Ergodic Theory and Dynamical Systems

A procedure is developed for constructing C1 diffeomorphisms of the two sphere having inverse limits of certain interval maps as attractors. The method is carried out for a particular interval map yielding a diffeomorphism with a transitive non-hyperbolic attractor.

Let T1,…,Tn be continuous representations of a σ-compact separable locally compact amenable group G as measure-preserving transformations of a non-atomic separable probability space (X, β, m). Let (Kn) be a right Følner sequence of compact sets in G. If T1,…,Tn are pairwise commuting in the sense that Ti(g)Tj(h) = Tj(h)Ti(g) for i ≠ j and g, h ∈ G, then necessary and sufficient conditions can be given, in terms of the ergodicity of certain tensor products, for the following to hold: for all F1,…,Fn∈L∞, the sequence AN(x) where

converges in L2(X) to . The necessary and sufficient conditions are that each of the following representations are ergodic: Tn, Tn−1⊗Tn−1Tn,…,T2⊗T2T3⊗…⊗T2…Tn, T1⊗T1T2⊗…⊗T1…Tn.

In order to prove this theorem, specific properties of the decomposition of L2(X) into its weakly mixing and compact subspaces with respect to a representation Ti are needed. These properties are also used to prove some generalizations of wellknown facts from ergodic theory in the case where G is the integer group Z.

A highly developed branch of the modern theory of dynamical systems is the study of deterministic ones with statistical properties in behaviour. During the last decade such systems were discovered in various domains of physics, chemistry, biology and technology. It is due to their complexity, that only the simplest of such systems have been analytically investigated (Lorenz system, Rikitake dynamo, billiard systems), and that is why numerical methods are widely used, especially in applied investigations. In numerical modelling we have no true trajectory of a dynamical system f, but an approximation = (x1, x2,…) such that the sequence of distances is small in some sense. For the case of round off errors in computer modelling, such a sequence is uniformly small, i.e. there exists some ε > 0, such that supnρ(xn+1, fxn)<ε. The sequence in this case is called an ε-trajectory of the dynamical system f[1]. In a series of investigations [1–14] a study was made of the properties and applications of ε-trajectories.

We consider interval exchange transformations T for which the lengths of the exchanged intervals have linear rank 2 over the field of rationals. We prove that, for such T, minimality implies unique ergodicity. We also provide an algorithm which tests T for aperiodicity and minimality.

Let f be a C0 circle map of degree one with precisely one local minimum and one local maximum, and let [ρ−(f), ρ+(f)] be the interval of rotation numbers of f. We study the structure of the function ρ(λ) = ρ+(Rλ∘f), where Rλ is the rotation through the angle λ.

We show that a full solenoid is locally the product of a euclidean component and p-adic components for each rational prime p. An automorphism of a solenoid preserves these components, and its topological entropy is shown to be the sum of the euclidean and p-adic contributions. The p-adic entropy of the corresponding rational matrix is computed using its p-adic eigenvalues, and this is used to recover Yuzvinskii's calculation of entropy for solenoidal automorphisms. The proofs apply Bowen's investigation of entropy for uniformly continuous transformations to linear maps over the adele ring of the rationals.

We find a class of C∞ maps of an interval with zero topological entropy and chaotic in the sense of Li and Yorke.

A positive Liapunov exponent for the critical value of an S-unimodal mapping implies a positive Liapunov exponent of the backward orbit of the critical point, uniform hyperbolic structure on the set of periodic points and an exponential diminution of the length of the intervals of monotonicity. This is the proof of the Collet-Eckmann conjecture from 1981 in the general case.

We define an ergodic ℤ-foliation and show that it can be realized as a quotient space of the ‘covering space’. The covering space has two actions, T and S, where T is a ℤ-action, S is a map of order two, and S and T skew-commute; that is, STS = T−1. We study the isometry between two foliations via the isomorphism between two bigger group actions in the covering spaces. Properties of an ergodic foliation are studied in a way similar to the study of an ergodic action. We construct a counterexample of a K-automorphism to show that, unlike Bernoulli automorphisms, ℤ-actions do not completely determine ℤ-foliations.

All subshifts of finite type are known to appear as basic parts of the non-wandering sets of Smale diffeomorphisms in dimensions three or more. This paper concerns the symmetries of subshifts of finite type; that is, the homeomorphisms of the shift space which commute with the shift. The group of symmetries is known to be very large for aperiodic shifts. For certain (structually stable) Smale diffeomorphisms of the sphere of dimension five or more, we show each symmetry can be extended to a homeomorphism of the sphere commuting with the diffeomorphism on the whole sphere.

We exhibit a transitive sofic system for which the core matrix has negative trace, and hence cannot share the nonzero spectrum of any subshift of finite type cover. We also show that every transitive sofic system has an integral core matrix.

We regret to say that there is an error in the proof of theorem 1 of [F]. The algebras E*(U), E*(S) used there are not preserved by d, as claimed, save in the restricted case when U and S commute. A counterexample is the nilmanifold automorphism of [AA], Appendix 23. Thus the results of §1 and §2 are dubious, although the third section is unaffected.