scholarly journals Möbius disjointness along ergodic sequences for uniquely ergodic actions

2018 ◽  
Vol 39 (10) ◽  
pp. 2793-2826 ◽  
Author(s):  
JOANNA KUŁAGA-PRZYMUS ◽  
MARIUSZ LEMAŃCZYK
Keyword(s):  
Invariant Measure ◽  
Lebesgue Measure ◽  
Metric Space ◽  
Irrational Rotation ◽  
Compact Metric Space ◽  
Irrational Rotations ◽  
Ergodic Flow ◽  
Weakly Mixing ◽  
Image Position ◽  
Uniquely Ergodic

We show that there is an irrational rotation $Tx=x+\unicode[STIX]{x1D6FC}$ on the circle $\mathbb{T}$ and a continuous $\unicode[STIX]{x1D711}:\mathbb{T}\rightarrow \mathbb{R}$ such that for each (continuous) uniquely ergodic flow ${\mathcal{S}}=(S_{t})_{t\in \mathbb{R}}$ acting on a compact metric space $Y$, the automorphism $T_{\unicode[STIX]{x1D711},{\mathcal{S}}}$ acting on $(X\times Y,\unicode[STIX]{x1D707}\otimes \unicode[STIX]{x1D708})$ by the formula $T_{\unicode[STIX]{x1D711},{\mathcal{S}}}(x,y)=(Tx,S_{\unicode[STIX]{x1D711}(x)}(y))$, where $\unicode[STIX]{x1D707}$ stands for the Lebesgue measure on $\mathbb{T}$ and $\unicode[STIX]{x1D708}$ denotes the unique ${\mathcal{S}}$-invariant measure, has the property of asymptotically orthogonal powers. This gives a class of relatively weakly mixing extensions of irrational rotations for which Sarnak’s conjecture on the Möbius disjointness holds for all uniquely ergodic models of $T_{\unicode[STIX]{x1D711},{\mathcal{S}}}$. Moreover, we obtain a class of ‘random’ ergodic sequences $(c_{n})\subset \mathbb{Z}$ such that if $\boldsymbol{\unicode[STIX]{x1D707}}$ denotes the Möbius function, then $$\begin{eqnarray}\lim _{N\rightarrow \infty }\frac{1}{N}\mathop{\sum }_{n\leq N}g(S_{c_{n}}y)\boldsymbol{\unicode[STIX]{x1D707}}(n)=0\end{eqnarray}$$ for all (continuous) uniquely ergodic flows ${\mathcal{S}}$, all $g\in C(Y)$ and $y\in Y$.

A Remark on Invariant Scrambled Sets

2012 ◽  
Vol 204-208 ◽  
pp. 4776-4779
Author(s):  
Lin Huang ◽  
Huo Yun Wang ◽  
Hong Ying Wu
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Periodic Point ◽  
Metric Space ◽  
Cantor Sets ◽  
Countable Union ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Isolated Points ◽  
Weakly Mixing

By a dynamical system we mean a compact metric space together with a continuous map . This article is devoted to study of invariant scrambled sets. A dynamical system is a periodically adsorbing system if there exists a fixed point and a periodic point such that and are dense in . We show that every topological weakly mixing and periodically adsorbing system contains an invariant and dense Mycielski scrambled set for some , where has no isolated points. A subset is a Myceilski set if it is a countable union of Cantor sets.

Mixing properties of (α,β)-expansions

2009 ◽  
Vol 29 (4) ◽  
pp. 1119-1140 ◽  
Author(s):  
KARMA DAJANI ◽  
YUSUF HARTONO ◽  
COR KRAAIKAMP
Keyword(s):  
Dynamical System ◽  
Invariant Measure ◽  
Lebesgue Measure ◽  
Natural Extension ◽  
Mixing Properties ◽  
Special Values ◽  
Image Position ◽  
Relationship Of ◽  
The Relationship

AbstractLet 0<α<1 andβ>1. We show that everyx∈[0,1] has an expansion of the formwherehi=hi(x)∈{0,α/β}, andpi=pi(x)∈{0,1}. We study the dynamical system underlying this expansion and give the density of the invariant measure that is equivalent to the Lebesgue measure. We prove that the system is weakly Bernoulli, and we give a version of the natural extension. For special values ofα, we give the relationship of this expansion with the greedyβ-expansion.

scholarly journals A circle diffeomorphism with breaks that is absolutely continuously linearizable

2016 ◽  
Vol 38 (1) ◽  
pp. 371-383 ◽  
Author(s):  
ALEXEY TEPLINSKY
Keyword(s):  
Invariant Measure ◽  
Lebesgue Measure ◽  
Piecewise Linear ◽  
Irrational Rotation ◽  
Rigid Rotation ◽  
Absolutely Continuous ◽  
Circle Homeomorphism ◽  
Break Points ◽  
Circle Diffeomorphism ◽  
Irrational Rotation Number

