Examples of minimal diffeomorphisms on 𝕋² semiconjugate to an ergodic translation

Alejandro Passeggi; Martín Sambarino

Displaying similar documents to “Examples of minimal diffeomorphisms on 𝕋² semiconjugate to an ergodic translation”

Ergodic theorems in fully symmetric spaces of τ-measurable operators

Vladimir Chilin, Semyon Litvinov (2015)

Studia Mathematica

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Junge and Xu (2007), employing the technique of noncommutative interpolation, established a maximal ergodic theorem in noncommutative $L_{p}$ -spaces, 1 < p < ∞, and derived corresponding maximal ergodic inequalities and individual ergodic theorems. In this article, we derive maximal ergodic inequalities in noncommutative $L_{p}$ -spaces directly from the results of Yeadon (1977) and apply them to prove corresponding individual and Besicovitch weighted ergodic theorems. Then we extend these...

Hamiltonian loops from the ergodic point of view

Leonid Polterovich (1999)

Journal of the European Mathematical Society

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Let $G$ be the group of Hamiltonian diffeomorphisms of a closed symplectic manifold $Y$ . A loop $h : S^{1} \to G$ is called strictly ergodic if for some irrational number the associated skew product map $T : S^{1} \times Y \to S^{1} \times Y$ defined by $T (t, y) = (t + α; h (t) y)$ is strictly ergodic. In the present paper we address the following question. Which elements of the fundamental group of $G$ can be represented by strictly ergodic loops? We prove existence of contractible strictly ergodic loops for a wide class of symplectic manifolds (for instance for simply...

Pointwise convergence for subsequences of weighted averages

Patrick LaVictoire (2011)

Colloquium Mathematicae

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We prove that if μₙ are probability measures on ℤ such that μ̂ₙ converges to 0 uniformly on every compact subset of (0,1), then there exists a subsequence $n_{k}$ such that the weighted ergodic averages corresponding to $μ_{n_{k}}$ satisfy a pointwise ergodic theorem in L¹. We further discuss the relationship between Fourier decay and pointwise ergodic theorems for subsequences, considering in particular the averages along n² + ⌊ρ(n)⌋ for a slowly growing function ρ. Under some monotonicity assumptions,...

Ergodic transforms associated to general averages

H. Aimar, A. L. Bernardis, F. J. Martín-Reyes (2010)

Studia Mathematica

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Jones and Rosenblatt started the study of an ergodic transform which is analogous to the martingale transform. In this paper we present a unified treatment of the ergodic transforms associated to positive groups induced by nonsingular flows and to general means which include the usual averages, Cesàro-α averages and Abel means. We prove the boundedness in $L^{p}$ , 1 < p < ∞, of the maximal ergodic transforms assuming that the semigroup is Cesàro bounded in $L^{p}$ . For p = 1 we find that the...

JOP's counting function and Jones' square function

Karin Reinhold (2006)

Studia Mathematica

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We study a class of square functions in a general framework with applications to a variety of situations: samples along subsequences, averages of $ℤ ₊^{d}$ actions and of positive L¹ contractions. We also study the relationship between a counting function first introduced by Jamison, Orey and Pruitt, in a variety of situations, and the corresponding ergodic averages. We show that the maximal counting function is not dominated by the square functions.

Mixing via families for measure preserving transformations

Rui Kuang, Xiangdong Ye (2008)

Colloquium Mathematicae

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In topological dynamics a theory of recurrence properties via (Furstenberg) families was established in the recent years. In the current paper we aim to establish a corresponding theory of ergodicity via families in measurable dynamical systems (MDS). For a family ℱ (of subsets of ℤ₊) and a MDS (X,,μ,T), several notions of ergodicity related to ℱ are introduced, and characterized via the weak topology in the induced Hilbert space L²(μ). T is ℱ-convergence ergodic of order k if for any...

Infinite measure preserving flows with infinite ergodic index

Alexandre I. Danilenko, Anton V. Solomko (2009)

Colloquium Mathematicae

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We construct a rank-one infinite measure preserving flow ${(T_{r})}_{r \in ℝ}$ such that for each p > 0, the “diagonal” flow ${(T_{r} \times \dots \times T_{r})}_{r \in ℝ} (p t i m e s)$ on the product space is ergodic.

Non-Typical Points for β-Shifts

David Färm, Tomas Persson (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

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We study sets of non-typical points under the map $f_{β} \mapsto β x$ mod 1 for non-integer β and extend our results from [Fund. Math. 209 (2010)] in several directions. In particular, we prove that sets of points whose forward orbit avoid certain Cantor sets, and the set of points for which ergodic averages diverge, have large intersection properties. We observe that the technical condition β > 1.541 found in the above paper can be removed.

