Continuous maps from the real line to itself give, in a natural way, a partial ordering of permutations. This ordering restricted to cycles is studied.

Necessary and sufficient conditions are given for a cycle to have an immediate predecessor. When a cycle has an immediate predecessor it is unique; it is shown how to construct it. Every cycle has immediate successors; it is shown how to construct them.

Relationships between the permutations associated to periodic orbits of maps of the interval and the topology of the space of these maps are given. These extend the author's previous work on periodic orbits to orbit types.

We develop simple topological criteria for the existence of periodic orbits in maps of the annulus. These are applied to one-parameter families of dissipative twist maps of the annulus and their attractors. It follows that many of the motions found by variational methods in area preserving twist maps also occur in the dissipative case.

An extension from Riemann to Lorentz manifolds is proved for Anosov's theorem on the hyperbolicity (or exponential sensitivity to initial conditions) of the geodesic flow.

Let A be an n × n real matrix; sufficient conditions were previously worked out, assuming non-commensurability of eigenvectors, for A to be SL(n, )-conjugate to a matrix all sufficiently large powers of which have strictly positive entries. We show that when the ‘large’ eigenvectors are commensurable and satisfy the obvious necessary conditions, then A is also going to be so conjugate. In particular, we deduce, if A is a rational matrix with large eigenvalue exceeding 1 and of multiplicity one, then A is algebraically shift equivalent to an eventually positive matrix, using only integer rectangular matrices.

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

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.

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.

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].

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.

Using elementary methods we show that for every unitary representation π7 of G = SL(2, ℝ) with no non-zero invariant vectors the matrix coefficients 〈 π(a(t))v, w 〉 of

decay exponentially fast for any vectors v, w Holder continuous in the direction of the rotation subgroup of G.

Let M be a compact manifold of not necessarily constant negative curvature. We observe that π1(M) acts amenably on the sphere at infinity of the universal cover of M with respect to a natural measure class. We also note that this action is of type III1.

In this paper, we prove that μ/λ is a modulus for a Šilnikov system with eigenvalues λ and −μ ± iω. To prove this we define a number using knot and link invariants of periodic orbits, which is related to the ratio of eigenvalues μ/λ.

With the purpose of understanding when two subshifts of finite type are equivalent from the point of view of their spaces of Markov measures we propose the notion of Markov equivalence. We show that a Markov equivalence must respect the cycles (periodic orbits) of the subshifts. In particular, Markov equivalent subshifts of finite type have the same zeta function.