scholarly journals Equidistribution of singular measures on nilmanifolds and skew products

2011 ◽  
Vol 31 (6) ◽  
pp. 1785-1817
Author(s):  
FABRIZIO POLO
Keyword(s):  
Haar Measure ◽  
Uniform Density ◽  
Unipotent Group ◽  
Affine Transformations ◽  
Skew Products ◽  
Singular Measures ◽  
Ergodic Properties ◽  
Push Forward ◽  
Multiplicative Ergodic Theorem ◽  
Uniquely Ergodic

AbstractWe prove that for a minimal rotationTon a two-step nilmanifold and any measureμ, the push-forwardTn⋆μofμunderTntends toward Haar measure if and only ifμprojects to Haar measure on the maximal torus factor. For an arbitrary nilmanifold we get the same result along a sequence of uniform density one. These results strengthen Parry’s result [Ergodic properties of affine transformations and flows on nilmanifolds.Amer. J. Math.91(1968), 757–771] that such systems are uniquely ergodic. Extending the work of Furstenberg [Strict ergodicity and transformations of the torus.Amer. J. Math.83(1961), 573–601], we get the same result for a large class of iterated skew products. Additionally we prove a multiplicative ergodic theorem for functions taking values in the upper unipotent group. Finally we characterize limits ofTn⋆μfor some skew product transformations with expansive fibers. All results are presented in terms of twisting and weak twisting, properties that strengthen unique ergodicity in a way analogous to that in which mixing and weak mixing strengthen ergodicity for measure-preserving systems.

scholarly journals ℤd-actions with prescribed topological and ergodic properties

2011 ◽  
Vol 32 (1) ◽  
pp. 191-209 ◽  
Author(s):  
YURI LIMA
Keyword(s):  
Topological Entropy ◽  
Topological Dynamics ◽  
Ergodic Transformations ◽  
Positive Topological Entropy ◽  
Ergodic Properties ◽  
Uniquely Ergodic

AbstractWe extend constructions of Hahn and Katznelson [On the entropy of uniquely ergodic transformations. Trans. Amer. Math. Soc.126 (1967), 335–360] and Pavlov [Some counterexamples in topological dynamics. Ergod. Th. & Dynam. Sys.28 (2008), 1291–1322] to ℤd-actions on symbolic dynamical spaces with prescribed topological and ergodic properties. More specifically, we describe a method to build ℤd-actions which are (totally) minimal, (totally) strictly ergodic and have positive topological entropy.

scholarly journals Propriétés ergodiques, en mesure infinie, de certains systèmes dynamiques fibrés

1989 ◽  
Vol 9 (3) ◽  
pp. 433-453 ◽  
Author(s):  
Y. Guivarc'h
Keyword(s):  
Dynamical Systems ◽  
Invariant Measure ◽  
Geodesic Flow ◽  
Negative Curvature ◽  
Compact Manifolds ◽  
Constant Negative Curvature ◽  
Skew Products ◽  
Ergodic Properties ◽  
Anosov Systems

AbstractWe study the ergodic properties of a class of dynamical systems with infinite invariant measure. This class contains skew-products of Anosov systems with ℝd. The results are applied to theKproperty of skew-products and also to the ergodicity of the geodesic flow on abelian coverings of compact manifolds with constant negative curvature.

Continuous singular measures equivalent to their convolution squares

1976 ◽  
Vol 80 (2) ◽  
pp. 249-268 ◽  
Author(s):  
Gavin Brown ◽  
Edwin Hewitt
Keyword(s):  
Abelian Group ◽  
Haar Measure ◽  
Measure Algebra ◽  
Locally Compact ◽  
Character Group ◽  
Singular Measures ◽  
Continuous Measures ◽  
Discrete Abelian Group

Throughout this paper, G will denote a locally compact, non-discrete, Abelian group (subjected to various conditions) and X wi11 denote the character group of G. All terminology and notation are as in (7). The measure algebra M(G), as is known, is a very complicated entity. We address ourselves here to some novel peculiarities of the subspace Ms(G) of continuous measures in M(G) that are singular with respect to Haar measure λ.

scholarly journals Commutator methods for the spectral analysis of uniquely ergodic dynamical systems

2014 ◽  
Vol 35 (3) ◽  
pp. 944-967 ◽  
Author(s):  
R. TIEDRA DE ALDECOA
Keyword(s):  
Spectral Analysis ◽  
Dynamical Systems ◽  
Absolute Continuity ◽  
Unitary Operators ◽  
Skew Products ◽  
Lebesgue Spectrum ◽  
The Absolute ◽  
Time Changes ◽  
Continuity Of The Spectrum ◽  
Uniquely Ergodic

AbstractWe present a method, based on commutator methods, for the spectral analysis of uniquely ergodic dynamical systems. When applicable, it leads to the absolute continuity of the spectrum of the corresponding unitary operators. As an illustration, we consider time changes of horocycle flows, skew products over translations and Furstenberg transformations. For time changes of horocycle flows we obtain absolute continuity under assumptions weaker than those to be found in the literature, and for skew products over translations and Furstenberg transformations we obtain countable Lebesgue spectrum under assumptions not previously covered in the literature.

