Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-26T13:28:08.743Z Has data issue: false hasContentIssue false
  • English
  • Français

Applications of (a,b)-continued fraction transformations

Published online by Cambridge University Press:  19 September 2011

SVETLANA KATOK  and
ILIE UGARCOVICI
SVETLANA KATOK
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA (email: [email protected])
ILIE UGARCOVICI
Affiliation:
Department of Mathematical Sciences, DePaul University, Chicago, IL 60614, USA (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We describe a general method of arithmetic coding of geodesics on the modular surface based on the study of one-dimensional Gauss-like maps associated to a two-parameter family of continued fractions introduced in [Katok and Ugarcovici. Structure of attractors for (a,b)-continued fraction transformations. J. Modern Dynamics4 (2010), 637–691]. The finite rectangular structure of the attractors of the natural extension maps and the corresponding ‘reduction theory’ play an essential role. In special cases, when an (a,b)-expansion admits a so-called ‘dual’, the coding sequences are obtained by juxtaposition of the boundary expansions of the fixed points, and the set of coding sequences is a countable sofic shift. We also prove that the natural extension maps are Bernoulli shifts and compute the density of the absolutely continuous invariant measure and the measure-theoretic entropy of the one-dimensional map.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 2: Daniel J. Rudolph – in Memoriam , April 2012 , pp. 739 - 761
Copyright
Copyright © Cambridge University Press 2011

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

[1]Abramov, L. M.. On the entropy of a flow. Sov. Math. Dokl. 128(5) (1959), 873–875.Google Scholar
[2]Adler, R.. Symbolic dynamics and Markov partitions. Bull. Amer. Math. Soc. 35(1) (1998), 1–56.CrossRefGoogle Scholar
[3]Adler, R. and Flatto, L.. Cross Section Maps for Geodesic Flows, I (The Modular Surface) (Progress in Mathematics, 21). Ed. Katok, A.. Birkhäuser, Boston, MA, 1982, pp. 103–161.Google Scholar
[4]Adler, R. and Flatto, L.. Geodesic flows, interval maps, and symbolic dynamics. Bull. Amer. Math. Soc. 25(2) (1991), 229–334.CrossRefGoogle Scholar
[5]Ambrose, W. and Kakutani, S.. Structure and continuity of measurable flows. Duke Math. J. 9 (1942), 25–42.CrossRefGoogle Scholar
[6]Artin, E.. Ein mechanisches System mit quasiergodischen Bahnen. Abh. Math. Semin. Univ. Hambg. 3 (1924), 170–175.CrossRefGoogle Scholar
[7]Bowen, R. and Series, C.. Markov maps associated with Fuchsian groups. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 153–170.CrossRefGoogle Scholar
[8]Gurevich, B. and Katok, S.. Arithmetic coding and entropy for the positive geodesic flow on the modular surface. Moscow Math. J. 1(4) (2001), 569–582.CrossRefGoogle Scholar
[9]Hurwitz, A.. Über eine besondere Art der Kettenbruch-Entwicklung reeler Grossen. Acta Math. 12 (1889), 367–405.CrossRefGoogle Scholar
[10]Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
[11]Katok, S.. Fuchsian Groups. University of Chicago Press, Chicago, 1992.Google Scholar
[12]Katok, S.. Coding of closed geodesics after Gauss and Morse. Geom. Dedicata 63 (1996), 123–145.CrossRefGoogle Scholar
[13]Katok, S. and Ugarcovici, I.. Geometrically Markov geodesics on the modular surface. Moscow Math. J. 5 (2005), 135–151.CrossRefGoogle Scholar
[14]Katok, S. and Ugarcovici, I.. Arithmetic Coding of Geodesics on the Modular Surface via Continued Fractions (CWI Tract 135). Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 2005, pp. 59–77.Google Scholar
[15]Katok, S. and Ugarcovici, I.. Symbolic dynamics for the modular surface and beyond. Bull. Amer. Math. Soc. 44 (2007), 87–132.CrossRefGoogle Scholar
[16]Katok, S. and Ugarcovici, I.. Structure of attractors for (a,b)-continued fraction transformations. J. Mod. Dyn. 4 (2010), 637–691.CrossRefGoogle Scholar
[17]Ornstein, D.. Ergodic Theory, Randomness, and Dynamical Systems. Yale University Press, New Haven, CT, 1973.Google Scholar
[18]Rychlik, M.. Bounded variation and invariant measures. Studia Math. 76 (1983), 69–80.CrossRefGoogle Scholar
[19]Series, C.. On coding geodesics with continued fractions. Enseign. Math. 29 (1980), 67–76.Google Scholar
[20]Sullivan, D.. Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. Acta Math. 153 (1984), 259–277.CrossRefGoogle Scholar
[21]Walters, P.. Invariant measures and equilibrium states for some mappings which expand distances. Trans. Amer. Math. Soc. 236 (1978), 121–153.CrossRefGoogle Scholar
[22]Zagier, D.. Zetafunkionen und quadratische Körper: eine Einführung in die höhere Zahlentheorie. Springer, Berlin–New York, 1982.Google Scholar
[23]Zweimüller, R.. Ergodic structure and invariant densities of non-Markovian interval maps with indifferent fixed points. Nonlinearity 11 (1998), 1263–1276.CrossRefGoogle Scholar