scholarly journals Multifractal analysis of Birkhoff averages for countable Markov maps

2015 ◽  
Vol 35 (8) ◽  
pp. 2559-2586 ◽  
Author(s):  
GODOFREDO IOMMI ◽  
THOMAS JORDAN
Keyword(s):  
Number Theory ◽  
Hausdorff Dimension ◽  
Multifractal Analysis ◽  
Multifractal Formalism ◽  
Interval Maps ◽  
Real Analytic ◽  
New Phenomena ◽  
Set Of Points

In this paper we prove a multifractal formalism of Birkhoff averages for interval maps with countably many branches. Furthermore, we prove that under certain assumptions the Birkhoff spectrum is real analytic. We also show that new phenomena occur; indeed, the spectrum can be constant or it can have points where it is not analytic. Conditions for these to happen are obtained. Applications of these results to number theory are also given. Finally, we compute the Hausdorff dimension of the set of points for which the Birkhoff average is infinite.

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 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.

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.

scholarly journals Hölder differentiability of self-conformal devil's staircases

2014 ◽  
Vol 156 (2) ◽  
pp. 295-311 ◽  
Author(s):  
SASCHA TROSCHEIT
Keyword(s):  
Distribution Function ◽  
Probability Distribution ◽  
Hausdorff Dimension ◽  
Probability Distribution Function ◽  
Iterated Function System ◽  
Gibbs Measures ◽  
General Sense ◽  
Multifractal Formalism ◽  
Iterated Function ◽  
Set Of Points

AbstractIn this paper we consider the probability distribution function of a Gibbs measure supported on a self-conformal set given by an iterated function system (devil's staircase) applied to a compact subset of ${\mathbb R}$. We use thermodynamic multifractal formalism to calculate the Hausdorff dimension of the sets Sα0, Sα∞ and Sα, the set of points at which this function has, respectively, Hölder derivative 0, ∞ or no derivative in the general sense. This extends recent work by Darst, Dekking, Falconer, Kesseböhmer and Stratmann, and Yao, Zhang and Li by considering arbitrary such Gibbs measures given by a potential function independent of the geometric potential.

scholarly journals Multifractal analysis in non-uniformly hyperbolic interval maps

Nonlinearity ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 110-133
Author(s):  
Guanzhong Ma ◽  
Wenqiang Shen ◽  
Xiao Yao
Keyword(s):  
Hausdorff Dimension ◽  
Level Set ◽  
Multifractal Analysis ◽  
Ergodic Measure ◽  
Moran Set ◽  
Interval Maps ◽  
The Given

Abstract In this paper, we establish a framework for the construction of Moran set driven by dynamics. Under this framework, we study the Hausdorff dimension of the generalized intrinsic level set with respect to the given ergodic measure in a class of non-uniformly hyperbolic interval maps with finitely many branches.

scholarly journals The singularity spectrumf(α) for cookie-cutters

1989 ◽  
Vol 9 (3) ◽  
pp. 527-541 ◽  
Author(s):  
D. A. Rand
Keyword(s):  
Hausdorff Dimension ◽  
Gibbs State ◽  
Thermodynamic Formalism ◽  
Legendre Transform ◽  
Pointwise Dimension ◽  
Large Fluctuations ◽  
Rényi Dimension ◽  
Real Analytic ◽  
Set Of Points

AbstractI use a thermodynamic formalism to study the spectrumf(α) which characterises the large fluctuations of pointwise dimension in a Gibbs state supported on a hyperbolic cookie-cutter. Amongst other things, it is proved thatf(α) is the Hausdorff dimension of the set of points with pointwise dimension α, thatf(α) is real-analytic and that its Legendre transform τ(q) is related to the Renyi dimensionDqof the Gibbs state by the formula (1 −q)Dq= τ(q).

scholarly journals Lyapunov spectrum for Hénon-like maps at the first bifurcation

2016 ◽  
Vol 38 (3) ◽  
pp. 1168-1200
Author(s):  
HIROKI TAKAHASI
Keyword(s):  
Lyapunov Exponent ◽  
Hausdorff Dimension ◽  
Level Sets ◽  
Multifractal Analysis ◽  
Lyapunov Spectrum ◽  
Probability Measures ◽  
Limit Set ◽  
Uniform Hyperbolicity ◽  
Partial Description ◽  
Set Of Points

For a strongly dissipative Hénon-like map at the first bifurcation parameter at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set, we effect a multifractal analysis, i.e. decompose the set of non-wandering points on the unstable manifold into level sets of an unstable Lyapunov exponent, and give a partial description of the Lyapunov spectrum which encodes this decomposition. We derive a formula for the Hausdorff dimension of the level sets in terms of the entropy and unstable Lyapunov exponent of invariant probability measures, and show the continuity of the Lyapunov spectrum. We also show that the set of points for which the unstable Lyapunov exponents do not exist carries the full Hausdorff dimension.

scholarly journals On shrinking targets for piecewise expanding interval maps

2015 ◽  
Vol 37 (2) ◽  
pp. 646-663 ◽  
Author(s):  
TOMAS PERSSON ◽  
MICHAŁ RAMS
Keyword(s):  
Hausdorff Dimension ◽  
Invariant Measure ◽  
Gibbs Measure ◽  
Intersection Property ◽  
Interval Maps ◽  
Large Intersection ◽  
Set Of Points

For a map $T:[0,1]\rightarrow [0,1]$ with an invariant measure $\unicode[STIX]{x1D707}$, we study, for a $\unicode[STIX]{x1D707}$-typical $x$, the set of points $y$ such that the inequality $|T^{n}x-y|<r_{n}$ is satisfied for infinitely many $n$. We give a formula for the Hausdorff dimension of this set, under the assumption that $T$ is piecewise expanding and $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D719}}$ is a Gibbs measure. In some cases we also show that the set has a large intersection property.

scholarly journals Non-dense orbits on homogeneous spaces and applications to geometry and number theory

2021 ◽  
pp. 1-46
Author(s):  
JINPENG AN ◽  
LIFAN GUAN ◽  
DMITRY KLEINBOCK
Keyword(s):  
Number Theory ◽  
Hausdorff Dimension ◽  
Symmetric Spaces ◽  
Homogeneous Spaces ◽  
Discrete Subgroup ◽  
Locally Symmetric Spaces ◽  
Affine Map ◽  
Integer Points ◽  
Dense Orbits ◽  
Set Of Points

Abstract Let G be a Lie group, let $\Gamma \subset G$ be a discrete subgroup, let $X=G/\Gamma $ and let f be an affine map from X to itself. We give conditions on a submanifold Z of X that guarantee that the set of points $x\in X$ with f-trajectories avoiding Z is hyperplane absolute winning (a property which implies full Hausdorff dimension and is stable under countable intersections). A similar result is proved for one-parameter actions on X. This has applications in constructing exceptional geodesics on locally symmetric spaces and in non-density of the set of values of certain functions at integer points.

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