scholarly journals On a family of self-affine sets: Topology, uniqueness, simultaneous expansions

2015 ◽  
Vol 37 (1) ◽  
pp. 193-227 ◽  
Author(s):  
KEVIN G. HARE ◽  
NIKITA SIDOROV
Keyword(s):  
Hausdorff Dimension ◽  
Positive Lebesgue Measure ◽  
Compact Set ◽  
Annual Conference ◽  
Topological Properties ◽  
Empty Interior ◽  
Affine Maps ◽  
Simply Connected ◽  
Set Of Points ◽  
Connectedness Locus

Let$\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2}>1$and$T_{i}(x,y)=((x+i)/\unicode[STIX]{x1D6FD}_{1},(y+i)/\unicode[STIX]{x1D6FD}_{2}),i\in \{\pm 1\}$. Let$A:=A_{\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2}}$be the unique compact set satisfying$A=T_{1}(A)\cup T_{-1}(A)$. In this paper, we give a detailed analysis of$A$and the parameters$(\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2})$where$A$satisfies various topological properties. In particular, we show that if$\unicode[STIX]{x1D6FD}_{1}<\unicode[STIX]{x1D6FD}_{2}<1.202$, then$A$has a non-empty interior, thus significantly improving the bound from Dajaniet al[Self-affine sets with positive Lebesgue measure.Indag. Math. (N.S.)25(2014), 774–784]. In the opposite direction, we prove that the connectedness locus for this family studied in Solomyak [Connectedness locus for pairs of affine maps and zeros of power series.Preprint, 2014, arXiv:1407.2563] is not simply connected. We prove that the set of points of$A$which have a unique address has positive Hausdorff dimension for all$(\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2})$. Finally, we investigate simultaneous$(\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2})$-expansions of reals, which were the initial motivation for studying this family in Güntürk [Simultaneous and hybrid beta-encodings.Information Sciences and Systems, 2008. CISS 2008. 42nd Annual Conference2008, pp. 743–748].

scholarly journals Wiman–Valiron Discs and the Dimension of Julia Sets

2020 ◽  
Author(s):  
James Waterman
Keyword(s):  
Hausdorff Dimension ◽  
Julia Set ◽  
Julia Sets ◽  
Singular Values ◽  
Simply Connected ◽  
Set Of Points ◽  
Meromorphic Map

Abstract We show that the Hausdorff dimension of the set of points of bounded orbit in the Julia set of a meromorphic map with a simply connected direct tract and a certain restriction on the singular values is strictly greater than one. This result is obtained by proving new results related to Wiman–Valiron theory.

Simultaneous dense and non-dense orbits for certain partially hyperbolic diffeomorphisms

2018 ◽  
Vol 40 (4) ◽  
pp. 1083-1107
Author(s):  
WEISHENG WU
Keyword(s):  
Hausdorff Dimension ◽  
Tangent Space ◽  
Anosov Diffeomorphism ◽  
Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Dense Orbits ◽  
Set Of Points ◽  
Partially Hyperbolic Diffeomorphism ◽  
Stable Subspace

Let$g:M\rightarrow M$be a$C^{1+\unicode[STIX]{x1D6FC}}$-partially hyperbolic diffeomorphism preserving an ergodic normalized volume on$M$. We show that, if$f:M\rightarrow M$is a$C^{1+\unicode[STIX]{x1D6FC}}$-Anosov diffeomorphism such that the stable subspaces of$f$and$g$span the whole tangent space at some point on$M$, the set of points that equidistribute under$g$but have non-dense orbits under$f$has full Hausdorff dimension. The same result is also obtained when$M$is the torus and$f$is a toral endomorphism whose center-stable subspace does not contain the stable subspace of$g$at some point.

scholarly journals On the Baire Generic Validity of the t-Multifractal Formalism in Besov and Sobolev Spaces

2019 ◽  
Vol 2019 ◽  
pp. 1-14
Author(s):  
Moez Ben Abid ◽  
Mourad Ben Slimane ◽  
Ines Ben Omrane ◽  
Borhen Halouani
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Besov Space ◽  
Sobolev Spaces ◽  
Wavelet Coefficients ◽  
Multifractal Formalism ◽  
Spectrum Of A Function ◽  
Set Of Points

The t-multifractal formalism is a formula introduced by Jaffard and Mélot in order to deduce the t-spectrum of a function f from the knowledge of the (p,t)-oscillation exponent of f. The t-spectrum is the Hausdorff dimension of the set of points where f has a given value of pointwise Lt regularity. The (p,t)-oscillation exponent is measured by determining to which oscillation spaces Op,ts (defined in terms of wavelet coefficients) f belongs. In this paper, we first prove embeddings between oscillation and Besov-Sobolev spaces. We deduce a general lower bound for the (p,t)-oscillation exponent. We then show that this lower bound is actually equality generically, in the sense of Baire’s categories, in a given Sobolev or Besov space. We finally investigate the Baire generic validity of the t-multifractal formalism.

scholarly journals Amount of failure of upper-semicontinuity of entropy in non-compact rank-one situations, and Hausdorff dimension

2015 ◽  
Vol 37 (2) ◽  
pp. 539-563 ◽  
Author(s):  
S. KADYROV ◽  
A. POHL
Keyword(s):  
Hausdorff Dimension ◽  
Homogeneous Spaces ◽  
Upper Semicontinuity ◽  
Metric Entropy ◽  
Real Rank ◽  
Semisimple Lie Group ◽  
Uniform Lattice ◽  
Finite Center ◽  
Rank One ◽  
Set Of Points

