Bi-invariant sets and measures have integer Hausdorff dimension

1999 ◽  
Vol 19 (2) ◽  
pp. 523-534 ◽  
Author(s):  
DAVID MEIRI ◽  
YUVAL PERES
Keyword(s):  
Hausdorff Dimension ◽  
Invariant Measure ◽  
Ergodic Measure ◽  
Invariant Set ◽  
Invariant Sets ◽  
Dimensional Torus ◽  
Uniformly Distributed Sequence ◽  
Invariant Ergodic Measure

Let $A,B$ be two diagonal endomorphisms of the $d$-dimensional torus with corresponding eigenvalues relatively prime. We show that for any $A$-invariant ergodic measure $\mu$, there exists a projection onto a torus ${\mathbb T}^r$ of dimension $r\ge\dim\mu$, that maps $\mu$-almost every $B$-orbit to a uniformly distributed sequence in ${\mathbb T}^r$. As a corollary we obtain that the Hausdorff dimension of any bi-invariant measure, as well as any closed bi-invariant set, is an integer.

Invariant measures of full dimension for some expanding maps

1997 ◽  
Vol 17 (1) ◽  
pp. 147-167 ◽  
Author(s):  
DIMITRIOS GATZOURAS ◽  
YUVAL PERES
Keyword(s):  
Hausdorff Dimension ◽  
Invariant Measure ◽  
Large Class ◽  
Open Problem ◽  
Invariant Measures ◽  
Thermodynamic Formalism ◽  
Invariant Set ◽  
Invariant Sets ◽  
Expanding Maps ◽  
Full Dimension

It is an open problem to determine for which maps $f$, any compact invariant set $K$ carries an ergodic invariant measure of the same Hausdorff dimension as $K$. If $f$ is conformal and expanding, then it is a known consequence of the thermodynamic formalism that such measures do exist. (We give a proof here under minimal smoothness assumptions.) If $f$ has the form $f(x_1,x_2)=(f_1(x_1),f_2(x_2))$, where $f_1$ and $f_2$ are conformal and expanding maps satisfying $\inf \vert Df_1\vert\geq\sup\vert Df_2\vert$, then for a large class of invariant sets $K$, we show that ergodic invariant measures of dimension arbitrarily close to the dimension of $K$ do exist. The proof is based on approximating $K$ by self-affine sets.

scholarly journals Some compact invariant sets for hyperbolic linear automorphisms of torii

1988 ◽  
Vol 8 (2) ◽  
pp. 191-204 ◽  
Author(s):  
Albert Fathi
Keyword(s):  
Hausdorff Dimension ◽  
Homology Group ◽  
Invariant Set ◽  
Finite Union ◽  
Invariant Sets ◽  
Pisot Number ◽  
Toral Automorphism ◽  
Lower Capacity

AbstractIf the action induced by a pseudo-Anosov map on the first homology group is hyperbolic, it is possible, by a theorem of Franks, to find a compact invariant set for the toral automorphism associated with this action. If the stable and unstable foliations of the Pseudo-Anosov map are orientable, we show that the invariant set is a finite union of topological 2-discs. Using some ideas of Urbański, it is possible to prove that the lower capacity of the associated compact invariant set is >2; in particular, the invariant set is fractal. When the dilatation coefficient is a Pisot number, we can compute the Hausdorff dimension of the compact invariant set.

Fractals in the Large

1998 ◽  
Vol 50 (3) ◽  
pp. 638-657 ◽  
Author(s):  
Robert S. Strichartz
Keyword(s):  
Invariant Measure ◽  
Power Series ◽  
Metric Space ◽  
Iterated Function System ◽  
Invariant Set ◽  
Invariant Sets ◽  
Iterated Function ◽  
Positive Weights ◽  
Self Similar ◽  
Expansive Maps

AbstractA reverse iterated function system (r.i.f.s.) is defined to be a set of expansive maps ﹛T1,…, Tm﹜ on a discrete metric space M. An invariant set F is defined to be a set satisfying , and an invariant measure μ is defined to be a solution of for positive weights pj. The structure and basic properties of such invariant sets and measures is described, and some examples are given. A blowup ℱ of a self-similar set F in ℝn is defined to be the union of an increasing sequence of sets, each similar to F. We give a general construction of blowups, and show that under certain hypotheses a blowup is the sum set of F with an invariant set for a r.i.f.s. Some examples of blowups of familiar fractals are described. If μ is an invariant measure on ℤ+ for a linear r.i.f.s., we describe the behavior of its analytic transform, the power series on the unit disc.

scholarly journals Countable contraction mappings in metric spaces: invariant sets and measure

2014 ◽  
Vol 12 (4) ◽  
Author(s):  
María Barrozo ◽  
Ursula Molter
Keyword(s):  
Invariant Measure ◽  
Metric Space ◽  
Metric Spaces ◽  
Complete Metric Space ◽  
Invariant Set ◽  
Invariant Sets ◽  
Countable Number ◽  
Contraction Mappings ◽  
Classic Result ◽  
System F

AbstractWe consider a complete metric space (X, d) and a countable number of contraction mappings on X, F = {F i: i ∈ ℕ}. We show the existence of a smallest invariant set (with respect to inclusion) for F. If the maps F i are of the form F i(x) = r i x + b i on X = ℝd, we prove a converse of the classic result on contraction mappings, more precisely, there exists a unique bounded invariant set if and only if r = supi r i is strictly smaller than 1.Further, if ρ = {ρ k}k∈ℕ is a probability sequence, we show that if there exists an invariant measure for the system (F, ρ), then its support must be precisely this smallest invariant set. If in addition there exists any bounded invariant set, this invariant measure is unique, even though there may be more than one invariant set.

