scholarly journals Ledrappier–Young formulae for a family of nonlinear attractors

2021 ◽  
Author(s):  
Natalia Jurga ◽  
Lawrence D. Lee
Keyword(s):  
Invariant Measures ◽  
Gibbs Measures ◽  
Iterated Function Systems ◽  
Hölder Continuous ◽  
Natural Class ◽  
Iterated Function ◽  
Bernoulli Measures

AbstractWe study a natural class of invariant measures supported on the attractors of a family of nonlinear, non-conformal iterated function systems introduced by Falconer, Fraser and Lee. These are pushforward quasi-Bernoulli measures, a class which includes the well-known class of Gibbs measures for Hölder continuous potentials. We show that these measures are exact dimensional and that their exact dimensions satisfy a Ledrappier–Young formula.

A new class of markov processes for image encoding

1988 ◽  
Vol 20 (01) ◽  
pp. 14-32 ◽  
Author(s):  
Michael F. Barnsley ◽  
John H. Elton
Keyword(s):  
Invariant Measures ◽  
Metric Spaces ◽  
Iterated Function Systems ◽  
Interesting Case ◽  
Image Encoding ◽  
New Class ◽  
Affine Maps ◽  
Iterated Maps ◽  
Iterated Function ◽  
Infinite Extent

A new class of iterated function systems is introduced, which allows for the computation of non-compactly supported invariant measures, which may represent, for example, greytone images of infinite extent. Conditions for the existence and attractiveness of invariant measures for this new class of randomly iterated maps, which are not necessarily contractions, in metric spaces such as , are established. Estimates for moments of these measures are obtained. Special conditions are given for existence of the invariant measure in the interesting case of affine maps on . For non-singular affine maps on , the support of the measure is shown to be an infinite interval, but Fourier transform analysis shows that the measure can be purely singular even though its distribution function is strictly increasing.

A new class of markov processes for image encoding

1988 ◽  
Vol 20 (1) ◽  
pp. 14-32 ◽  
Author(s):  
Michael F. Barnsley ◽  
John H. Elton
Keyword(s):  
Invariant Measures ◽  
Metric Spaces ◽  
Iterated Function Systems ◽  
Interesting Case ◽  
Image Encoding ◽  
New Class ◽  
Affine Maps ◽  
Iterated Maps ◽  
Iterated Function ◽  
Infinite Extent

A new class of iterated function systems is introduced, which allows for the computation of non-compactly supported invariant measures, which may represent, for example, greytone images of infinite extent. Conditions for the existence and attractiveness of invariant measures for this new class of randomly iterated maps, which are not necessarily contractions, in metric spaces such as , are established. Estimates for moments of these measures are obtained.Special conditions are given for existence of the invariant measure in the interesting case of affine maps on . For non-singular affine maps on , the support of the measure is shown to be an infinite interval, but Fourier transform analysis shows that the measure can be purely singular even though its distribution function is strictly increasing.

scholarly journals ITERATED FUNCTION SYSTEMS WITH PLACE-DEPENDENT PROBABILITIES AND THE INVERSE PROBLEM OF MEASURE APPROXIMATION USING MOMENTS

Fractals ◽  
2018 ◽  
Vol 26 (05) ◽  
pp. 1850076 ◽  
Author(s):  
D. LA TORRE ◽  
E. MAKI ◽  
F. MENDIVIL ◽  
E. R. VRSCAY
Keyword(s):  
Invariant Measures ◽  
Iterated Function Systems ◽  
Probability Measures ◽  
Moment Matching ◽  
Computational Results ◽  
Compact Metric Space ◽  
Moment Vector ◽  
Iterated Function ◽  
Approximation Of Probability Measures ◽  
Vector Formula

We are concerned with the approximation of probability measures on a compact metric space [Formula: see text] by invariant measures of iterated function systems with place-dependent probabilities (IFSPDPs). The approximation is performed by moment matching. Associated with an IFSPDP is a linear operator [Formula: see text], where [Formula: see text] denotes the set of all infinite moment vectors of probability measures on [Formula: see text]. Let [Formula: see text] be a probability measure that we desire to approximate, with moment vector [Formula: see text]. We then look for an IFSPDP which maps [Formula: see text] as close to itself as possible in terms of an appropriate metric on [Formula: see text]. Some computational results are presented.