Positive L¹ operators associated with nonsingular mappings and an example of E. Hille

Isaac Kornfeld, Wojciech Kosek (2003)

Colloquium Mathematicae

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E. Hille [Hi1] gave an example of an operator in L¹[0,1] satisfying the mean ergodic theorem (MET) and such that supₙ||Tⁿ|| = ∞ (actually, $| | T ⁿ | | \sim n^{1 / 4}$ ). This was the first example of a non-power bounded mean ergodic L¹ operator. In this note, the possible rates of growth (in n) of the norms of Tⁿ for such operators are studied. We show that, for every γ > 0, there are positive L¹ operators T satisfying the MET with $l i m_{n \to \infty} | | T ⁿ | | / n^{1 - γ} = \infty . I n t h e c l a s s o f p o s i t i v e o p e r a t o r s t h e s e e x a m p l e s a r e t h e b e s t p o s s i b l e i n t h e s e n s e t h a t f o r e v e r y s u c h o p e r a t o r T t h e r e e x i s t s a γ ₀ > 0 s u c h t h a t$ lim supn→ ∞ ||Tⁿ||/n1-γ₀ = 0 $.$ A class of numerical sequences αₙ, intimately...

Multiple disjointness and invariant measures on minimal distal flows

Juho Rautio (2015)

Studia Mathematica

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We examine multiple disjointness of minimal flows, that is, we find conditions under which the product of a collection of minimal flows is itself minimal. Our main theorem states that, for a collection ${X_{i}}_{i \in I}$ of minimal flows with a common phase group, assuming each flow satisfies certain structural and algebraic conditions, the product $\prod_{i \in I} X_{i}$ is minimal if and only if $\prod_{i \in I} X_{i}^{e q}$ is minimal, where $X_{i}^{e q}$ is the maximal equicontinuous factor of $X_{i}$ . Most importantly, this result holds when each $X_{i}$ is distal. When...

Example of a mean ergodic L¹ operator with the linear rate of growth

Wojciech Kosek (2011)

Colloquium Mathematicae

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The rate of growth of an operator T satisfying the mean ergodic theorem (MET) cannot be faster than linear. It was recently shown (Kornfeld-Kosek, Colloq. Math. 98 (2003)) that for every γ > 0, there are positive L¹[0,1] operators T satisfying MET with $l i m_{n \to \infty} | | T ⁿ | | / n^{1 - γ} = \infty$ . In the class of positive L¹ operators this is the most one can hope for in the sense that for every such operator T, there exists a γ₀ > 0 such that $l i m s u p | | T ⁿ | | / n^{1 - γ ₀} = 0 .$ In this note we construct an example of a nonpositive L¹ operator with the...

An anti-classification theorem for ergodic measure preserving transformations

Matthew Foreman, Benjamin Weiss (2004)

Journal of the European Mathematical Society

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Despite many notable advances the general problem of classifying ergodic measure preserving transformations (MPT) has remained wide open. We show that the action of the whole group of MPT’s on ergodic actions by conjugation is turbulent in the sense of G. Hjorth. The type of classifications ruled out by this property include countable algebraic objects such as those that occur in the Halmos–von Neumann theorem classifying ergodic MPT’s with pure point spectrum. We treat both the classical...

Spectral decompositions, ergodic averages, and the Hilbert transform

Earl Berkson, T. A. Gillespie (2001)

Studia Mathematica

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Let U be a trigonometrically well-bounded operator on a Banach space , and denote by ${ₙ (U)}_{n = 1}^{\infty}$ the sequence of (C,2) weighted discrete ergodic averages of U, that is, $ₙ (U) = 1 / n \sum_{0 < | k | \leq n} (1 - | k | / (n + 1)) U^{k}$ . We show that this sequence ${ₙ (U)}_{n = 1}^{\infty}$ of weighted ergodic averages converges in the strong operator topology to an idempotent operator whose range is x ∈ : Ux = x, and whose null space is the closure of (I - U). This result expands the scope of the traditional Ergodic Theorem, and thereby serves as a link between Banach space spectral...

Minimal generators for aperiodic endomorphisms

Zbigniew S. Kowalski (1995)

Commentationes Mathematicae Universitatis Carolinae

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Every aperiodic endomorphism $f$ of a nonatomic Lebesgue space which possesses a finite 1-sided generator has a 1-sided generator $β$ such that $k_{f} \leq card β \leq k_{f} + 1$ . This is the best estimate for the minimal cardinality of a 1-sided generator. The above result is the generalization of the analogous one for ergodic case.

Dispersing cocycles and mixing flows under functions

Klaus Schmidt (2002)

Fundamenta Mathematicae

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Let T be a measure-preserving and mixing action of a countable abelian group G on a probability space (X,,μ) and A a locally compact second countable abelian group. A cocycle c: G × X → A for T disperses if $l i m_{g \to \infty} c (g, \cdot) - α (g) = \infty$ in measure for every map α: G → A. We prove that such a cocycle c does not disperse if and only if there exists a compact subgroup A₀ ⊂ A such that the composition θ ∘ c: G × X → A/A₀ of c with the quotient map θ: A → A/A₀ is trivial (i.e. cohomologous to a homomorphism η: G → A/A₀). This...