APPLICATIONS OF A RELATIVE VARIATIONAL PRINCIPLE TO DIMENSIONS OF NONCONFORMAL EXPANDING MAPS

2011 ◽  
Vol 11 (04) ◽  
pp. 643-679 ◽  
Author(s):  
YUKI YAYAMA
Keyword(s):  
Dynamical Systems ◽  
Variational Principle ◽  
Hausdorff Dimension ◽  
Ergodic Measure ◽  
Sierpinski Carpet ◽  
3 Dimensional ◽  
Sofic Shift ◽  
Full Shift ◽  
Sierpiński Carpet ◽  
Invariant Ergodic Measure

Zhao and Cao (2008) showed the relative variational principle for subadditive potentials in random dynamical systems. Applying their result, we find the Hausdorff dimension of an n (≥3)-dimensional general Sierpiński carpet which has an irreducible sofic shift in symbolic representation and study an invariant ergodic measure of full Hausdorff dimension. These generalize the results of Kenyon and Peres (1996) on the Hausdorff dimension of an n-dimensional general Sierpiński carpet represented by a full shift.

MULTIPLIERS OF HOMOCLINIC TANGENCIES AND A THEOREM OF GONCHENKO AND SHILNIKOV ON AREA PRESERVING MAPS

2005 ◽  
Vol 15 (11) ◽  
pp. 3589-3594 ◽  
Author(s):  
VALENTIN AFRAIMOVICH ◽  
TODD YOUNG
Keyword(s):  
Hausdorff Dimension ◽  
Invariant Set ◽  
Differential Invariants ◽  
Invariant Sets ◽  
Area Preserving Maps ◽  
Homoclinic Tangencies ◽  
Area Preserving

We investigate differential invariants for homoclinic tangencies and discuss the role of these invariants in the Hausdorff dimension of invariant sets associated with the tangency and its unfoldings. Further, we give a streamlined proof of a theorem of Goncheko and Shilnikov on the case of a tangency in an area preserving map. Specifically, the invariants determine whether or not a hyperbolic invariant set is formed near the tangency.

scholarly journals Dimensions of compact invariant sets of some expanding maps

2009 ◽  
Vol 29 (1) ◽  
pp. 281-315 ◽  
Author(s):  
YUKI YAYAMA
Keyword(s):  
Hausdorff Dimension ◽  
Finite Type ◽  
Invariant Set ◽  
Invariant Sets ◽  
Conformal Map ◽  
Expanding Maps ◽  
Sierpinski Carpet ◽  
Shift Of Finite Type ◽  
Unique Measure ◽  
Sierpiński Carpet

AbstractWe study the Hausdorff dimension and measures of full Hausdorff dimension for a compact invariant set of an expanding non-conformal map on the torus given by an integer-valued diagonal matrix. The Hausdorff dimension of a ‘general Sierpiński carpet’ was found by McMullen and Bedford and the uniqueness of the measure of full Hausdorff dimension in some cases was proved by Kenyon and Peres. We extend these results by using compensation functions to study a general Sierpiński carpet represented by a shift of finite type. We give some conditions under which a general Sierpiński carpet has a unique measure of full Hausdorff dimension and study the properties of the unique measure.

scholarly journals Minimal F2-flows

1985 ◽  
Vol 39 (2) ◽  
pp. 187-193
Author(s):  
Daniel Berend
Keyword(s):  
Hausdorff Dimension ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Affine Transformations ◽  
Dimensional Torus ◽  
Minimal Sets ◽  
Necessary And Sufficient

AbstractLet σ be an ergodic endomorphism of the r–dimensional torus and Π a semigroup generated by two affine transformations lying above σ. We show that the flow defined by Π admits minimal sets of positive Hausdorff dimension and we give necessary and sufficient conditions for this flow to be minimal.

scholarly journals On orbits of endomorphisms of tori and the Schmidt game

1988 ◽  
Vol 8 (4) ◽  
pp. 523-529 ◽  
Author(s):  
S. G. Dani
Keyword(s):  
Hausdorff Dimension ◽  
Periodic Points ◽  
Dimensional Torus

AbstractWe show that there exists a subset F of the n-dimensional torus n such that F has Hausdorff dimension n and for any x∈F and any semisimple automorphism σ of n the closure of the σ-orbit of x contains no periodic points.

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