Unique Ergodicity Research Articles

Corrections to ‘Invariant measures for higher-rank hyperbolic abelian actions’

The proofs of Theorems 5.1 and 7.1 of [2] contain a gap. We will show below how to close it under some suitable additional assumptions in these theorems and their corollaries. We will assume the notation of [2] throughout. In particular, $\mu$ is a measure invariant and ergodic under an $R^k$-action $\alpha$. Let us first explain the gap. Both theorems are proved by establishing a dichotomy for the conditional measures of $\mu$ along the intersection of suitable stable manifolds. They were either atomic or invariant under suitable translation or unipotent subgroups $U$. Atomicity eventually led to zero entropy. Invariance of the conditional measures showed invariance of $\mu$ under $U$. We then claimed that $\mu$ was algebraic using, respectively, unique ergodicity of the translation subgroup on a rational subtorus or Ratner's theorem (cf. [2, Lemma 5.7]). This conclusion, however, only holds for the $U$-ergodic components of $\mu$ which may not equal $\mu$. In fact, in the toral case, the $R^k$-action may have a zero-entropy factor such that the conditional measures along the fibers are Haar measures along a foliation by rational subtori. Since invariant measures with zero entropy have not been classified, we cannot conclude algebraicity of the total measure $\mu$ at this time. In the toral case, the existence of zero entropy factors turns out to be precisely the obstruction to our methods. The case of Weyl chamber flows is somewhat different as the ‘Haar’ direction of the measure may not be integrable. In this case, we need to use additional information coming from the semisimplicity of the ambient Lie group to arrive at the versions of Theorem 7.1 presented below.

Unique ergodicity and the approximation of attractors and their invariant measures using Ulam's method

Asymptotically stable attractors supporting an invariant measure, for which the ergodic theorem holds almost everywhere with respect to Lebesgue measure, can be approximated by a space discretization procedure called Ulam's method. As an application of this result we propose the use of this method to approximate the `chaotic' attractors of flows in lower dimensions. A Monte Carlo implementation makes this feasible. The approximation method can be extended to attractors whose neighbourhoods contain positively invariant compact sets called blocks. Note that such attractors can fail to have open basins of attraction. When the attractor is uniquely ergodic, we also prove the weak convergence of the approximate measures constructed by the method and as an application, we show the weak convergence of Ulam's method for the logistic map at the Feigenbaum parameter value. More generally, using the work of Buescu and Stewart on transitive attractors of continuous maps, we prove weak convergence of the approximate measures and convergence of their supports to classes of Lyapounov stable attracting Cantor sets.

Minimality and unique ergodicity for subgroup actions

Minimality and unique ergodicity for subgroup actions

The cohomological equation for area-preserving flows on compact surfaces

We study the equation Xu = f where X belongs to a class of area-preserving vector fields, having saddle-type singularities, on a compact orientable surface M of genus g ≥ 2. For a “full measure” set of such vector fields we prove the existence, for any sufficiently smooth complex valued function f in a finite codimensional subspace, of a finitely differentiable solution u. The loss of derivatives is finite, but the codimension increases as the differentiability required for the solution increases, so that there are a countable number of necessary and sufficient conditions which must be imposed on f , in addition to infinite differentiability, to obtain infinitely differentiable solutions. This is related to the fact that the ”Keane conjecture” (proved by several authors such as H.Masur, W.Veech, M.Rees, S.Kerckhoff, M.Boshernitzan), which implies that for ”almost all” X the unique ergodicity of the flow generated by X on the complement of its singularity set, does not extend to distributions. Indeed, our approach proves that, for “almost all” X, the vector space of invariant distributions not supported at the singularities has infinite (countable) dimension, while according to the Keane conjecture the cone of invariant measures is generated by the invariant area form ω.

MARKOV SEMIGROUPS ON UHF ALGEBRAS

We consider a class of Markov semigroups on UHF algebras. We establish the existence of dynamics for long range interactions. Our idea is a non-commutative extension of the argument for classical interacting particle systems. As a by-product we obtain sufficient conditions for unique ergodicity.

A Characterization of the Uniquely Ergodic Endomorphisms of the Circle

A Characterization of the Uniquely Ergodic Endomorphisms of the Circle

A condition for unique ergodicity of minimal symbolic flows

AbstractA sufficient condition for unique ergodicity of symbolic flows is provided. In an important but special case of interval exchange transformations, the condition has already been validated by W. Veech.

Unique ergodicity of continuous self-maps of the circle

Let ƒ: S 1 → S 1 be a continuous self-map of the circle. We show that ƒ is uniquely ergodic iff ƒ has at most one periodic orbit.

