scholarly journals Hyperbolic polygonal billiards with finitely many ergodic SRB measures

2017 ◽  
Vol 38 (6) ◽  
pp. 2062-2085 ◽  
Author(s):  
GIANLUIGI DEL MAGNO ◽  
JOÃO LOPES DIAS ◽  
PEDRO DUARTE ◽  
JOSÉ PEDRO GAIVÃO
Keyword(s):  
Lebesgue Measure ◽  
Srb Measures ◽  
Polygonal Billiards ◽  
Angle Of Reflection ◽  
Full Lebesgue Measure

We study polygonal billiards with reflection laws contracting the angle of reflection towards the normal. It is shown that if a polygon does not have parallel sides facing each other, then the corresponding billiard map has finitely many ergodic Sinai–Ruelle–Bowen measures whose basins cover a set of full Lebesgue measure.

scholarly journals On the PropertyN-1

2016 ◽  
Vol 2016 ◽  
pp. 1-5
Author(s):  
Stanisław Kowalczyk ◽  
Małgorzata Turowska
Keyword(s):  
Continuous Function ◽  
Lebesgue Measure ◽  
Continuous Functions ◽  
Approximate Derivative ◽  
Banach's Theorem ◽  
Full Lebesgue Measure

We construct a continuous functionf:[0,1]→Rsuch thatfpossessesN-1-property, butfdoes not have approximate derivative on a set of full Lebesgue measure. This shows that Banach’s Theorem concerning differentiability of continuous functions with Lusin’s property(N)does not hold forN-1-property. Some relevant properties are presented.

scholarly journals Approximation properties of -expansions II

2017 ◽  
Vol 38 (5) ◽  
pp. 1627-1641
Author(s):  
SIMON BAKER
Keyword(s):  
Hausdorff Dimension ◽  
Lebesgue Measure ◽  
Measure Zero ◽  
Hausdorff Measures ◽  
Approximation Properties ◽  
Real Numbers ◽  
Positive Real ◽  
Transference Principle ◽  
Significant Step ◽  
Full Lebesgue Measure

Let $\unicode[STIX]{x1D6FD}\in (1,2)$ be a real number. For a function $\unicode[STIX]{x1D6F9}:\mathbb{N}\rightarrow \mathbb{R}_{\geq 0}$, define $W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6F9})$ to be the set of $x\in \mathbb{R}$ such that for infinitely many $n\in \mathbb{N},$ there exists a sequence $(\unicode[STIX]{x1D716}_{i})_{i=1}^{n}\in \{0,1\}^{n}$ satisfying $0\leq x-\sum _{i=1}^{n}(\unicode[STIX]{x1D716}_{i}/\unicode[STIX]{x1D6FD}^{i})\leq \unicode[STIX]{x1D6F9}(n)$. In Baker [Approximation properties of $\unicode[STIX]{x1D6FD}$-expansions. Acta Arith. 168 (2015), 269–287], the author conjectured that for Lebesgue almost every $\unicode[STIX]{x1D6FD}\in (1,2)$, the condition $\sum _{n=1}^{\infty }2^{n}\unicode[STIX]{x1D6F9}(n)=\infty$ implies that $W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6F9})$ is of full Lebesgue measure within $[0,1/(\unicode[STIX]{x1D6FD}-1)]$. In this paper we make a significant step towards proving this conjecture. We prove that given a sequence of positive real numbers $(\unicode[STIX]{x1D714}_{n})_{n=1}^{\infty }$ satisfying $\lim _{n\rightarrow \infty }\unicode[STIX]{x1D714}_{n}=\infty$, for Lebesgue almost every $\unicode[STIX]{x1D6FD}\in (1.497,\ldots ,2)$, the set $W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D714}_{n}\cdot 2^{-n})$ is of full Lebesgue measure within $[0,1/(\unicode[STIX]{x1D6FD}-1)]$. We also study the case where $\sum _{n=1}^{\infty }2^{n}\unicode[STIX]{x1D6F9}(n)<\infty$ in which the set $W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6F9})$ has Lebesgue measure zero. Applying the mass transference principle developed by Beresnevich and Velani in [A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2) 164(3) (2006), 971–992], we obtain some results on the Hausdorff dimension and the Hausdorff measure of $W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6F9})$.

SRB Measures for Polygonal Billiards with Contracting Reflection Laws

2014 ◽  
Vol 329 (2) ◽  
pp. 687-723 ◽  
Author(s):  
Gianluigi Del Magno ◽  
João Lopes Dias ◽  
Pedro Duarte ◽  
José Pedro Gaivão ◽  
Diogo Pinheiro
Keyword(s):  
Srb Measures ◽  
Polygonal Billiards

Mass transference principle for limsup sets generated by rectangles

2015 ◽  
Vol 158 (3) ◽  
pp. 419-437 ◽  
Author(s):  
BAO-WEI WANG ◽  
JUN WU ◽  
JIAN XU
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Lebesgue Measure ◽  
Hausdorff Measure ◽  
Unit Cube ◽  
Transference Principle ◽  
Theoretical Statement ◽  
Formula Group ◽  
Image Position ◽  
Full Lebesgue Measure

