Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-24T14:42:06.196Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-archimedean connected Julia sets with branching

Published online by Cambridge University Press:  12 October 2015

DVIJ BAJPAI ,
ROBERT L. BENEDETTO ,
RUQIAN CHEN ,
EDWARD KIM ,
OWEN MARSCHALL ,
DARIUS ONUL  and
YANG XIAO
DVIJ BAJPAI
Affiliation:
Amherst College, Amherst, MA 01002, USA email [email protected], [email protected], [email protected], [email protected], [email protected]
ROBERT L. BENEDETTO
Affiliation:
Amherst College, Amherst, MA 01002, USA email [email protected], [email protected], [email protected], [email protected], [email protected]
RUQIAN CHEN
Affiliation:
University of Illinois, Urbana, IL 68101, USA email [email protected]
EDWARD KIM
Affiliation:
Amherst College, Amherst, MA 01002, USA email [email protected], [email protected], [email protected], [email protected], [email protected]
OWEN MARSCHALL
Affiliation:
Amherst College, Amherst, MA 01002, USA email [email protected], [email protected], [email protected], [email protected], [email protected]
DARIUS ONUL
Affiliation:
Amherst College, Amherst, MA 01002, USA email [email protected], [email protected], [email protected], [email protected], [email protected]
YANG XIAO
Affiliation:
Brown University, Providence, RI 02912, USA email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We construct the first examples of rational functions defined over a non-archimedean field with a certain dynamical property: the Julia set in the Berkovich projective line is connected but not contained in a line segment. We also show how to compute the measure-theoretic and topological entropy of such maps. In particular, we give an example for which the measure-theoretic entropy is strictly smaller than the topological entropy, thus answering a question of Favre and Rivera-Letelier.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 1 , February 2017 , pp. 59 - 78
Copyright
© Cambridge University Press, 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Baker, M. H. and Rumely, R. S.. Equidistribution of small points, rational dynamics, and potential theory. Ann. Inst. Fourier (Grenoble) 56 (2006), 625–688.Google Scholar
Baker, M. H. and Rumely, R. S.. Potential Theory and Dynamics on the Berkovich Projective Line. American Mathematical Society, Providence, RI, 2010.Google Scholar
Benedetto, R. L.. Non-archimedean dynamics in dimension one: Lecture notes, Arizona Winter School, 2010; available at http://swc.math.arizona.edu/aws/2010/2010BenedettoNotes-09Mar.pdf.Google Scholar
Benedetto, R. L.. A non-archimedean quartic map with connected Julia set and non-maximal entropy, in preparation.Google Scholar
Berkovich, V. G.. Spectral Theory and Analytic Geometry Over Non-archimedean Fields. American Mathematical Society, Providence, RI, 1990.Google Scholar
Boyle, M., Buzzi, J. and Gómez, R.. Almost isomorphism for countable state Markov shifts. J. Reine Angew. Math. 592 (2006), 23–47.Google Scholar
DeMarco, L. and Faber, X.. Degenerations of complex dynamical systems. Forum of Mathematics, Sigma. 2 (2014), e6, 36 pages.Google Scholar
DeMarco, L. and Faber, X.. Degenerations of complex dynamical systems II: analytic and algebraic stability. Preprint, 2013, arXiv:1309.7103.Google Scholar
Favre, C. and Jonsson, M.. The Valuative Tree (Lecture Notes in Mathematics, 1853) . Springer, Berlin, 2004.Google Scholar
Favre, C. and Rivera-Letelier, J.. Théorème d’équidistribution de Brolin en dynamique p-adique. C. R. Math. Acad. Sci. Paris 339 (2004), 271–276.Google Scholar
Favre, C. and Rivera-Letelier, J.. Théorie ergodique des fractions rationnelles sur un corps ultramétrique. Proc. Lond. Math. Soc. (3) 100 (2010), 116–154.Google Scholar
Favre, C. and Rivera-Letelier, J.. Expansion et entropie en dynamique non-archimédienne, in preparation, 2015.Google Scholar
Gurevich, B. M.. Topological entropy of enumerable Markov chains. Dokl. Akad. Nauk SSSR 187 (1969), 715–718 Engl. Transl. in Sov. Math. Dokl. 10 (1969), 911–915.Google Scholar
Gurevich, B. M.. Shift entropy and Markov measures in the math space of a denumerable graph. Dokl. Akad. Nauk SSSR 192 (1970), 963–965 Engl. Transl. in Sov. Math. Dokl. 11 (1970), 744–747.Google Scholar
Gurevich, B. M. and Savchenko, S. V.. Thermodynamic formalism for countable symbolic Markov shifts. Uspekhi Mat. Nauk 53(2) (1998), 3–106 Engl. Transl. in Russian Math. Surveys 53(2) (1998), 245–344.Google Scholar
Kitchens, B. P.. Symbolic Dynamics: One-sided, Two-sided, and Countable State Markov Shifts. Springer, Berlin, 1998.Google Scholar
Kiwi, J.. Puiseux series polynomial dynamics and iteration of complex cubic polynomials. Ann. Inst. Fourier (Grenoble) 56 (2006), 1337–1404.Google Scholar
Przytycki, F. and Urbański, M.. Conformal Fractals: Ergodic Theory Methods. Cambridge University Press, Cambridge, 2010.Google Scholar
Rivera-Letelier, J.. Dynamique des fonctions rationnelles sur des corps locaux. Astérisque 287 (2003), 147–230.Google Scholar
Rivera-Letelier, J.. Espace hyperbolique p-adique et dynamique des fonctions rationnelles. Compos. Math. 138 (2003), 199–231.Google Scholar
Rivera-Letelier, J.. Points périodiques des fonctions rationnelles dans l’espace hyperbolique p-adique. Comment. Math. Helv. 80 (2005), 593–629.Google Scholar
Royden, H. L.. Real Analysis, 3rd edn. Macmillan, New York, 1988.Google Scholar
Ruette, S.. On the Vere-Jones classification and existence of maximal measures for countable topological Markov chains. Pacific J. Math. 209 (2003), 365–380.Google Scholar
Sarig, O. M.. Thermodynamic formalism for countable Markov shifts. Ergod. Th. & Dynam. Sys. 19 (1999), 1565–1593.Google Scholar
Vere-Jones, D.. Geometric ergodicity in denumerable Markov chains. Q. J. Math. Oxf. 2 13 (1962), 7–28.Google Scholar
Walters, P.. An Introduction to Ergodic Theory. Springer, New York, 1982.Google Scholar