Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-09T09:28:10.315Z Has data issue: false hasContentIssue false

Hölder-differentiability of Gibbs distribution functions

Published online by Cambridge University Press:  01 September 2009

MARC KESSEBÖHMER
Affiliation:
Fachbereich 3 – Mathematik und Informatik, Universität Bremen, D–28359 Bremen, Germany. e-mail: [email protected]
BERND O. STRATMANN
Affiliation:
Mathematical Institute, University of St Andrews, St Andrews KY16 9SS, Scotland. e-mail: [email protected]

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 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.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[Bes34]Besicovitch, A. S.Sets of fractional dimension(IV): On rational approximation to real numbers. J. London Math. Soc. 9 (1934), 126131.CrossRefGoogle Scholar
[Bil79]Billingsley, P.Probability and Measure (Wiley, 1979).Google Scholar
[Bow75]Bowen, R.Equilibrium states and the ergodic theory of Anosov diffeomorphism. Lecture Notes in Math., vol. 470 (Springer-Verlag, 1975).CrossRefGoogle Scholar
[Dar93]Darst, R.The Hausdorff dimension of the non–differentiability set of the Cantor function is [ln2/ln3]2. Proc. Amer. Math. Soc. 119 (1993), 105108.Google Scholar
[Dar95]Darst, R.Hausdorff dimension of sets of non-differentiability points of Cantor functions. Math. Proc. Camb. Phil. Soc. 117 no. 1 (1995), 185191.CrossRefGoogle Scholar
[Den05]Denker, M.Introduction to Analysis of Dynamical Systems (Einführung in die Analysis Dynamischer Systeme) (Springer-Lehrbuch, 2005) (German), p. 285.Google Scholar
[DL03]Dekking, F. M. and Li, WenxiaHow smooth is a devil's staircase? Fractals 11 no. 1 (2003), 101107.CrossRefGoogle Scholar
[Fal97]Falconer, K. J.Techniques in Fractal Geometry (John Wiley & Sons Ltd., 1997).Google Scholar
[Fal04]Falconer, K. J.One-sided multifractal analysis and points of non-differentiability of devil's staircases. Math. Proc. Camb. Phil. Soc. 136 (2004), 67174.CrossRefGoogle Scholar
[Gil32]Gilman, R. E.A class of functions continuous but not absolutely continuous. Ann. Math. 33 (1932), 433442.CrossRefGoogle Scholar
[Gut90]Gutzwiller, M. Chaos in classical and quantum mechanics. Interdis. Appl. Math. (Springer Verlag, 1990).Google Scholar
[GM88]Gutzwiller, M. and Mandelbrot, B. B.Invariant multifractal measures in chaotic Hamiltonian systems, and related structures. Phys. Rev. Lett. 60 (1988), 673676.CrossRefGoogle ScholarPubMed
[HV98]Hill, R. and Velani, S. L.The Jarník–Besicovitch theorem for geometrically finite Kleinian groups. Proc. London Math. Soc. 77 no. 3 (1998), 524550.CrossRefGoogle Scholar
[Jar29]Jarník, V.Diophantische Approximationen and Hausdorff Mass. Math. Sbornik 36 (1929), 371382.Google Scholar
[JKPS]Jordan, T., Kesseböhmer, M., Pollicott, M. and Stratmann, B. O. Sets of non-differentiability for conjugacies between expanding interval maps, preprint: arXiv:0807.0115v1 (2008).CrossRefGoogle Scholar
[KS07]Kesseböhmer, M. and Stratmann, B. O.Fractal analysis for sets of non-differentiability of Minkowski's question mark function. J. Number Theory 128 (2008), 26632686.CrossRefGoogle Scholar
[Li07]Wenxia, LiNon-differentiability points of Cantor functions. Math. Nachr. 280 no. 1–2 (2007), 140151.Google Scholar
[LXD02]Wenxia, Li, Xiao, Dongmei and Dekking, F. M.Non-differentiability of devil's staircases and dimensions of subsets of Moran sets. Math. Proc. Camb. Phil. Soc. 133 no. 2 (2002), 345355.Google Scholar
[Mo02]Morris, J.The Hausdorff dimension of the non-differentiability set of a non-symmetric Cantor function. Rocky Mountain J. Math. 32 (2002), 357370.CrossRefGoogle Scholar
[Pes97]Pesin, Ya.Dimension theory in dynamical systems: Contemporary views and applications. Chicago Lectures in Mathematics (The University of Chicago Press, 1997).CrossRefGoogle Scholar
[Rue78]Ruelle, D. Thermodynamic formalism. Encyclopedia of Mathematics and its Application, vol. 5 (Addison-Wesley, 1978).Google Scholar
[Str95]Stratmann, B. O.Fractal dimensions for Jarník limit sets of geometrically finite Kleinian groups; the semi-classical approach. Arkiv för Mat. 33 (1995), 385403.CrossRefGoogle Scholar
[Str99]Stratmann, B. O.Weak singularity spectra of the Patterson measure for geometrically finite Kleinian groups with parabolic elements. Michigan Math. J. 46 (1999), 573587.CrossRefGoogle Scholar
[SU02]Stratmann, B. O. and Urbański, M.Jarník and Julia; a Diophantine analysis for parabolic rational maps. Math. Scan. 91 (2002), 2754.CrossRefGoogle Scholar