AbstractWe generalise the mass transference principle established by Beresnevich and Velani to limsup sets generated by rectangles. More precisely, let {xn}n⩾1 be a sequence of points in the unit cube [0, 1]d with d ⩾ 1 and {rn}n⩾1 a sequence of positive numbers tending to zero. Under the assumption of full Lebesgue measure theoretical statement of the set \begin{equation*}\big\{x\in [0,1]^d: x\in B(x_n,r_n), \ {{\rm for}\, {\rm infinitely}\, {\rm many}}\ n\in \mathbb{N}\big\},\end{equation*} we determine the lower bound of the Hausdorff dimension and Hausdorff measure of the set \begin{equation*}\big\{x\in [0,1]^d: x\in B^{a}(x_n,r_n), \ {{\rm for}\, {\rm infinitely}\, {\rm many}}\ n\in \mathbb{N}\big\},\end{equation*} where a = (a1, . . ., ad) with 1 ⩽ a1 ⩽ a2 ⩽ . . . ⩽ ad and Ba(x, r) denotes a rectangle with center x and side-length (ra1, ra2,. . .,rad). When a1 = a2 = . . . = ad, the result is included in the setting considered by Beresnevich and Velani.

scholarly journals SRB measures for almost Axiom A diffeomorphisms

2015 ◽  
Vol 36 (7) ◽  
pp. 2015-2043 ◽  
Author(s):  
JOSÉ F. ALVES ◽  
RENAUD LEPLAIDEUR
Keyword(s):  
Probability Measure ◽  
Lebesgue Measure ◽  
Compact Manifold ◽  
Invariant Set ◽  
Singular Set ◽  
Axiom A ◽  
Srb Measures ◽  
Hyperbolic Points

We consider a diffeomorphism $f$ of a compact manifold $M$ which is almost Axiom A, i.e. $f$ is hyperbolic in a neighborhood of some compact $f$-invariant set, except in some singular set of neutral points. We prove that if there exists some $f$-invariant set of hyperbolic points with positive unstable Lebesgue measure such that for every point in this set the stable and unstable leaves are ‘long enough’, then $f$ admits an SRB (probability) measure.

scholarly journals Dimension Bound for Badly Approximable Grids

2018 ◽  
Vol 2019 (20) ◽  
pp. 6317-6346 ◽  
Author(s):  
Seonhee Lim ◽  
Nicolas de Saxcé ◽  
Uri Shapira
Keyword(s):  
Lebesgue Measure ◽  
Main Tool ◽  
Maximal Entropy ◽  
Homogeneous Dynamics ◽  
Relative Version ◽  
Badly Approximable ◽  
Full Lebesgue Measure

Abstract We show that there exists a subset of full Lebesgue measure $V\subset \mathbb{R}^{n}$ such that for every ϵ > 0 there exists δ > 0 such that for any v ∈ V the dimension of the set of vectors w satisfying $$ \liminf_{k\to\infty} k^{1/n}\langle kv-w\rangle\geqslant \epsilon$$ (where 〈⋅〉 denotes the distance from the nearest integer) is bounded above by n − δ. This result is obtained as a corollary of a discussion in homogeneous dynamics and the main tool in the proof is a relative version of the principle of uniqueness of measures with maximal entropy.

scholarly journals The solar Julia sets of basic quadratic Cremer polynomials

2009 ◽  
Vol 30 (1) ◽  
pp. 51-65 ◽  
Author(s):  
A. BLOKH ◽  
X. BUFF ◽  
A. CHÉRITAT ◽  
L. OVERSTEEGEN
Keyword(s):  
Lebesgue Measure ◽  
Topological Structure ◽  
Julia Set ◽  
Julia Sets ◽  
Full Lebesgue Measure

AbstractIn general, little is known about the exact topological structure of Julia sets containing a Cremer point. In this paper we show that there exist quadratic Cremer Julia sets of positive area such that for a full Lebesgue measure set of angles the impressions are degenerate, the Julia set is connected im kleinen at the landing points of these rays, and these points are contained in no other impression.

A chaotic function with a distributively scrambled set of full Lebesgue measure

2007 ◽  
Vol 66 (10) ◽  
pp. 2274-2280 ◽  
Author(s):  
Gongfu Liao ◽  
Lidong Wang ◽  
Xiaodong Duan
Keyword(s):  
Lebesgue Measure ◽  
Full Lebesgue Measure

scholarly journals Physical measures for certain partially hyperbolic attractors on 3-manifolds

2017 ◽  
Vol 39 (1) ◽  
pp. 74-104 ◽  
Author(s):  
RICARDO T. BORTOLOTTI
Keyword(s):  
Finite Number ◽  
Lebesgue Measure ◽  
Gibbs States ◽  
Stable Foliation ◽  
Ergodic Properties ◽  
Central Direction ◽  
Hyperbolic Attractors ◽  
Partially Hyperbolic ◽  
Physical Measures ◽  
Full Lebesgue Measure

In this work, we analyze ergodic properties of certain partially hyperbolic attractors whose central direction has a neutral behavior; the main feature is a condition of transversality between the projections of unstable leaves, projecting through the stable foliation. We prove that partial hyperbolic attractors satisfying this condition of transversality, neutrality in the central direction and regularity of the stable foliation admit a finite number of physical measures, coinciding with the ergodic u-Gibbs States, whose union of the basins has full Lebesgue measure. Moreover, we describe the construction of robustly non-hyperbolic attractors satisfying these properties.

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