scholarly journals Ergodic properties of the Anzai skew-product for the non-commutative torus

2020 ◽  
pp. 1-22 ◽  
Author(s):  
SIMONE DEL VECCHIO ◽  
FRANCESCO FIDALEO ◽  
LUCA GIORGETTI ◽  
STEFANO ROSSI
Keyword(s):  
Dynamical Systems ◽  
Unit Circle ◽  
Systematic Study ◽  
Cartesian Product ◽  
Skew Product ◽  
Quantum Dynamical ◽  
Ergodic Properties ◽  
Pointwise Limit ◽  
Quantum Dynamical Systems ◽  
Uniquely Ergodic

We provide a systematic study of a non-commutative extension of the classical Anzai skew-product for the cartesian product of two copies of the unit circle to the non-commutative 2-tori. In particular, some relevant ergodic properties are proved for these quantum dynamical systems, extending the corresponding ones enjoyed by the classical Anzai skew-product. As an application, for a uniquely ergodic Anzai skew-product $\unicode[STIX]{x1D6F7}$ on the non-commutative $2$ -torus $\mathbb{A}_{\unicode[STIX]{x1D6FC}}$ , $\unicode[STIX]{x1D6FC}\in \mathbb{R}$ , we investigate the pointwise limit, $\lim _{n\rightarrow +\infty }(1/n)\sum _{k=0}^{n-1}\unicode[STIX]{x1D706}^{-k}\unicode[STIX]{x1D6F7}^{k}(x)$ , for $x\in \mathbb{A}_{\unicode[STIX]{x1D6FC}}$ and $\unicode[STIX]{x1D706}$ a point in the unit circle, and show that there are examples for which the limit does not exist, even in the weak topology.

scholarly journals Spectral gap property and strong ergodicity for groups of affine transformations of solenoids

2018 ◽  
Vol 40 (5) ◽  
pp. 1180-1193
Author(s):  
BACHIR BEKKA ◽  
CAMILLE FRANCINI
Keyword(s):  
Abelian Group ◽  
Probability Measure ◽  
Solvable Group ◽  
Haar Measure ◽  
Discrete Group ◽  
Spectral Gap ◽  
Canonical Projection ◽  
Affine Transformations ◽  
Finitely Generated ◽  
Finite Dimensional

Let $X$ be a solenoid, i.e. a compact, finite-dimensional, connected abelian group with normalized Haar measure $\unicode[STIX]{x1D707}$, and let $\unicode[STIX]{x1D6E4}\rightarrow \operatorname{Aff}(X)$ be an action of a countable discrete group $\unicode[STIX]{x1D6E4}$ by continuous affine transformations of $X$. We show that the probability measure preserving action $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ does not have the spectral gap property if and only if there exists a $p_{\text{a}}(\unicode[STIX]{x1D6E4})$-invariant proper subsolenoid $Y$ of $X$ such that the image of $\unicode[STIX]{x1D6E4}$ in $\operatorname{Aff}(X/Y)$ is a virtually solvable group, where $p_{\text{a}}:\operatorname{Aff}(X)\rightarrow \operatorname{Aut}(X)$ is the canonical projection. When $\unicode[STIX]{x1D6E4}$ is finitely generated or when $X$ is the $a$-adic solenoid for an integer $a\geq 1$, the subsolenoid $Y$ can be chosen so that the image $\unicode[STIX]{x1D6E4}$ in $\operatorname{Aff}(X/Y)$ is a virtually abelian group. In particular, an action $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ by affine transformations on a solenoid $X$ has the spectral gap property if and only if $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ is strongly ergodic.

scholarly journals On the construction of smooth ergodic skew-products

1988 ◽  
Vol 8 (2) ◽  
pp. 311-326 ◽  
Author(s):  
Mahesh G. Nerurkar
Keyword(s):  
Dynamical System ◽  
Coefficient Matrix ◽  
Small Divisor ◽  
Almost Periodic ◽  
Qualitative Behaviour ◽  
Skew Products ◽  
Ergodic Properties ◽  
Periodic Approximation ◽  
Linear Differential ◽  
Smooth Dynamical System

AbstractIn this paper we prove results about lifting dynamical and ergodic properties of a given smooth dynamical system to its skew-product extensions by smooth cocycles. The classical small divisor argument shows that in general such results are not possible. However, using the notion of the ‘fast periodic approximation’ introduced by A. Katok, we will show that if the dynamical system admits such a ‘fast periodic approximation’ then indeed a certain qualitative behaviour which is prohibited by small divisor type conditions is now in fact generic. The techniques are also applied to show that ‘recurrent-proximal’ behaviour of solutions of linear differential equations with almost periodic coefficients is generic under suitable conditions on the coefficient matrix.

Sign in / Sign up

Export Citation Format

Share Document