scholarly journals Dimension estimates for iterated function systems and repellers. Part II

2021 ◽  
pp. 1-36
Author(s):  
DE-JUN FENG ◽  
KÁROLY SIMON
Keyword(s):  
Invariant Measures ◽  
Transversality Condition ◽  
Iterated Function Systems ◽  
Upper Bounds ◽  
Dimension Theory ◽  
Lyapunov Dimension ◽  
Part D ◽  
Box Counting ◽  
Box Counting Dimension ◽  
Iterated Function

Abstract This is the second part of our study on the dimension theory of $C^1$ iterated function systems (IFSs) and repellers on $\mathbb {R}^d$ . In the first part [D.-J. Feng and K. Simon. Dimension estimates for $C^1$ iterated function systems and repellers. Part I. Preprint, 2020, arXiv:2007.15320], we proved that the upper box-counting dimension of the attractor of every $C^1$ IFS on ${\Bbb R}^d$ is bounded above by its singularity dimension, and the upper packing dimension of every ergodic invariant measure associated with this IFS is bounded above by its Lyapunov dimension. Here we introduce a generalized transversality condition (GTC) for parameterized families of $C^1$ IFSs, and show that if the GTC is satisfied, then the dimensions of the IFS attractor and of the ergodic invariant measures are given by these upper bounds for almost every (in an appropriate sense) parameter. Moreover, we verify the GTC for some parameterized families of $C^1$ IFSs on ${\Bbb R}^d$ .

scholarly journals Sampling and interpolation of cumulative distribution functions of Cantor sets in [0, 1]

2021 ◽  
Vol 54 (1) ◽  
pp. 85-109
Author(s):  
Allison Byars ◽  
Evan Camrud ◽  
Steven N. Harding ◽  
Sarah McCarty ◽  
Keith Sullivan ◽  
...  
Keyword(s):  
Cumulative Distribution Function ◽  
Invariant Measures ◽  
Borel Probability Measure ◽  
Iterated Function Systems ◽  
Distribution Functions ◽  
Cantor Sets ◽  
Cumulative Distribution ◽  
Sampling Schemes ◽  
Iterated Function ◽  
Borel Probability

Abstract Cantor sets are constructed from iteratively removing sections of intervals. This process yields a cumulative distribution function (CDF), constructed from the invariant Borel probability measure associated with their iterated function systems. Under appropriate assumptions, we identify sampling schemes of such CDFs, meaning that the underlying Cantor set can be reconstructed from sufficiently many samples of its CDF. To this end, we prove that two Cantor sets have almost-nowhere intersection with respect to their corresponding invariant measures.

scholarly journals Hölder-differentiability of Gibbs distribution functions

2009 ◽  
Vol 147 (2) ◽  
pp. 489-503 ◽  
Author(s):  
MARC KESSEBÖHMER ◽  
BERND O. STRATMANN
Keyword(s):  
Distribution Function ◽  
Gibbs Measures ◽  
Thermodynamic Formalism ◽  
Iterated Function Systems ◽  
Gibbs Distribution ◽  
Distribution Functions ◽  
Limit Sets ◽  
Iterated Function ◽  
Natural Home ◽  
Set Of Points

AbstractIn this paper we give non-trivial applications of the thermodynamic formalism to the theory of distribution functions of Gibbs measures (devil's staircases) supported on limit sets of finitely generated conformal iterated function systems in ℝ. For a large class of these Gibbs states we determine the Hausdorff dimension of the set of points at which the distribution function of these measures is not α-Hölder-differentiable. The obtained results give significant extensions of recent work by Darst, Dekking, Falconer, Li, Morris and Xiao. In particular, our results clearly show that the results of these authors have their natural home within the thermodynamic formalism.

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