Ergodic Theory and Dynamical Systems

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.

On the 20th of October 1987 one of the greatest scholars of our time, Andrei Nikolaevich Kolmogorov, passed away.

The totally minimal flow (X, T) is said to have finite almost periodic rank if there is a positive integer n such that whenever (x1, x2,…, xn+1) is an almost periodic point of the product flow (Xn+1, T×…×T) then, for some i≠j, xi, and xj are in the same orbit. The rank of (X, T) is the smallest such integer. If (Y, S) is a graphic flow, (Y, Sn) has rank |n| and it is shown that every finite rank flow has, modulo a proximal extension, a graphic power factor. Various classes of finite rank flows are defined, and characterized in terms of their Ellis groups. There are four disjoint types which have basic structural differences.

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 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).

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.

When considering an action α of a compact group G on a C*-algebra A, the notion of an α-invariant Hilbert space in A has proved extremely useful [1, 4, 8, 14, 17, 18]. Following Roberts [13] a Hilbert space in (a unital algebra) A is a closed subspace H of A such that x*y is a scalar for all x, y in H. For example if G is abelian, and α is ergodic in the sense that the fixed point algebra Aα is trivial, then A is generated as a Banach space by a unitary in each of the spectral subspaces

which are then invariant one dimensional Hilbert spaces. If G is not abelian, then Hilbert spaces (which are always assumed to be invariant) do not necessarily exist, even for ergodic actions. For non-ergodic actions, it is also desirable to relax the requirement to x*y being a constant multiple of some positive element of Aα. More generally, if γ is a finite dimensional matrix representation of G and n is a positive integer, we define to be the subspace

where d is the dimension d(γ) of γ, and Mnd denotes n×d complex matrices. (Usually we will denote the extended action of αg to αg ⊗ 1 on A⊗Mnd again by αg.) Let .

In the fall of 1957 A. N. Kolmogorov started lecturing on the theory of dynamical systems and supervised a seminar on the same theme at the Mechanical–Mathematical Department of Moscow State University. He began his lectures with the theory of systems with a pure point spectrum, which he approached from a probabilistic point of view. This approach, undoubtedly, has many advantages. In the seminar we studied Ito's theory of multiple stochastic integrals and, under the supervision of A. N. Kolmogorov, I. V. Girsanov constructed an example of a Gaussian dynamical system with a simple continuous spectrum. At one of the meetings of the seminar, still before the advent of entropy, Kolmogorov suggested a proof of an assertion, which today would read: the unitary operator induced by a K-automorphism has a countable Lebesgue spectrum. At this point Kolmogorov was studying Shannon's theory of information and the concept of the capacity of functional spaces. Judging by his well known article, we can say that the first of these, and all that is related to it, played a big role in the development of information theory in our country. The investigation of capacity is connected with Kolmogorov's work on Hilbert's 13th problem and was summarized in his well-known survey co-authored by V. M. Tihomirov.

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.

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.

If f is a transcendental entire function and D is a non-wandering component of the set of normality of the iterates of f such that fn → ∞ in D then log |fn(z)| = O(n) as n → ∞ for z in D. For a wandering component the convergence of fn to ∞ in D may be arbitrarily fast.

Each flow dominates a unique gradient flow (the gradient part) which in turn dominates any other gradient flow dominated by the original flow. The set of points carried to rest points of the gradient part (the chain recurrent set) is characterized in terms of a recurrence relation. Filtrations of the given flow correspond to those of the gradient part.

If the action induced by a pseudo-Anosov map on the first homology group is hyperbolic, it is possible, by a theorem of Franks, to find a compact invariant set for the toral automorphism associated with this action. If the stable and unstable foliations of the Pseudo-Anosov map are orientable, we show that the invariant set is a finite union of topological 2-discs. Using some ideas of Urbański, it is possible to prove that the lower capacity of the associated compact invariant set is >2; in particular, the invariant set is fractal. When the dilatation coefficient is a Pisot number, we can compute the Hausdorff dimension of the compact invariant set.

This paper concerns estimates for periodic solutions of a very general class of Hamiltonian systems of prescribed energy. The estimtes are a priori upper and lower bounds for the action integral in terms of the period.

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.

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.

We define the notions of the pseudo-rotation set and rotation shadowing of pseudo-orbits for endomorphisms of the circle and for homeomorphisms of the annulus. The results include: the rotation shadowing property holds for all endomorphisms of the circle; the pseudo-rotation set equals the closure of the rotation set; the closure of the rotation set varies upper-semicontinuously.

Let ƒ denote a continuous map of a compact interval I to itself. A point x ∈ I is called a γ-limit point of ƒ if it is both an ω-limit point and an α-limit point of some point y ∈ I. Let Γ denote the set of γ-limit points. In the present paper, we show that (1) −Γ is either empty or countably infinite, where denotes the closure of the set P of periodic points, (2) x ∈ I is a γ-limit point if and only if there exist y1 and y2 in I such that x is an ω-limit point of y1, and y1 is an ω-limit point of y2, and if and only if there exists a sequence y1, y2,…of points in I such that x is an ω-limit point of y1, and yi is an ω-limit point of yi+1 for every i ≥ 1, and (3) the period of each periodic point of ƒ is a power of 2 if and only if every γ-limit point is recurrent.