Ergodic Translation Research Articles

Topological obstructions to dominated splitting for ergodic translations on the higher dimensional torus

Consider the space of analytic, quasi-periodic on the higher dimensional torus. We provide examples of with nontrivial Lyapunov spectrum, whose homotopy classes do not contain any satisfying the dominated splitting property. This shows that the main result in the recent work Complex one-frequency cocycles by A. Avila, S. Jitomirskaya and C. Sadel does not hold in the higher dimensional torus setting.

The absolute continuous spectrum of skew products of compact Lie groups

Let X and G be compact Lie groups, F 1: X → X the time-one map of a C ∞ measure-preserving flow, ϕ: X → G a continuous function and π a finite-dimensional irreducible unitary representation of G. Then, we prove that the skew products $${T_\phi }:X \times G \to X \times G,(x,g) \mapsto ({F_1}(x),g\phi (x))$$ , have purely absolutely continuous spectrum in the subspace associated to π if π po ϕ has a Dini-continuous Lie derivative along the flow and if a matrix multiplication operator related to the topological degree of πpoϕ has nonzero determinant. This result provides a simple, but general, criterion for the presence of an absolutely continuous component in the spectrum of skew products of compact Lie groups. As an illustration, we consider the cases where F 1 is an ergodic translation on T d and X × G = T d × T dʹ , X × G = T d × SU(2) and X × G = T d × U(2). Our proofs rely on recent results on positive commutator methods for unitary operators.

On manifolds supporting distributionally uniquely ergodic diffeomorphisms

A smooth diffeomorphism is said to be distributionally uniquely ergodic (DUE for short) when it is uniquely ergodic and its unique invariant probability measure is the only invariant distribution (up to multiplication by a constant). Ergodic translations on tori are classical examples of DUE diffeomorphisms. In this article we construct DUE diffeomorphisms supported on closed manifolds different from tori, providing some counterexamples to a conjecture proposed by Forni in [For08].

Asymptotics of random lozenge tilings via Gelfand–Tsetlin schemes

A Gelfand–Tsetlin scheme of depth $$N$$ is a triangular array with $$m$$ integers at level $$m$$ , $$m=1,\ldots ,N$$ , subject to certain interlacing constraints. We study the ensemble of uniformly random Gelfand–Tsetlin schemes with arbitrary fixed $$N$$ th row. We obtain an explicit double contour integral expression for the determinantal correlation kernel of this ensemble (and also of its $$q$$ -deformation). This provides new tools for asymptotic analysis of uniformly random lozenge tilings of polygons on the triangular lattice; or, equivalently, of random stepped surfaces. We work with a class of polygons which allows arbitrarily large number of sides. We show that the local limit behavior of random tilings (as all dimensions of the polygon grow) is directed by ergodic translation invariant Gibbs measures. The slopes of these measures coincide with the ones of tangent planes to the corresponding limit shapes described by Kenyon and Okounkov (Acta Math 199(2):263–302, 2007). We also prove that at the edge of the limit shape, the asymptotic behavior of random tilings is given by the Airy process. In particular, our results cover the most investigated case of random boxed plane partitions (when the polygon is a hexagon).

Nonstandard smooth realization of translations on the torus

Let $M$ be a smooth compact connected manifold of dimension greaterthan two, on which there exists a free (modulo zero) smooth circleaction that preserves a positive smooth volume. In this article, weconstruct volume-preserving diffeomorphisms on $M$ that are metricallyisomorphic to ergodic translations on the torus of dimension greaterthan two, where one given coordinate of the translation is anarbitrary Liouville number. To obtain this result, we determinesufficient conditions on translation vectors of the torus that allow usto explicitly construct the sequence of successive conjugacies inAnosov--Katok's method, with suitable estimates of their norm.

Examples of minimal diffeomorphisms on T2semiconjugate to an ergodic translation

We prove that for every $\epsilon >0$ there exists a minimal diffeomorphism $f:\mathbb {T}^{2}\rightarrow \mathbb {T}^{2}$ of class $C^{3-\epsilon }$ and semiconjugate to an ergodic translation with the following properties: zero entropy, sensitivity to i

Q-Distributions on boxed plane partitions

We introduce elliptic weights of boxed plane partitions and prove that they give rise to a generalization of MacMahon’s product formula for the number of plane partitions in a box. We then focus on the most general positive degenerations of these weights that are related to orthogonal polynomials; they form three 2-D families. For distributions from these families, we prove two types of results. First, we construct explicit Markov chains that preserve these distributions. In particular, this leads to a relatively simple exact sampling algorithm. Second, we consider a limit when all dimensions of the box grow and plane partitions become large and prove that the local correlations converge to those of ergodic translation invariant Gibbs measures. For fixed proportions of the box, the slopes of the limiting Gibbs measures (that can also be viewed as slopes of tangent planes to the hypothetical limit shape) are encoded by a single quadratic polynomial.

RANDOM WALK IN QUASI-PERIODIC RANDOM ENVIRONMENT

We consider a one-dimensional random walk with finite range in a random medium described by an ergodic translation on a torus. For regular data and under a Diophantine condition on the translation, we prove a central limit theorem with deterministic centering.

