Metric diophantine approximation in Julia sets of expanding rational maps

Richard Hill; Sanju L. Velani

Publications Mathématiques de l'IHÉS (1997)

  • Volume: 85, page 193-216
  • ISSN: 0073-8301

How to cite

top

Hill, Richard, and Velani, Sanju L.. "Metric diophantine approximation in Julia sets of expanding rational maps." Publications Mathématiques de l'IHÉS 85 (1997): 193-216. <http://eudml.org/doc/104119>.

@article{Hill1997,
author = {Hill, Richard, Velani, Sanju L.},
journal = {Publications Mathématiques de l'IHÉS},
keywords = {metric diophantine approximation; well-approximable point; backward orbit; Hausdorff dimension; expanding rational map; Riemann sphere; Julia set; Hölder function; conformal measures},
language = {eng},
pages = {193-216},
publisher = {Institut des Hautes Études Scientifiques},
title = {Metric diophantine approximation in Julia sets of expanding rational maps},
url = {http://eudml.org/doc/104119},
volume = {85},
year = {1997},
}

TY - JOUR
AU - Hill, Richard
AU - Velani, Sanju L.
TI - Metric diophantine approximation in Julia sets of expanding rational maps
JO - Publications Mathématiques de l'IHÉS
PY - 1997
PB - Institut des Hautes Études Scientifiques
VL - 85
SP - 193
EP - 216
LA - eng
KW - metric diophantine approximation; well-approximable point; backward orbit; Hausdorff dimension; expanding rational map; Riemann sphere; Julia set; Hölder function; conformal measures
UR - http://eudml.org/doc/104119
ER -

References

top
  1. [1] A. F. BEARDON, Iteration of Rational Functions, Springer GTM 132, 1991. Zbl0742.30002MR92j:30026
  2. [2] A. S. BESICOVITCH, Sets of fractional dimension (IV): On rational approximation to real numbers, J. London Math. Soc. 9 (1934), 126-131. Zbl0009.05301JFM60.0204.01
  3. [3] R. BOWEN, Equilibrium States and the Ergodic Theory for Anosov Diffeomorphisms, III, Springer LNM 470, 1975. Zbl0308.28010MR56 #1364
  4. [4] H. BROLIN, Invariant sets under iteration of rational functions, Ark. Mat. 6 (1965), 103-144. Zbl0127.03401MR33 #2805
  5. [5] J. W. S. CASSELS, An Introduction to Diophantine Approximation, Cambridge Univ. Press, 1957. Zbl0077.04801MR19,396h
  6. [6] M. DENKER, Ch. GRILLENBERGER and K. SIGMUND, Ergodic Theory on Compact Spaces, IV, Springer LNM 527, 1976. Zbl0328.28008MR56 #15879
  7. [7] M. DENKER and M. URBAŃSKI, Ergodic theory of equilibrium states of rational maps, Nonlinearity 4 (1991), 103-134. Zbl0718.58035MR92a:58112
  8. [8] K. J. FALCONER, Fractal Geometry - Mathematical Foundations and Applications, J. Wiley, Chichester, 1990. Zbl0689.28003
  9. [9] R. HILL and S. L. VELANI, The Ergodic Theory of Shrinking Targets, Invent. Math. 119 (1995), 175-198. Zbl0834.28009MR96e:58088
  10. [10] R. HILL and S. L. VELANI, Markov maps and moving, shrinking targets, in preparation. Zbl0834.28009
  11. [11] R. HILL and S. L. VELANI, The Shrinking Target Problem for Matrix Transformations of Tori, J. Lond. Math. Soc. (to appear). Zbl0987.37008
  12. [12] R. HILL and S. L. VELANI, The Jarník-Besicovitch theorem for geometrically finite Kleinian groups, Proc. Lond. Math. Soc. (to appear). Zbl0924.11063
  13. [13] E. HILLE, Analytic function theory, Ginn and Company: Boston - New York - Chicago - Atlanta - Dallas - Palo Alto - Toronto, 1962. Zbl0102.29401MR34 #1490
  14. [14] V. JARNÍK, Diophantische Approximationen und Hausdorffsches Mass, Math. Sb. 36 (1929), 371-382. Zbl55.0719.01JFM55.0719.01
  15. [15] V. LJUBICH, Entropy properties of rational endomorphisms of the Riemann sphere, Ergod. Th. Dynam. Sys. 3 (1983), 351-386. Zbl0537.58035MR85k:58049
  16. [16] M. V. MELIAN and S. L. VELANI, Geodesic excursions into cusps in infinite volume hyperbolic manifolds, Mathematica Gottingensis 45, 1993. Zbl0793.53052
  17. [17] W. PARRY and M. POLLICOTT, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, Soc. Math. France, 187-188, 1990. Zbl0726.58003MR92f:58141
  18. [18] S. J. PATTERSON, The limit set of a Fuchsian group, Acta Math. 136 (1976), 241-273. Zbl0336.30005MR56 #8841
  19. [19] D. RUELLE, Thermodynamical Formalism, Encycl. Math. Appl. 5, 1978. Zbl0401.28016MR80g:82017
  20. [20] V. G. SPRINDŽUK, Metric theory of Diophantine approximation (translated by R. A. SILVERMAN), V. H. Winston & Sons, Washington D.C., 1979. Zbl0306.10037MR80k:10048
  21. [21] D. SULLIVAN, Conformal Dynamical Systems, in Proc. Conf. on Geometric Dynamics, Rio de Janeiro, 1981, Springer LNM 1007, 725-752. Zbl0524.58024MR85m:58112
  22. [22] P. WALTERS, A variational principle for the pressure of continuous transformations, Am. J. Math. 97 (1975), 937-971. Zbl0318.28007MR52 #11006

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.