On fixed points of C 1 extensions of expanding maps in the unit disc

Roberto Tauraso

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1994)

  • Volume: 5, Issue: 4, page 303-308
  • ISSN: 1120-6330

Abstract

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Using a result due to M. Shub, a theorem about the existence of fixed points inside the unit disc for C 1 extensions of expanding maps defined on the boundary is established. An application to a special class of rational maps on the Riemann sphere and some considerations on ergodic properties of these maps are also made.

How to cite

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Tauraso, Roberto. "On fixed points of \( C^{1} \) extensions of expanding maps in the unit disc." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 5.4 (1994): 303-308. <http://eudml.org/doc/244274>.

@article{Tauraso1994,
abstract = {Using a result due to M. Shub, a theorem about the existence of fixed points inside the unit disc for \( C^\{1\} \) extensions of expanding maps defined on the boundary is established. An application to a special class of rational maps on the Riemann sphere and some considerations on ergodic properties of these maps are also made.},
author = {Tauraso, Roberto},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Fixed point; Expanding map; Blaschke product; Invariant measure; expending map; map; Julia set; Futou set; invariant measure; index of the map; fixed points},
language = {eng},
month = {12},
number = {4},
pages = {303-308},
publisher = {Accademia Nazionale dei Lincei},
title = {On fixed points of \( C^\{1\} \) extensions of expanding maps in the unit disc},
url = {http://eudml.org/doc/244274},
volume = {5},
year = {1994},
}

TY - JOUR
AU - Tauraso, Roberto
TI - On fixed points of \( C^{1} \) extensions of expanding maps in the unit disc
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1994/12//
PB - Accademia Nazionale dei Lincei
VL - 5
IS - 4
SP - 303
EP - 308
AB - Using a result due to M. Shub, a theorem about the existence of fixed points inside the unit disc for \( C^{1} \) extensions of expanding maps defined on the boundary is established. An application to a special class of rational maps on the Riemann sphere and some considerations on ergodic properties of these maps are also made.
LA - eng
KW - Fixed point; Expanding map; Blaschke product; Invariant measure; expending map; map; Julia set; Futou set; invariant measure; index of the map; fixed points
UR - http://eudml.org/doc/244274
ER -

References

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  1. ABATE, M., Iteration Theory of Holomorphic Maps on Taut Manifolds. Mediterranean Press, Commenda di Rende1989. Zbl0747.32002MR1098711
  2. BROLIN, H., Invariant sets under iteration of rational function. Ark. Mat., vol. 6, 1965, 103-144. Zbl0127.03401MR194595
  3. BROWN, R. F., The Lefschetz Fixed Point Theorem. Scott, Foresman and Co., Glenview1971. Zbl0216.19601MR283793
  4. BROWN, R. F. - GREENE, R. E., An interior fixed point property of the disc. Amer. Math. Monthly, vol. 101, 1994, 39-47. Zbl0813.54030MR1252704DOI10.2307/2325123
  5. BROWN, R. F. - GREENE, R. E. - SCHIRMER, H., Fixed points of map extension. In: Topological Fixed Point Theory and Applications. Prooceedings (Tianjin, 1988). Lecture Notes in Mathematics, vol. 1411, Springer-Verlag, Berlin1989. Zbl0688.55002MR1031780DOI10.1007/BFb0086438
  6. BURCKEL, R. B., Iterating analytic self-maps of discs. Amer. Math. Monthly, vol. 88, 1981, 387-460. Zbl0466.30001MR622955DOI10.2307/2321822
  7. GAMELIN, T. W. - GREENE, R. E., Introduction to Topology. Saunders College Publ., Philadelphia1983. Zbl0625.54002MR696379
  8. MARTIN, N. F. G., On Finite Blaschke Products whose restriction to the unit circle are exact endomorphisms. Bull. London Math. Soc., vol. 15, 1983, 343-348. Zbl0487.28017MR703758DOI10.1112/blms/15.4.343
  9. NITECKI, Z., Differentiable Dynamics. The M.I.T. Press, Cambridge, Mass.1971. Zbl0246.58012MR649788
  10. POMMERENKE, C., Boundary behaviour of Conformal Maps. Springer-Verlag, New York1992. Zbl0762.30001MR1217706
  11. RUDIN, W., Real and Complex Analysis. McGraw-Hill, New York1966. Zbl0925.00005MR210528
  12. SHUB, M., Endomorphisms of compact differentiable manifolds. Amer. J. Math., vol. 91, 1969, 175-199. Zbl0201.56305MR240824
  13. STEINMETZ, N., Rational Iteration. De Gruyter, Berlin1993. Zbl0773.58010MR1224235DOI10.1515/9783110889314
  14. WALTERS, P., Invariant measures and equilibrium states for some mappings which expand distances. Trans. Amer. Math. Soc., vol. 236, 1978, 121-153. Zbl0375.28009MR466493
  15. WALTERS, P., An Introduction to Ergodic Theory. Springer-Verlag, New York1982. Zbl0299.28012MR648108

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