Complexity of Hartman sequences

Let T:x↦x+g be an ergodic translation on the compact group C and M⊆C a continuity set, i.e. a subset with topological boundary of Haar measure 0. An infinite binary sequence a:ℤ↦{0,1} defined by a(k)=1 if T k (0 C )∈M and a(k)=0 otherwise, is called a Hartman sequence. This paper studies the growth rate of P a (n), where P a (n) denotes the number of binary words of length n∈ℕ occurring in a. The growth rate is always subexponential and this result is optimal. If T is an ergodic translation x↦x+α (α=(α 1 ,...,α s )) on 𝕋 s and M is a box with side lengths ρ j not equal α j ℤ+ℤ for all j=1,...,s, we show that lim n P a (n)/n s =2 s ∏ j=1 s ρ j s-1 .

The Convergence Result for Critical Reversible Nearest Particle Systems

We consider critical reversible nearest particle systems. We assume that the associated renewal measure has large moments as well as some regularity conditions. It is shown that such processes, started from a nontrivial ergodic translation invariant distribution, converge in distribution to the upper invariant measure.

Diffeomorphisms of thek-torus with wandering domains

AbstractIt is shown that diffeomorphisms analogous to a classical example on the circle due to Denjoy can be constructed on the generalk-torus. Such a diffeomorphism has the property that it is semiconjugate to an ergodic translation but has a wandering domain with dense orbit. The construction on thek-torus can be madeCr, and by aCrsmall perturbation of a translation, for anyr<k+ 1.

Diffeomorphisms of the Torus with Wandering Domains

Examples of ${C^{3 - \varepsilon }}$ diffeomorphisms of the $2$-torus are constructed, each of which is semiconjugate to an ergodic translation but has a wandering domain with dense orbit. This is a generalization of a classical example on the circle due to Denjoy.

Diffeomorphisms of the torus with wandering domains

Examples of ${C^{3 - \varepsilon }}$ diffeomorphisms of the $2$-torus are constructed, each of which is semiconjugate to an ergodic translation but has a wandering domain with dense orbit. This is a generalization of a classical example on the circle due to Denjoy.

A state space isomorphism theorem for minimal dynamical systems

Consider a discrete time dynamical systemxk+1=f(x k ) on a compact metric spaceM, wheref:M→M is a continuous map. Leth:M→B k be a continuous output function. Suppose that all of the positive orbits off are dense and that the system is observable. We prove that any output trajectory of the system determinesf andh andM up to a homeomorphism. IfM is a compact Abelian topological group andf is an ergodic translation, then any output trajectory determines the system up to a translation and a group isomorphism of the group.

Observing Ergodic Translations on Compact Abelian Groups with Discontinuous Functions

Several authors have considered the problem of global observability for minimal dynamical systems, and particularly the example of ergodic (equivalently, minimal) translations on compact abelian groups. The global observability problem for ergodic translations is considered here in the case where the output function may be discontinuous. The result for continuous output functions, essentially due to McMahon, is described. This result shows that a continuous output function observes all ergodic translations if and only if it has no nontrivial symmetries. This result is extended here in two directions. First, functions continuous except on a meagre set and symmetries modulo meagre sets are considered. A result analogous to the continuous case is obtained that includes and generalizes the continuous case and results of Balogh, Bennett, and Martin on observing ergodic translations with the characteristic functions of certain subsets of the group. Functions continuous except on a set of measure zero are also considered. Secondly, ergodic theory and harmonic analysis techniques are employed to consider the case of intergrable output functions and symmetries modulo sets of measure zero. It is shown that an integrable function observes ergodic translations modulo sets of measure zero if and only if it has no nontrivial symmetries modulo sets of measure zero. The group of symmetries of the output function modulo sets of measure zero can be determined from its Fourier series. The results here show that classical observability can be determined from the Fourier series when the function is continuous almost everywhere, but only observability modulo sets of measure zero can be determined when the output function is merely integrable.

Weakly isomorphic transformations that are not isomorphic

A new method for construction of transformations T i: (X i, B i, μ i) ↷, i=1,2, that are factors of each other but that are not measuretheoretically isomorphic is provided. This method uses ergodic product cocycles of the form ϕ∘S i 1xϕ∘S i 2x...,, where ϕ: X→Z 2 is a cocycle, S belongs to the centralizer of T and T is an ergodic translation on a compact, monothetic group X.

Subsequence generators for ergodic group translations

LetT be an ergodic translation on a compact abelian group. For every infinite set of integers {ni} and e >0 there is a setA of measure less than e such that {TniA} generates the σ-algebra of measurable sets.

Eigen operators of ergodic transformations

The importance of eigenvalues as a tool in the study of m.p.t.'s suggests that eigen operators, which include eigenvalues as a special case, may prove a yet finer tool in this work. In this paper, we will investigate the nature of eigen operators and their relation to their corresponding ergodic transformations. It will be shown that the solutions of this equation (1) depend on the theory of ergodic translations in a monothetic measure group. DEFINITIONS. We take (S, 1, m) to be a a-finite measure space in the sense of Halmos [3]. A measure-preserving transformation (m.p.t.) h is a oneto-one mapping of S onto itself so that if A1 = h(A2) = { h(s) I sEA2 }, then A1 is :-measurable if and only if A2 is and m(Ai)=m(A2). If h(A) =A, ACEZ, necessarily implies that either m(A)=O or m(S-A)=0, then h is said to be ergodic. Let X be a complex B-space. Then the set of all open sets in X generates a a-field of subsets of X known as the Borelfield of X, and each set in the Borel field is called a Borel set. A mapping X from S into X is called strongly measurable if