Multifractal analysis on the level sets described by moving averages

2015 ◽  
Vol 11 (07) ◽  
pp. 2175-2189
Author(s):  
Haibo Chen ◽  
Xiaohua Wang ◽  
Zhixiong Wen
Keyword(s):  
Ergodic Theory ◽  
Level Sets ◽  
Multifractal Analysis ◽  
Ergodic Transformation ◽  
Moving Averages ◽  
Hausdorff Dimensions ◽  
Ergodic Processes

In this paper, the Hausdorff dimensions of level sets described by two kinds of moving averages are determined. The dissimilar results complement the work of Pfaffelhuber [Moving shift averages for ergodic transformation, Metrika22 (1975) 97–101] and del Junco and Steele [Moving averages of ergodic processes, Metrika24 (1977) 35–43], and reveal simultaneously that the two moving averages are of different convergence processes in the ergodic theory.

STERN–BROCOT PRESSURE AND MULTIFRACTAL SPECTRA IN ERGODIC THEORY OF NUMBERS

2004 ◽  
Vol 04 (01) ◽  
pp. 77-84 ◽  
Author(s):  
MARC KESSEBÖHMER ◽  
BERND O. STRATMANN
Keyword(s):  
Ergodic Theory ◽  
Growth Rates ◽  
Level Sets ◽  
Multifractal Analysis ◽  
Legendre Transformation ◽  
Multifractal Spectra ◽  
Theory Of Numbers

In this note we apply the general multifractal analysis for growth rates derived in [10], and show that this leads to some new results in ergodic theory and the theory of multifractals of numbers. Namely, we consider Stern–Brocot growth rates and introduce the Stern–Brocot pressure P. We then obtain the results that P is differentiable everywhere and that its Legendre transformation governs the multifractal spectra arising from level sets of Stern–Brocot rates.

Multifractal analysis for disintegrations of Gibbs measures and conditional Birkhoff averages

2009 ◽  
Vol 29 (3) ◽  
pp. 885-918 ◽  
Author(s):  
DE-JUN FENG ◽  
LIN SHU
Keyword(s):  
Convex Analysis ◽  
Level Sets ◽  
Multifractal Analysis ◽  
Gibbs Measures ◽  
Thermodynamic Formalism

AbstractThe paper is devoted to the study of the multifractal structure of disintegrations of Gibbs measures and conditional (random) Birkhoff averages. Our approach is based on the relativized thermodynamic formalism, convex analysis and, especially, the delicate constructions of Moran-like subsets of level sets.

Slow increasing functions and the largest partial quotients in continued fraction expansions

2016 ◽  
Vol 164 (1) ◽  
pp. 1-14 ◽  
Author(s):  
JINHUA CHANG ◽  
HAIBO CHEN
Keyword(s):  
Level Sets ◽  
Limit Theorems ◽  
Continued Fraction ◽  
Positive Function ◽  
Continued Fraction Expansion ◽  
Hausdorff Dimensions ◽  
Partial Quotients ◽  
Continued Fraction Expansions ◽  
Increasing Functions

AbstractLet 0 ⩽ α ⩽ ∞ and ψ be a positive function defined on (0, ∞). In this paper, we will study the level sets L(α, {ψ(n)}), B(α, {ψ(n)}) and T(α, {ψ(n)}) which are related respectively to the sequence of the largest digits among the first n partial quotients {Ln(x)}n≥1, the increasing sequence of the largest partial quotients {Bn(x)}n⩾1 and the sequence of successive occurrences of the largest partial quotients {Tn(x)}n⩾1 in the continued fraction expansion of x ∈ [0,1) ∩ ℚc. Under suitable assumptions of the function ψ, we will prove that the sets L(α, {ψ(n)}), B(α, {ψ(n)}) and T(α, {ψ(n)}) are all of full Hausdorff dimensions for any 0 ⩽ α ⩽ ∞. These results complement some limit theorems given by J. Galambos [4] and D. Barbolosi and C. Faivre [1].

scholarly journals Strong renewal theorems and Lyapunov spectra forα-Farey andα-Lüroth systems

2011 ◽  
Vol 32 (3) ◽  
pp. 989-1017 ◽  
Author(s):  
MARC KESSEBÖHMER ◽  
SARA MUNDAY ◽  
BERND O. STRATMANN
Keyword(s):  
Number Theory ◽  
Ergodic Theory ◽  
Level Sets ◽  
Gauss Map ◽  
Unit Interval ◽  
Elementary Number Theory ◽  
The Core ◽  
Lyapunov Spectra ◽  
Infinite Ergodic Theory ◽  
Farey Map