Invariant jets of a smooth dynamical system

Sophie Lemaire (2001)

Bulletin de la Société Mathématique de France

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The local deformations of a submanifold under the effect of a smooth dynamical system are studied with the help of Oseledets’ multiplicative ergodic theorem. Equivalence classes of submanifolds, called jets, are introduced in order to describe these local deformations. They identify submanifolds having the same approximations up to some order at a given point. For every order $k$ , a condition on the Lyapunov exponents of the dynamical system is established insuring the convergence of the...

Spectral theory and operator ergodic theory on super-reflexive Banach spaces

Earl Berkson (2010)

Studia Mathematica

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On reflexive spaces trigonometrically well-bounded operators have an operator-ergodic-theory characterization as the invertible operators U such that $s u p_{n \in ℕ, z \in} | | \sum_{0 < | k | \leq n} (1 - | k | / (n + 1)) k^{- 1} z^{k} U^{k} | | < \infty$ . (*) Trigonometrically well-bounded operators permeate many settings of modern analysis, and this note highlights the advances in both their spectral theory and operator ergodic theory made possible by a recent rekindling of interest in the R. C. James inequalities for super-reflexive spaces. When the James inequalities are combined with...

Generalizations of Cesàro means and poles of the resolvent

Laura Burlando (2004)

Studia Mathematica

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An improvement of the generalization-obtained in a previous article [Bu1] by the author-of the uniform ergodic theorem to poles of arbitrary order is derived. In order to answer two natural questions suggested by this result, two examples are also given. Namely, two bounded linear operators T and A are constructed such that $n^{- 2} T ⁿ$ converges uniformly to zero, the sum of the range and the kernel of 1-T being closed, and $n^{- 3} \sum_{k = 0}^{n - 1} A^{k}$ converges uniformly, the sum of the range of 1-A and the kernel of (1-A)²...

Multiparameter admissible superadditive processes

Doğan Çömez (2005)

Colloquium Mathematicae

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In this article some properties of Markovian mean ergodic operators are studied. As an application of the tools developed, and using the admissibility feature, a “reduction of order” technique for multiparameter admissible superadditive processes is obtained. This technique is later utilized to obtain a.e. convergence of averages $n^{- 2} \sum_{i, j = 0}^{n - 1} f_{(i, j)}$ as well as their weighted version.

Bohr local properties of $C_{Λ} (T)$

Françoise Lust-Piquard (1989)

Colloquium Mathematicae

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On invariant measures for piecewise $C^{2}$ -transformations of the n-dimensional cube

M. Jabłoński (1983)

Annales Polonici Mathematici

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Eigenvalues and simplicity of interval exchange transformations

Sébastien Ferenczi, Luca Q. Zamboni (2011)

Annales scientifiques de l'École Normale Supérieure

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For a class of $d$ -interval exchange transformations, which we call the symmetric class, we define a new self-dual induction process in which the system is successively induced on a union of sub-intervals. This algorithm gives rise to an underlying graph structure which reflects the dynamical behavior of the system, through the Rokhlin towers of the induced maps. We apply it to build a wide assortment of explicit examples on four intervals having different dynamical properties: these include...

Decay of covariances, uniqueness of ergodic component and scaling limit for a class of $\nabla φ$ systems with non-convex potential

Codina Cotar, Jean-Dominique Deuschel (2012)

Annales de l'I.H.P. Probabilités et statistiques

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We consider a gradient interface model on the lattice with interaction potential which is a non-convex perturbation of a convex potential. Using a technique which decouples the neighboring vertices into even and odd vertices, we show for a class of non-convex potentials: the uniqueness of ergodic component for $\nabla φ$ -Gibbs measures, the decay of covariances, the scaling limit and the strict convexity of the surface tension.

Strictly ergodic, uniform positive entropy models

Eli Glasner, Benjamin Weiss (1994)

Bulletin de la Société Mathématique de France

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Correspondence between diffeomorphism groups and singular foliations

Tomasz Rybicki (2012)

Annales Polonici Mathematici

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It is well-known that any isotopically connected diffeomorphism group G of a manifold determines a unique singular foliation $ℱ_{G}$ . A one-to-one correspondence between the class of singular foliations and a subclass of diffeomorphism groups is established. As an illustration of this correspondence it is shown that the commutator subgroup [G,G] of an isotopically connected, factorizable and non-fixing $C^{r}$ diffeomorphism group G is simple iff the foliation $ℱ_{[G, G]}$ defined by [G,G] admits no proper...