In this paper we answer positively to a question of whether it is possible for a circle diffeomorphism with breaks to be smoothly conjugate to a rigid rotation in the case where its breaks are lying on pairwise distinct trajectories. An example constructed is a piecewise linear circle homeomorphism that has four break points lying on distinct trajectories and whose invariant measure is absolutely continuous with respect to the Lebesgue measure. The irrational rotation number for our example can be chosen to be a Roth number, but not of bounded type.

scholarly journals Spaces of sections of Banach algebra bundles

2012 ◽  
Vol 10 (2) ◽  
pp. 279-298 ◽  
Author(s):  
Emmanuel Dror Farjoun ◽  
Claude L. Schochet
Keyword(s):  
Spectral Sequence ◽  
Banach Algebra ◽  
Metric Space ◽  
Compact Metric Space ◽  
The Real ◽  
Finite Dimensional ◽  
Image Position

AbstractSuppose thatBis a G-Banach algebra over= ℝ or ℂXis a finite dimensional compact metric space, ζ :P → Xis a standard principalG-bundle, andAζ= Γ(X,P×GB) is the associated algebra of sections. We produce a spectral sequence which converges to π*(GLoAζ) withA related spectral sequence converging toK*+1(Aζ) (the real or complex topologicalK-theory) allows us to conclude that ifBis Bott-stable, (i.e., if π*(GLoB) →K*+1(B) is an isomorphism for all * > 0) then so isAζ.

Minimal entropy and Mostow's rigidity theorems

1996 ◽  
Vol 16 (4) ◽  
pp. 623-649 ◽  
Author(s):  
Gérard Besson ◽  
Gilles Courtois ◽  
Sylvestre Gallot
Keyword(s):  
Riemannian Manifold ◽  
Topological Entropy ◽  
Metric Space ◽  
Universal Cover ◽  
Compact Metric Space ◽  
Entropy Functional ◽  
Volume Entropy ◽  
Image Position ◽  
A Minor ◽  
Special Metrics

Let (Y, g) be a compact connected n-dimensional Riemannian manifold and let () be its universal cover endowed with the pulled-back metric. If y ∈ , we definewhere B(y, R) denotes the ball of radius R around y in . It is a well known fact that this limit exists and does not depend on y ([Man]). The invariant h(g) is called the volume entropy of the metric g but, for the sake of simplicity, we shall use the term entropy. The idea of recognizing special metrics in terms of this invariant looks at first glance very optimistic. First the entropy, which behaves like the inverse of a distance, is sensitive to changes of scale which makes it a bad invariant: however, this is a minor drawback that can be circumvented by looking at the behaviour of the entropy functional on the space of metrics with fixed volume (equal to one for example). Nevertheless, it seems very unlikely that two numbers, the entropy and the volume, might characterize any metric. The very first person to consider such a possibility was Katok ([Kat1]). In this article the entropy is thought of as a dynamical invariant which actually is suggested by its name. More precisely, let us define this dynamical invariant, which is called the topological entropy: let (M, d) be a compact metric space and ψt, a flow on it, we define.

Embedding Smooth Dendroids in Hyperspaces

1979 ◽  
Vol 31 (1) ◽  
pp. 130-138 ◽  
Author(s):  
J. Grispolakis ◽  
E. D. Tymchatyn
Keyword(s):  
Continuous Function ◽  
Partial Order ◽  
Metric Space ◽  
Hausdorff Metric ◽  
Vietoris Topology ◽  
Compact Metric Space ◽  
Partially Ordered ◽  
Arcwise Connected ◽  
Ordered Space ◽  
Image Position

A continuum will be a connected, compact, metric space. By a mapping we mean a continuous function. By a partially ordered space X we mean a continuum X together with a partial order which is closed when regarded as a subset of X × X. We let 2x (resp. C(X)) denote the hyperspace of closed subsets (resp. subcontinua) of X with the Vietoris topology which coincides with the topology induced by the Hausdorff metric. The hyperspaces 2X and C(X) are arcwise connected metric continua (see [3, Theorem 2.7]). If A ⊂ X we let C(A) denote the subspace of subcontinua of X which lie in A.If X is a partially ordered space we define two functions L, M : X → 2X by setting for each x ∊ X

Conjugation between circle maps with several break points

2015 ◽  
Vol 36 (8) ◽  
pp. 2351-2383 ◽  
Author(s):  
ABDELHAMID ADOUANI
Keyword(s):  
Invariant Measure ◽  
Lebesgue Measure ◽  
Break Point ◽  
Irrational Rotation ◽  
Singular Function ◽  
Absolutely Continuous ◽  
Circle Maps ◽  
Almost Everywhere ◽  
Rotation Numbers ◽  
Break Points

