Metric diophantine approximation in Julia sets of expanding rational maps
Publications Mathématiques de l'IHÉS, Volume 85  (1997), p. 193-216
@article{PMIHES_1997__85__193_0,
     author = {Hill, Richard and Velani, Sanju L.},
     title = {Metric diophantine approximation in Julia sets of expanding rational maps},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     publisher = {Institut des Hautes \'Etudes Scientifiques},
     volume = {85},
     year = {1997},
     pages = {193-216},
     zbl = {0885.11051},
     mrnumber = {99b:58143},
     language = {en},
     url = {http://www.numdam.org/item/PMIHES_1997__85__193_0}
}
Hill, Richard; Velani, Sanju L. Metric diophantine approximation in Julia sets of expanding rational maps. Publications Mathématiques de l'IHÉS, Volume 85 (1997) , pp. 193-216. http://www.numdam.org/item/PMIHES_1997__85__193_0/

[1] A. F. Beardon, Iteration of Rational Functions, Springer GTM 132, 1991. | MR 92j:30026 | Zbl 0742.30002

[2] A. S. Besicovitch, Sets of fractional dimension (IV): On rational approximation to real numbers, J. London Math. Soc. 9 (1934), 126-131. | JFM 60.0204.01 | Zbl 0009.05301

[3] R. Bowen, Equilibrium States and the Ergodic Theory for Anosov Diffeomorphisms, III, Springer LNM 470, 1975. | MR 56 #1364 | Zbl 0308.28010

[4] H. Brolin, Invariant sets under iteration of rational functions, Ark. Mat. 6 (1965), 103-144. | MR 33 #2805 | Zbl 0127.03401

[5] J. W. S. Cassels, An Introduction to Diophantine Approximation, Cambridge Univ. Press, 1957. | MR 19,396h | Zbl 0077.04801

[6] M. Denker, Ch. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, IV, Springer LNM 527, 1976. | MR 56 #15879 | Zbl 0328.28008

[7] M. Denker and M. Urbański, Ergodic theory of equilibrium states of rational maps, Nonlinearity 4 (1991), 103-134. | MR 92a:58112 | Zbl 0718.58035

[8] K. J. Falconer, Fractal Geometry - Mathematical Foundations and Applications, J. Wiley, Chichester, 1990. | Zbl 0689.28003

[9] R. Hill and S. L. Velani, The Ergodic Theory of Shrinking Targets, Invent. Math. 119 (1995), 175-198. | MR 96e:58088 | Zbl 0834.28009

[10] R. Hill and S. L. Velani, Markov maps and moving, shrinking targets, in preparation.

[11] R. Hill and S. L. Velani, The Shrinking Target Problem for Matrix Transformations of Tori, J. Lond. Math. Soc. (to appear). | Zbl 0987.37008

[12] R. Hill and S. L. Velani, The Jarník-Besicovitch theorem for geometrically finite Kleinian groups, Proc. Lond. Math. Soc. (to appear). | Zbl 0924.11063

[13] E. Hille, Analytic function theory, Ginn and Company: Boston - New York - Chicago - Atlanta - Dallas - Palo Alto - Toronto, 1962. | MR 34 #1490 | Zbl 0102.29401

[14] V. Jarník, Diophantische Approximationen und Hausdorffsches Mass, Math. Sb. 36 (1929), 371-382. | JFM 55.0719.01

[15] V. Ljubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergod. Th. Dynam. Sys. 3 (1983), 351-386. | MR 85k:58049 | Zbl 0537.58035

[16] M. V. Melian and S. L. Velani, Geodesic excursions into cusps in infinite volume hyperbolic manifolds, Mathematica Gottingensis 45, 1993. | Zbl 0793.53052

[17] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, Soc. Math. France, 187-188, 1990. | MR 92f:58141 | Zbl 0726.58003

[18] S. J. Patterson, The limit set of a Fuchsian group, Acta Math. 136 (1976), 241-273. | MR 56 #8841 | Zbl 0336.30005

[19] D. Ruelle, Thermodynamical Formalism, Encycl. Math. Appl. 5, 1978. | MR 80g:82017 | Zbl 0401.28016

[20] V. G. Sprindžuk, Metric theory of Diophantine approximation (translated by R. A. SILVERMAN), V. H. Winston & Sons, Washington D.C., 1979. | MR 80k:10048

[21] D. Sullivan, Conformal Dynamical Systems, in Proc. Conf. on Geometric Dynamics, Rio de Janeiro, 1981, Springer LNM 1007, 725-752. | MR 85m:58112 | Zbl 0524.58024

[22] P. Walters, A variational principle for the pressure of continuous transformations, Am. J. Math. 97 (1975), 937-971. | MR 52 #11006 | Zbl 0318.28007