dimensional cellular automata into classes which seem to correspond to different types of behaviour.

It is shown that for generic domains D in n, n ≥ 2, every periodic billiard trajectory in D passes only once through each of its reflection points, and any two different periodic billiard trajectories in D have no common reflection point.

Let T be an ergodic measure preserving point transformation from a probability space X onto itself. Assume that is an increasing sequence of subsets of the positive integers. Conditions are given which are sufficient for the ergodic maximal function associated with these subsets to be weak type (p, p). These conditions are shown to be both necessary and sufficient for a larger two-sided maximal function. The conditions are in the form of covering lemmas for the integers.

Ruelle and Capocaccia gave a new definition of Gibbs states on Smale spaces. Equilibrium states of suitable functions thereon are known to be Gibbs states. The converse is discussed in this paper, where the problem is reduced to shift spaces and solved there by constructing suitable conjugating homeomorphisms in order to verify the conditions for Gibbs states which Bowen gave for shift spaces, where the equivalence to equilibrium states is known.

If X is a compact metric space and T a homeomorphism of X we say (X, T) has almost minimal power joinings (AMPJ) if there is a dense GδX* in X such that for each finite set k, x∈(X*)k and l:k → ℤ−{0}, the orbit closure cl {} is a product of off-diagonals (POOD) on Xk. By an offdiagonal on Xk′, k′k we mean a set of the form (⊗,j∈k′Tm(j))Δ, Δ the diagonal in Xk′, m:k′→ℤ any function, and by a POOD on Xk we mean that k is split into subsets k′, on each Xk′ we put an off-diagonal and then we take the product of these.

We show that examples of AMPJ exist and that this definition leads to a theory completely analogous to Rudolph's theory of minimal self-joinings in ergodic theory. In particular if (X, T) has AMPJ the automorphism group of T is {Tn}, T has only almost 1-1 factors (other than the trivial one) and the automorphism group and factors of ⊕i ∊ kT, k finite or countably infinite, can be very explicitly described. We also discuss ℝ-actions.

Let h(φ) be the topological entropy of a real continuous flow φ on a compact metric space X. Introducing an equivalent definition for the topological entropy on an expansive real flow enables us to investigate the topological entropies of mutually conjugate expansive flows and estimate the periodic orbits of an expansive flow which has the pseudo-orbit tracing property.

According to M. Gromov, any sequence of Riemann manifolds with uniformly bounded geometry has a subsequence that converges to a limit. It is shown here that this limit Riemann structure is Lipschitz, generates a Lipschitz geodesic flow, and consequently, as Gromov asserted, the limit distance function is of class C1,1. Sharpness of the results is discussed. A simple, extrinsic proof of Gromov's Theorem is included.

The continuity of Hausdorff dimension of closed invariant subsets K of a C2-expanding mapping g of the circle is investigated. If g/K satisfies the specification property then the equilibrium states of Hölder continuous functions are studied. It is proved that if f is a piecewise monotone continuous mapping of a compact interval and φ a continuous function with P(f,φ)> sup(φ), then the pressure P(f,φ) is attained on one-dimensional ‘Smale's horseshoes’, and some results of Misiurewicz and Szlenk [M−Sz] are extended to the case of pressure.

We construct a rank-1 map T similar to Chacon's map [C], [JRS], but with 2n+1n-blocks in an (n + 1)-block and a single spacer in the middle. Hence T is rigid and the centralizer of T, C(T), is uncountable. We show T is simple [V], [JR], and hence any factor of T is the algebra of invariant sets of some weakly compact subgroup of C(T). We show C(T) has no such subgroups and hence T is prime. Lastly we show that T is graphic in the sense of [AM].

Confirming a conjecture by Boshernitzan, it is proved that if T is a minimal non-uniquely ergodic interval exchange, the minimum spacing of the partition determined by Tn is O(1/n).

If X is a compact, zero-dimensional group and T is an expansive, transitive automorphism then (X, T) is shown to be topologically conjugate to a full shift on finitely many symbols.

The problem of classifying such automorphisms up to simultaneous algebraic isomorphism and topological conjugacy is discussed but not solved. It is proved that for any entropy there are only finitely many such equivalence classes. When the entropy is log p for a prime p, there is only one equivalence class. All are then equivalent to

It is known that if Ω is a Lebesgue space, T:Ω→Ω is a loosely Bernoulli transformation, and L is a fixed non-zero integer, then the transformation S = TL will again be loosely Bernoulli on each ergodic component. In this note, the above stated result is extended to include the case where L is an arbitrary integrable integer-valued function on Ω.