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On the existence of conformal measures


Authors: Manfred Denker and Mariusz Urbański
Journal: Trans. Amer. Math. Soc. 328 (1991), 563-587
MSC: Primary 58F11; Secondary 28A12, 28A78, 28D05
DOI: https://doi.org/10.1090/S0002-9947-1991-1014246-4
MathSciNet review: 1014246
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Abstract: A general notion of conformal measure is introduced and some basic properties are studied. Sufficient conditions for the existence of these measures are obtained, using a general construction principle. The geometric properties of conformal measures relate equilibrium states and Hausdorff measures. This is shown for invariant subsets of $ {S^1}$ under expanding maps.


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  • [1] R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Math., Vol. 470, Springer-Verlag, 1975. MR 0442989 (56:1364)
  • [2] -, Hausdorff dimension of quasi circles, Inst. Hautes Études Sci. Publ. Math. 50 (1980), 11-25.
  • [3] M. Denker, C. Grillenberger, and K. Sigmund, Ergodic theory on compact spaces, Lecture Notes in Math., Vol. 527, Springer-Verlag, 1976. MR 0457675 (56:15879)
  • [4] M. Denker, G. Keller, and M. Urbański, On the uniqueness of equilibrium states for piecewise monotone mappings, Studia Math. 97 (1990), 27-36. MR 1074766 (92c:58061)
  • [5] K. J. Falconer, The geometry of fractal sets, Cambridge Univ. Press, New York, 1985. MR 867284 (88d:28001)
  • [6] F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations, Math. Z. 180 (1982), 119-140. MR 656227 (83h:28028)
  • [7] -, Equilibrium states for piecewise monotone maps, Ergodic Theory Dynamical Systems 2 (1982), 23-43. MR 684242 (85f:58069)
  • [8] I. A. Ibragimov and Y. V. Linnik, Independent and stationary sequences of random variables, Wolters-Noordhoff, Groningen, 1971. MR 0322926 (48:1287)
  • [9] N. G. Makarov, On the distortion of boundary sets under conformal mappings, Proc. London Math. Soc. 51 (1985), 369-384. MR 794117 (87d:30012)
  • [10] H. McClusky and A. Manning, Hausdorff dimension for horseshoes, Ergodic Theory Dynamical Systems 3 (1983), 251-260. MR 742227 (85j:58127)
  • [11] M. Misiurewicz, A short proof of the variational principle for $ {\mathbf{Z}}_ + ^N$ action on compact space, Bull. Acad. Polon. Sci. Ser. Math. 24 (1976), 1069-1075. MR 0430213 (55:3220)
  • [12] S. J. Patterson, The limit set of a Fuchsian group, Acta Math. 136 (1976), 241-273. MR 0450547 (56:8841)
  • [13] -, Lectures on measures on limit sets of Kleinian groups, Analytic and Geometric Aspects of Hyperbolic Space (D. B. A. Epstein, ed.), LMS Lecture Notes Ser., Vol. 111, Cambridge Univ. Press, 1987. MR 903855 (89b:58122)
  • [14 F] Przytycki, M. Urbański, and A. Zdunik, Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic maps. I, II, Ann. of Math. 130 (1989), 1-40 and Studia Math. 97 (1991), 189-225. MR 1100687 (93d:58140)
  • [15 C] A. Rogers, Hausdorff measures, Cambridge Univ. Press, 1970. MR 0281862 (43:7576)
  • [16 D] Ruelle, Repellers for real analytic maps, Ergodic Theory Dynamical Systems 2 (1982), 99-107. MR 684247 (84f:58095)
  • [17 M] Shub and D. Sullivan, Expanding endomorphisms of the circle revisited, Ergodic Theory Dynamical Systems 5 (1985), 285-289. MR 796755 (87g:58104)
  • [18 D] Sullivan, Conformal dynamical systems, Geometric Dynamics, Lecture Notes in Math., Vol. 1007, Springer-Verlag, 1983, pp. 725-752. MR 730296 (85m:58112)
  • [19] M. Urbański, Hausdorff dimension of invariant sets for expanding mappings of the circle, Ergodic Theory Dynamical Systems 6 (1986), 295-309. MR 857203 (87k:28017)
  • [20] -, Invariant subsets of expanding mappings of the circle, Ergodic Theory Dynamical Systems 7 (1987), 627-645. MR 922369 (89b:58142)
  • [21] -, Hausdorff dimension of invariant subsets for endomorphisms of the circle with an indifferent fixed point, J. London Math. Soc. 40 (1989), 158-170. MR 1028920 (91a:58106)
  • [22] -, On Hausdorff dimension of the Julia set with a rationally indifferent periodic point, Studia Math. 97 (1991), 167-188. MR 1100686 (93a:58146)
  • [23] P. Walters, Equilibrium states for $ \beta $-transformations and related transformations, Math. Z. 159 (1978), 65-88. MR 0466492 (57:6370)
  • [24] L. S. Young, Dimension, entropy and Lyapunov exponents, Ergodic Theory Dynamical Systems 2 (1982), 109-124. MR 684248 (84h:58087)

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DOI: https://doi.org/10.1090/S0002-9947-1991-1014246-4
Article copyright: © Copyright 1991 American Mathematical Society

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