A general condition for lifting theorems

We define a general condition, called stability on extensions T T of measure preserving transformations S S . Stability is defined in terms of relative unique ergodicity, and as a joining property. Ergodic compact group extensions are stable, and moreover stable extensions satisfy lifting theorems similar to those satisfied by group extensions. In general, stable extensions have relative entropy zero. In the class of continuous flow extensions over strictly ergodic homeomorphisms, stable extensions are generic.

Distal Functions and Unique Ergodicity

Distal Functions and Unique Ergodicity

Distal functions and unique ergodicity

A. Knapp [5] has shown that the set, D ( S ) D(S) , of all distal functions on a group S S is a norm closed subalgebra of l ∞ ( S ) {l^\infty }(S) that contains the constants and is closed under the complex conjugation and left translation by elements of S S . Also it is proved that [7] for any k ∈ N k \in \mathbb {N} and any λ ∈ R \lambda \in \mathbb {R} the function f : Z → C f:\mathbb {Z} \to \mathbb {C} defined by f ( n ) = e i λ n k f(n) = {e^{i\lambda {n^k}}} is distal on Z \mathbb {Z} . Now let W {\mathbf {W}} be the norm closure of the algebra generated by the set of functions \[ { n ↦ e i λ n k : k ∈ N , λ ∈ R } , \{ n \mapsto {e^{i\lambda {n^k}}}:k \in \mathbb {N},\;\lambda \in \mathbb {R}\} , \] which will be called the Weyl algebra. According to the facts mentioned above, all members of the Weyl Algebra are distal functions on Z \mathbb {Z} . In this paper, we will show that any element of W {\mathbf {W}} is uniquely ergodic (Theorem 2.13) and that the set W {\mathbf {W}} does not exhaust all the distal functions on Z \mathbb {Z} (Theorem 2.14). The latter will answer the question that has been asked (to the best of my knowledge) by P. Milnes [6]. The term Weyl algebra is suggested by S. Glasner. I would like to express my warmest gratitude to S. Glasner for his helpful advise, and to my advisor Professor Namioka for his enormous helps and contributions.

Unique ergodicity of flows on the torus and the rotation number

We prove in this paper that any flow on the 2- torus with no singular points and periodic orbits which is generated by a vector fieldV=(P, Q) satisfyingV∈C 1 orV∈C 0 andP≠0 is uniquely ergodic. Then we give an expression of the rotation number by using an invariant measure of a flow.

Rank two interval exchange transformations

AbstractWe 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.

Extensions of Q2R: Potts model and other lattices

The authors extend the Q2R rule from the Ising to the Potts model, and also to random lattices. Remedies against the lack of unique ergodicity of Q2R are also presented.

A unique ergodicity of minimal symbolic flows with linear block growth

A unique ergodicity of minimal symbolic flows with linear block growth

Unique ergodicity of irreducible Markov operators on C(X)

Unique ergodicity of irreducible Markov operators on C(X)

Unique ergodicity of flows on homogeneous spaces

LetG be a unimodular Lie group, Γ a co-compact discrete subgroup ofG and ‘a’ a semisimple element ofG. LetT a be the mapgΓ →ag Γ:G/Γ →G/Γ. The following statements are pairwise equivalent: (1) (T a, G/Γ,θ) is weak-mixing. (2) (T a, G/Γ) is topologically weak-mixing. (3) (G u, G/Γ) is uniquely ergodic. (4) (G u, G/Γ,θ) is ergodic. (5) (G u, G/Γ) is point transitive. (6) (G u, G/Γ) is minimal. If in additionG is semisimple with finite center and no compact factors, then the statement “(T a, G/Γ,θ) is ergodic” may be added to the above list.

R�partition de Suites et �quations Fonctionnelles Associ�es

LetX be a compact metric space, λ a probability measure onX, T a measure preserving transformation ofX into itself. Furthermore, letZ be a compact abelian group, andS the skew-product defined onX×Z by the formula:S(x,z)=(T x,z+ϕ(x)) where ϕ is some map fromX toZ. It is known that under the hypothesis of unique ergodicity for the triple (X, λ,T) and some suitable conditions of continuity onT and ϕ, the triple (X×Z,m,S) wherem is the product measure of λ by the Haar measure onZ is uniquely ergodic if and only if an associated functional equation has no non trivial solution.—We introduce here a generalization of unique ergodicity (which appears quite naturally when studying well distributed sequences) and prove about it the same kind of results.

Unique Ergodicity of Horospherical Flows

Unique Ergodicity of Horospherical Flows

Unique ergodicity for horocycle foliations

The notion of unique ergodicity is extended to multidimensional foliations, and it is shown that ifg is the strong stable or strong unstable foliation of a topologically mixing basic set Ωκ for an Axiom A diffeomorphism or flow theng is uniquely ergodic.