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The local Hölder exponent for the dimension of invariant subsets of the circle

Published online by Cambridge University Press:  08 March 2016

Dipartimento di Matematica, Università di Pisa, Largo Pontecorvo 5, 56127 Pisa, Italy email
Yale University, 10 Hillhouse Avenue, New Haven, CT 06511, USA email
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We consider for each $t$ the set $K(t)$ of points of the circle whose forward orbit for the doubling map does not intersect $(0,t)$, and look at the dimension function $\unicode[STIX]{x1D702}(t):=\text{H.dim}\,K(t)$. We prove that at every bifurcation parameter $t$, the local Hölder exponent of the dimension function equals the value of the function $\unicode[STIX]{x1D702}(t)$ itself. A similar statement holds for general expanding maps of the circle: namely, we consider the topological entropy of the map restricted to the survival set, and obtain bounds on its local Hölder exponent in terms of the value of the function.

Research Article
© Cambridge University Press, 2016 


Bandtlow, O. F. and Rugh, H. H.. Entropy-continuity for interval maps with holes, Preprint, 2015,arXiv:1510.06043.Google Scholar
de Melo, W. and van Strien, S.. One-dimensional dynamics. Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Springer, Berlin, 1993, p. 25.Google Scholar
Dettmann, C.. Open circle maps: small hole asymptotics. Nonlinearity 26(1) (2013), 307317.CrossRefGoogle Scholar
Guckenheimer, J.. The growth of topological entropy for one-dimensional maps. Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univertsity, Evanston, IL., 1979) (Lecture Notes in Mathematics, 819) . Springer, Berlin, 1980, pp. 216223.CrossRefGoogle Scholar
Hutchinson, J. E.. Fractals and self-similarity. Indiana Univ. Math. J. 30(5) (1981), 713747.CrossRefGoogle Scholar
Isola, S. and Politi, A.. Universal encoding for unimodal maps. J. Stat. Phys. 61(1–2) (1990), 263291.CrossRefGoogle Scholar
Keller, G. and Liverani, C.. Rare events, escape rates and quasistationarity: some exact formulae. J. Stat. Phys. 135(3) (2009), 519534.CrossRefGoogle Scholar
Lothaire, M.. Combinatorics on words. Encyclopedia of Mathematics and its Applications. Addison-Wesley, Reading, MA, 1983, 17.Google Scholar
Milnor, J. and Thurston, W.. On iterated maps of the interval. Dynamical systems (College Park, MD, 1986–1987) (Lecture Notes in Mathematics, 1342) . Springer, Berlin, 1988, pp. 465563.Google Scholar
Moran, M.. Hausdorff measure of infinitely generated self-similar sets. Monatsh. Math. 122(4) (1996), 387399.CrossRefGoogle Scholar
Nilsson, J.. On numbers badly approximable by dyadic rationals. Israel J. Math. 171 (2009), 93110.CrossRefGoogle Scholar
Tiozzo, G.. Topological entropy of quadratic polynomials and dimension of sections of the Mandelbrot set. Adv. Math. 273 (2015), 651715.CrossRefGoogle Scholar
Tiozzo, G.. Continuity of core entropy of quadratic polynomials. Invent. Math. (2015), to appear. doi:10.1007/s00222-015-0605-9.Google Scholar
Urbański, M.. On Hausdorff dimension of invariant sets for expanding maps of a circle. Ergod. Th. & Dynam. Sys. 6(2) (1986), 295309.CrossRefGoogle Scholar
Young, L. S.. Dimension, entropy and Lyapunov exponents. Ergod. Th. & Dynam. Sys. 2(1) (1982), 109124.CrossRefGoogle Scholar