Recently, Einsiedler and the authors provided a bound in terms of escape of mass for the amount by which upper-semicontinuity for metric entropy fails for diagonal flows on homogeneous spaces $\unicode[STIX]{x1D6E4}\setminus G$, where $G$ is any connected semisimple Lie group of real rank one with finite center, and $\unicode[STIX]{x1D6E4}$ is any non-uniform lattice in $G$. We show that this bound is sharp, and apply the methods used to establish bounds for the Hausdorff dimension of the set of points that diverge on average.

scholarly journals The Lyapunov spectrum of some parabolic systems

2009 ◽  
Vol 29 (3) ◽  
pp. 919-940 ◽  
Author(s):  
KATRIN GELFERT ◽  
MICHAŁ RAMS
Keyword(s):  
Lyapunov Exponent ◽  
Hausdorff Dimension ◽  
Lyapunov Exponents ◽  
Topological Entropy ◽  
Level Set ◽  
Level Sets ◽  
Parabolic Systems ◽  
Lyapunov Spectrum ◽  
Interval Maps ◽  
Set Of Points

AbstractWe study the Hausdorff dimension for Lyapunov exponents for a class of interval maps which includes several non-hyperbolic situations. We also analyze the level sets of points with given lower and upper Lyapunov exponents and, in particular, with zero lower Lyapunov exponent. We prove that the level set of points with zero exponent has full Hausdorff dimension, but carries no topological entropy.

Simultaneous equidistributing and non-dense points for non-commuting toral automorphisms

2018 ◽  
Vol 40 (1) ◽  
pp. 175-193
Author(s):  
MANFRED EINSIEDLER ◽  
ALEX MAIER
Keyword(s):  
Hausdorff Dimension ◽  
Dense Orbit ◽  
Set Of Points ◽  
Toral Automorphisms ◽  
Prime Dimension

We show in prime dimension that for two non-commuting totally irreducible toral automorphisms the set of points that equidistribute under the first map but have non-dense orbit under the second has full Hausdorff dimension. In non-prime dimension the argument fails only if the automorphisms have strong algebraic relations.

scholarly journals Directional Multifractal Analysis in the Lp Setting

2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Mourad Ben Slimane ◽  
Ines Ben Omrane ◽  
Moez Ben Abid ◽  
Borhen Halouani ◽  
Farouq Alshormani
Keyword(s):  
Hausdorff Dimension ◽  
Multifractal Analysis ◽  
Hölder Regularity ◽  
Wavelet Coefficients ◽  
Wide Range ◽  
Bounded Functions ◽  
Set Of Points ◽  
Holder Regularity

The classical Hölder regularity is restricted to locally bounded functions and takes only positive values. The local Lp regularity covers unbounded functions and negative values. Nevertheless, it has the same apparent regularity in all directions. In the present work, we study a recent notion of directional local Lp regularity introduced by Jaffard. We provide its characterization by a supremum of a wide range oriented anisotropic Triebel wavelet coefficients and leaders. In addition, we deduce estimates on the Hausdorff dimension of the set of points where the directional local Lp regularity does not exceed a given value. The obtained results are illustrated by some examples of self-affine cascade functions.

MIXED MULTIFRACTAL ANALYSIS FOR FUNCTIONS: GENERAL UPPER BOUND AND OPTIMAL RESULTS FOR VECTORS OF SELF-SIMILAR OR QUASI-SELF-SIMILAR OF FUNCTIONS AND THEIR SUPERPOSITIONS

Fractals ◽  
2016 ◽  
Vol 24 (04) ◽  
pp. 1650039 ◽  
Author(s):  
MOURAD BEN SLIMANE ◽  
ANOUAR BEN MABROUK ◽  
JAMIL AOUIDI
Keyword(s):  
Hausdorff Dimension ◽  
Upper Bound ◽  
Multifractal Analysis ◽  
Hölder Functions ◽  
Single Function ◽  
Cascade Models ◽  
Self Similar ◽  
Wavelet Decompositions ◽  
Vector Formula ◽  
Set Of Points

Mixed multifractal analysis for functions studies the Hölder pointwise behavior of more than one single function. For a vector [Formula: see text] of [Formula: see text] functions, with [Formula: see text], we are interested in the mixed Hölder spectrum, which is the Hausdorff dimension of the set of points for which each function [Formula: see text] has exactly a given value [Formula: see text] of pointwise Hölder regularity. We will conjecture a formula which relates the mixed Hölder spectrum to some mixed averaged wavelet quantities of [Formula: see text]. We will prove an upper bound valid for any vector of uniform Hölder functions. Then we will prove the validity of the conjecture for self-similar vectors of functions, quasi-self-similar vectors and their superpositions. These functions are written as the superposition of similar structures at different scales, reminiscent of some possible modelization of turbulence or cascade models. Their expressions look also like wavelet decompositions.

IMMOBILIZING A SHAPE

1999 ◽  
Vol 09 (02) ◽  
pp. 181-206 ◽  
Author(s):  
JUREK CZYZOWICZ ◽  
IVAN STOJMENOVIC ◽  
JORGE URRUTIA
Keyword(s):  
Large Class ◽  
Jordan Curve ◽  
Rigid Motion ◽  
Circular Disk ◽  
Convex Polygons ◽  
Connected Set ◽  
Simply Connected ◽  
Set Of Points

Let shape P be any simply-connected set in the plane, bounded by a Jordan curve, that is not a circular disk. We say that a set of points I on the boundary of P immobilize the shape if any rigid motion of P in the plane causes at least one point of I to penetrate the interior of P. We prove that four points always suffice to immobilize any shape. For a large class of shapes, which includes polygons without parallel edges, three points are sufficient to immobilize. An O(n log  n) algorithm is given that finds a set 3 points that immobilize a given polygon without parallel edges. The algorithm becomes linear for convex polygons. Some results are generalized for d-dimensional polytopes, where 2d points are always sufficient and sometimes necessary to immobilize.

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