Let ℋ denote the group of homeomorphisms of a σ-compact manifold M which preserve a locally positive non-atomic measure μ. Such a manifold can be compactified by adjoining ideal points called ‘ends’, collectively denoted by E. Every homeomorphism h in ℋ induces a measure preserving system (E, , μ*, h*) where is the algebra of clopen subsets of E, μ* is a 0-∞ measure induced on E by μ, and h*: E → E is a μ*-preserving homeomorphism. For any induced homeomorphism σ = f*, where f belongs to ℋ, define ℋσ = {h ∈ ℋ: h* = σ}. We prove that ergodicity is generic in ℋσ for the compact-open topology if and only if (E, , μ*, σ) is incompressible and ergodic. Furthermore ℋσ contains an ergodic homeomorphism if and only if (E, , μ*, σ) is incompressible. Since the identity on M induces the identity on E, which is incompressible, our results establish that every manifold (M, μ) supports an ergodic μ-preserving homeomorphism.

Let f be a continuous map of the circle into itself of degree one. We introduce the notion of rotation algorithms. One of these algorithms associates each z ∈ S1 with an interval, the so-called speed interval S(z, f), which is contained in the rotation interval ρ(f) of f. In contrast with the rotation set ρ(z, f), the interval S(z, f) sometimes allows us to ascertain that ρ(f) is non-degenerate, by using only finitely many elements of {fn (z) | n ≥ 0}. We further show that all choices for ρ(z, f) and S(z, f) occur, for certain z ∈ S1 provided that ρ(z, f) ⊂ S(z, f) ⊂ ρ(f).

We show that if messages of length one, two, three…of a code X which verifies ρx* < ρx are concatenated, we obtain a point generic for an invariant measure on the dynamical system associated to the code; this measure is induced by the maximal measure on the tower studied by Hansel and Blanchard.

Résumé. Nous montrons que si nous alignons les messages de longueur 1, puis 2, et ainsi de suite, d'un code X vérifiant ρx* < ρx, nous obtenons un point générique pour une mesure invariante sur le système dynamique associé au code en question; nous montrons que cette mesure se trouve être la mesure induite par la mesure d'entropie maximale sur la tour associée au code introduite par Hansel et Blanchard. Nous en déduisons des points génériques pour les systèmes sofiques et les θ-shifts munis de leur mesure maximale.

Associated to a rigid rank-1 transformation T is a semigroup ℒ(T) of natural numbers, closed under factors. If ℒ(S) ≠ ℒ(T) then S and T cannot be copied isomorphically onto the same space so that they commute. If ℒ(S) ⊅ ℒ(T) then S cannot be a factor of T. For each semigroup L we construct a weak mixing S such that ℒ(S) = L. The S where ℒ(S) = {l}, despite having uncountable commutant, has no roots.

Preceding and preparing for this example are two others: An uncountable abelian group G of weak mixing transformations for which any two (non-identity) members have identical self-joinings of all orders and powers. The second example, to contrast with the rank-1 property that the weak essential commutant must be the trivial group, is of a rank-2 transformation with uncountable weak essential commutant.

Let ϕ be a flow on a compact metric space Λ and let p ∈ Λ be chain recurrent. We show that (Λ; ℝ) ≠ 0 if dimp Λ = 1 or if p belongs to a section of ϕ. Applications to planar flows and to smooth flows are given.

We prove that for generic domains Ω ⊂ ℝn with smooth boundary X for every integer s ≥ 2 there is at most a finite number of periodic reflecting rays with just s reflections on X.

The author shows that any family C2-close to fα(x) = 1 − αx2(2 − ε ≤ α ≤ 2) satisfies Jakobson's theorem: For a positive measure set of α the transformation fα has an absolutely continuous invariant measure. He also indicates some generalizations.

In this paper the asymptotic behaviour of piecewise monotone functions f: I → I with a finite number of discontinuities is studied (where I ⊆ ℝ is a compact interval). It is shown that there is a finite number of f-almost-invariant subsets C1,…, Cr, R1,…, Rs, where each Ci is a disjoint union of closed intervals and each Rj is a Cantor-like subset of I, such that if x is a ‘typical’ point in I (in a topological sense) then exactly one of the following three possibilities will happen:

(1) {fn (x)}n ≥ 0 eventually ends up in some Ci.

(2) {fn (x)}n ≥ 0 is attracted to some Rj.

(3) {fn (x): n ≥ 0} is contained in an open, invariant set Z ⊆ I, which is such that for each n ≥ 1 fn is monotone and continuous on each connected component of Z.

Moreover, f acts topologically transitively on each Ci and minimally on each Rj. Furthermore, it is shown how the sets C1,…, Cr, R1,…, Rs can be constructed. Finally, our results are applied to some examples.

We estimate from below the measure theoretic entropy of the system of spheres interacting by elastic collisions. We find the asymptotics of the entropy in the case of two disks on atoms as the radius increases. We establish that high density leads to large entropy. We introduce a general estimate for Lyapunov exponents in the case of special symplectic matrices.