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 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.

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.

Ergodic Theory of Numbers

2002 ◽  
Author(s):  
Karma Dajani ◽  
Cor Kraaikamp
Keyword(s):  
Ergodic Theory ◽  
Theory Of Numbers

scholarly journals ON THE INVERSE MULTIFRACTAL FORMALISM

2009 ◽  
Vol 52 (1) ◽  
pp. 179-194 ◽  
Author(s):  
L. OLSEN
Keyword(s):  
Multifractal Analysis ◽  
Regularity Condition ◽  
Upper Bounds ◽  
Legendre Transform ◽  
Self Similarity ◽  
Multifractal Formalism ◽  
Multifractal Spectra ◽  
Physics Literature ◽  
Image Position ◽  
Legendre Transforms

AbstractTwo of the main objects of study in multifractal analysis of measures are the coarse multifractal spectra and the Rényi dimensions. In the 1980s it was conjectured in the physics literature that for ‘good’ measures the following result, relating the coarse multifractal spectra to the Legendre transform of the Rényi dimensions, holds, namely This result is known as the multifractal formalism and has now been verified for many classes of measures exhibiting some degree of self-similarity. However, it is also well known that there is an abundance of measures not satisfying the multifractal formalism and that, in general, the Legendre transforms of the Rényi dimensions provide only upper bounds for the coarse multifractal spectra. The purpose of this paper is to prove that even though the multifractal formalism fails in general, it is nevertheless true that all measures (satisfying a mild regularity condition) satisfy the inverse of the multifractal formalism, namely

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 Small-Angle Scattering and Multifractal Analysis of DNA Sequences

2020 ◽  
Vol 21 (13) ◽  
pp. 4651
Author(s):  
Eugen Mircea Anitas
Keyword(s):  
Dna Sequences ◽  
Small Angle ◽  
Multifractal Analysis ◽  
Small Angle Scattering ◽  
Spatial Inhomogeneity ◽  
Chaos Game Representation ◽  
Model Based ◽  
Angle Scattering ◽  
Box Counting Dimension ◽  
Multifractal Spectra

The arrangement of A, C, G and T nucleotides in large DNA sequences of many prokaryotic and eukaryotic cells exhibit long-range correlations with fractal properties. Chaos game representation (CGR) of such DNA sequences, followed by a multifractal analysis, is a useful way to analyze the corresponding scaling properties. This approach provides a powerful visualization method to characterize their spatial inhomogeneity, and allows discrimination between mono- and multifractal distributions. However, in some cases, two different arbitrary point distributions, may generate indistinguishable multifractal spectra. By using a new model based on multiplicative deterministic cascades, here it is shown that small-angle scattering (SAS) formalism can be used to address such issue, and to extract additional structural information. It is shown that the box-counting dimension given by multifractal spectra can be recovered from the scattering exponent of SAS intensity in the fractal region. This approach is illustrated for point distributions of CGR data corresponding to Escherichia coli, Phospholamban and Mouse mitochondrial DNA, and it is shown that for the latter two cases, SAS allows extraction of the fractal iteration number and the scaling factor corresponding to “ACGT” square, or to recover the number of bases. The results are compared with a model based on multiplicative deterministic cascades, and respectively with one which takes into account the existence of forbidden sequences in DNA. This allows a classification of the DNA sequences in terms of random and deterministic fractals structures emerging in CGR.

Reading multifractal spectra: Aging by multifractal analysis of heart rate

2011 ◽  
Vol 94 (6) ◽  
pp. 68005 ◽  
Author(s):  
D. Makowiec ◽  
A. Rynkiewicz ◽  
R. Gałaska ◽  
J. Wdowczyk-Szulc ◽  
M. Żarczyńska-Buchowiecka
Keyword(s):  
Heart Rate ◽  
Multifractal Analysis ◽  
Multifractal Spectra

MIXED MULTIFRACTAL ANALYSIS OF CRUDE OIL, GOLD AND EXCHANGE RATE SERIES

Fractals ◽  
2016 ◽  
Vol 24 (04) ◽  
pp. 1650046 ◽  
Author(s):  
MEIFENG DAI ◽  
SHUXIANG SHAO ◽  
JIANYU GAO ◽  
YU SUN ◽  
WEIYI SU
Keyword(s):  
Time Series ◽  
Fractal Dimension ◽  
Exchange Rate ◽  
Crude Oil ◽  
Multifractal Analysis ◽  
Multifractal Spectrum ◽  
Price Series ◽  
The Us ◽  
Multifractal Spectra ◽  
Us Dollar

The multifractal analysis of one time series, e.g. crude oil, gold and exchange rate series, is often referred. In this paper, we apply the classical multifractal and mixed multifractal spectrum to study multifractal properties of crude oil, gold and exchange rate series and their inner relationships. The obtained results show that in general, the fractal dimension of gold and crude oil is larger than that of exchange rate (RMB against the US dollar), reflecting a fact that the price series in gold and crude oil are more heterogeneous. Their mixed multifractal spectra have a drift and the plot is not symmetric, so there is a low level of mixed multifractal between each pair of crude oil, gold and exchange rate series.

Ergodic Theory of Numbers

2004 ◽  
Vol 111 (7) ◽  
pp. 633
Author(s):  
David W. Boyd ◽  
Karma Dajani ◽  
Cor Kraaikamp
Keyword(s):  
Ergodic Theory ◽  
Theory Of Numbers

Exponential Growth Rates of Periodic Asymmetric Oscillators

2008 ◽  
Vol 8 (4) ◽  
Author(s):  
Meirong Zhang ◽  
Zhe Zhou
Keyword(s):  
Lyapunov Exponents ◽  
Ergodic Theory ◽  
Growth Rates ◽  
Exponential Growth ◽  
Zero Solution ◽  
Unique Ergodicity ◽  
Topological Dynamics ◽  
Asymmetric Oscillator ◽  
Ergodicity Theorem

AbstractIn this paper we will study the dynamics of the periodic asymmetric oscillator xʺ + qdoes exist for each non-zero solution x(t) of the oscillator. The properties of these rates, or the Lyapunov exponents, will be given using the induced circle di®eomorphism of the oscillator. The proof is extensively based on the Denjoy theorem in topological dynamics and the unique ergodicity theorem in ergodic theory.

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