Hölder differentiability of self-conformal devil's staircases

@article{Troscheit2014HlderDO,
  title={H{\"o}lder differentiability of self-conformal devil's staircases},
  author={Sascha Troscheit},
  journal={Mathematical Proceedings of the Cambridge Philosophical Society},
  year={2014},
  volume={156},
  pages={295 - 311}
}
  • Sascha Troscheit
  • Published 2014
  • Mathematics
  • Mathematical Proceedings of the Cambridge Philosophical Society
Abstract In 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… Expand
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References

SHOWING 1-10 OF 14 REFERENCES
Hölder-differentiability of Gibbs distribution functions
Abstract In 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 ofExpand
Dimensions of non-differentiability points of Cantor functions
For a probability vector (p0, p1) there exists a corresponding self-similar Borel probability measure μ supported on the Cantor set C (with the strong separation property) in R generated by aExpand
One-sided multifractal analysis and points of non-differentiability of devil's staircases
  • K. Falconer
  • Mathematics
  • Mathematical Proceedings of the Cambridge Philosophical Society
  • 2004
We examine the multifractal spectra of one-sided local dimensions of Ahlfors regular measures on ${\bf R}$. This brings into a natural context a curious property that has been observed in a number ofExpand
Non‐differentiability points of Cantor functions
Let the Cantor set C in ℝ be defined by C = ∪rj =0hj (C) with a disjoint union, where the hj 's are similitude mappings with ratios 0 ai we determine the packing and box dimensions of S and give anExpand
Hausdorff dimension of sets of non-differentiability points of Cantor functions
Each number a in the segment (0, ½) produces a Cantor set, Ca, by putting b = 1 − 2a and recursively removing segments of relative length b from the centres of the interval [0, 1] and the intervalsExpand
The hausdorff dimension of the nondifferentiability set of a nonsymmetric cantor function
Each choice of numbers a and c in the segment (0, (1/2)) produces a Cantor set C ac by recursively removing segments from the interior of the interval [0, 1] so that intervals of relative length aExpand
The Hausdorff dimension of the nondifferentiability set of the Cantor function is [(2)/(3)]²
The main purpose of this note is to verify that the Hausdorff dimension of the set of points N* at which the Cantor function is not differentiable is [ln(2)/ln(3)]2. It is also shown that the imageExpand
The multifractal analysis of Gibbs measures: Motivation, mathematical foundation, and examples.
TLDR
A rigorous mathematical foundation is presented for the multifractal analysis of Gibbs measures invariant under dynamical systems for several classes of hyperbolic Dynamical systems. Expand
DIMENSION THEORY IN DYNAMICAL SYSTEMS: CONTEMPORARY VIEWS AND APPLICATIONS By YAKOV B. PESIN Chicago Lectures in Mathematics, University of Chicago Press, 312 pp. Price: hardback $56, paperback $19.95. ISBN 0 226 66222 5
New updated! The latest book from a very famous author finally comes out. Book of dimension theory in dynamical systems contemporary views and applications, as an amazing reference becomes what youExpand
Techniques in fractal geometry
Mathematical Background. Review of Fractal Geometry. Some Techniques for Studying Dimension. Cookie-cutters and Bounded Distortion. The Thermodynamic Formalism. The Ergodic Theorem and Fractals. TheExpand
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