AbstractIn this paper, we introduce and study theα-Farey map and its associated jump transformation, theα-Lüroth map, for an arbitrary countable partitionαof the unit interval with atoms which accumulate only at the origin. These maps represent linearized generalizations of the Farey map and the Gauss map from elementary number theory. First, a thorough analysis of some of their topological and ergodic theoretical properties is given, including establishing exactness for both types of these maps. The first main result then is to establish weak and strong renewal laws for what we have calledα-sum-level sets for theα-Lüroth map. Similar results have previously been obtained for the Farey map and the Gauss map by using infinite ergodic theory. In this respect, a side product of the paper is to allow for greater transparency of some of the core ideas of infinite ergodic theory. The second remaining result is to obtain a complete description of the Lyapunov spectra of theα-Farey map and theα-Lüroth map in terms of the thermodynamical formalism. We show how to derive these spectra and then give various examples which demonstrate the diversity of their behaviours in dependence on the chosen partitionα.

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.

Moving averages of ergodic processes

Metrika ◽  
1977 ◽  
Vol 24 (1) ◽  
pp. 35-43 ◽  
Author(s):  
A. del Junco ◽  
J. M. Steele
Keyword(s):  
Moving Averages ◽  
Ergodic Processes

scholarly journals Range-renewal structure in continued fractions

2016 ◽  
Vol 37 (4) ◽  
pp. 1323-1344
Author(s):  
JUN WU ◽  
JIAN-SHENG XIE
Keyword(s):  
Continued Fractions ◽  
Level Sets ◽  
Continued Fraction ◽  
Irrational Number ◽  
Hausdorff Dimensions ◽  
Partial Quotients ◽  
Image Position ◽  
Almost All

Let $\unicode[STIX]{x1D714}=[a_{1},a_{2},\ldots ]$ be the infinite expansion of a continued fraction for an irrational number $\unicode[STIX]{x1D714}\in (0,1)$, and let $R_{n}(\unicode[STIX]{x1D714})$ (respectively, $R_{n,k}(\unicode[STIX]{x1D714})$, $R_{n,k+}(\unicode[STIX]{x1D714})$) be the number of distinct partial quotients, each of which appears at least once (respectively, exactly $k$ times, at least $k$ times) in the sequence $a_{1},\ldots ,a_{n}$. In this paper, it is proved that, for Lebesgue almost all $\unicode[STIX]{x1D714}\in (0,1)$ and all $k\geq 1$, $$\begin{eqnarray}\displaystyle \lim _{n\rightarrow \infty }\frac{R_{n}(\unicode[STIX]{x1D714})}{\sqrt{n}}=\sqrt{\frac{\unicode[STIX]{x1D70B}}{\log 2}},\quad \lim _{n\rightarrow \infty }\frac{R_{n,k}(\unicode[STIX]{x1D714})}{R_{n}(\unicode[STIX]{x1D714})}=\frac{C_{2k}^{k}}{(2k-1)\cdot 4^{k}},\quad \lim _{n\rightarrow \infty }\frac{R_{n,k}(\unicode[STIX]{x1D714})}{R_{n,k+}(\unicode[STIX]{x1D714})}=\frac{1}{2k}.\end{eqnarray}$$ The Hausdorff dimensions of certain level sets about $R_{n}$ are discussed.

scholarly journals The packing spectrum for Birkhoff averages on a self-affine repeller

2011 ◽  
Vol 32 (4) ◽  
pp. 1444-1470 ◽  
Author(s):  
HENRY W. J. REEVE
Keyword(s):  
Variational Principle ◽  
Level Sets ◽  
Multifractal Analysis ◽  
Packing Dimension ◽  
Sufficient Condition ◽  
Hölder Continuous ◽  
Real Analytic

AbstractWe consider the multifractal analysis of Birkhoff averages of continuous potentials on a self-affine Sierpiński sponge. In particular, we give a variational principle for the packing dimension of the level sets. Furthermore, we prove that the packing spectrum is concave and continuous. We give a sufficient condition for the packing spectrum to be real analytic, but show that for general Hölder continuous potentials, this need not be the case. We also give a precise criterion for when the packing spectrum attains the full packing dimension of the repeller. Again, we present an example showing that this is not always the case.

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