Let$f$and$g$be two class$P$-homeomorphisms of the circle$S^{1}$with break point singularities, which are differentiable maps except at some singular points where the derivative has a jump. Assume that$f$and$g$have irrational rotation numbers and the derivatives$\text{Df}$and$\text{Dg}$are absolutely continuous on every continuity interval of$\text{Df}$and$\text{Dg}$, respectively. We prove that if the product of the$f$-jumps along all break points of$f$is distinct from that of$g$then the homeomorphism$h$conjugating$f$and$g$is a singular function, i.e. it is continuous on$S^{1}$, but$\text{Dh}(x)=0$ almost everywhere with respect to the Lebesgue measure. This result generalizes previous results for one and two break points obtained by Dzhalilov, Akin and Temir, and Akhadkulov, Dzhalilov and Mayer. As a consequence, we get in particular Dzhalilov–Mayer–Safarov’s theorem: if the product of the$f$-jumps along all break points of$f$is distinct from$1$, then the invariant measure$\unicode[STIX]{x1D707}_{f}$is singular with respect to the Lebesgue measure.

scholarly journals On the irregular points for systems with the shadowing property

2017 ◽  
Vol 38 (6) ◽  
pp. 2108-2131 ◽  
Author(s):  
YIWEI DONG ◽  
PIOTR OPROCHA ◽  
XUETING TIAN
Keyword(s):  
Lebesgue Measure ◽  
Metric Space ◽  
Compact Metric Space ◽  
Shadowing Property ◽  
Zero Entropy ◽  
Time Average

We prove that when $f$ is a continuous self-map acting on a compact metric space $(X,d)$ that satisfies the shadowing property, then the set of irregular points (i.e., points with divergent Birkhoff averages) has full entropy. Using this fact, we prove that, in the class of $C^{0}$-generic maps on manifolds, we can only observe (in the sense of Lebesgue measure) points with convergent Birkhoff averages. In particular, the time average of atomic measures along orbits of such points converges to some Sinai–Ruelle–Bowen-like measure in the weak$^{\ast }$ topology. Moreover, such points carry zero entropy. In contrast, irregular points are non-observable but carry infinite entropy.

Basins of measures on inverse limit spaces for the induced homeomorphism

2009 ◽  
Vol 30 (4) ◽  
pp. 1119-1130 ◽  
Author(s):  
JUDY KENNEDY ◽  
BRIAN E. RAINES ◽  
DAVID R. STOCKMAN
Keyword(s):  
Invariant Measure ◽  
Metric Space ◽  
Inverse Limit ◽  
Compact Metric Space ◽  
Srb Measure ◽  
Inverse Limit Spaces ◽  
Induced Measure

AbstractLet f:X→X be continuous and onto, where X is a compact metric space. Let $Y:=\invlim {X,f}$ be the inverse limit and F:Y →Y the induced homeomorphism. Suppose that μ is an f-invariant measure, and let m be the measure induced on Y by (μ,μ,…). We show that B is a basin of μ if and only if π−11(B) is a basin of m. From this it follows that if μ is an SRB measure for f on X, then the induced measure m on Y is an inverse-limit SRB measure for F. Conversely, if m is an inverse-limit SRB measure for F on Y, then the induced measure μ on X is an SRB measure for f.

Measures supported on the set of uniquely ergodic directions of an arbitrary holomorphic 1-form

1999 ◽  
Vol 19 (4) ◽  
pp. 1093-1109 ◽  
Author(s):  
WILLIAM A. VEECH
Keyword(s):  
Riemann Surface ◽  
Invariant Measure ◽  
Lebesgue Measure ◽  
Quadratic Differential ◽  
Probability Measures ◽  
Locally Finite ◽  
Horocycle Flow ◽  
Closed Riemann Surface ◽  
Borel Probability ◽  
Uniquely Ergodic

We introduce a set, $Q({\bf T})$, of Borel probability measures on the circle such that each $\mu\in Q({\bf T})$ obeys the conclusion of the Kerckhoff–Masur–Smillie theorem [3]: if $q$ is a meromorphic quadratic differential with at worst simple poles on a closed Riemann surface, then for each $\mu\in Q({\bf T})$ and $\mu$-a.e. $\zeta\in{\bf T}$, $\zeta q$ has uniquely ergodic vertical foliation. As an example, the normalized Cantor–Lebesgue measure belongs to $Q({\bf T})$. The analysis also yields an analogue, for the Teichmüller horocycle flow, of a theorem of Dani: every locally finite ergodic invariant measure for the Teichmüller horocycle flow is finite.

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