October  2010, 4(4): 637-691. doi: 10.3934/jmd.2010.4.637

Structure of attractors for $(a,b)$-continued fraction transformations

1. 

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802

2. 

Department of Mathematical Sciences, DePaul University, 2320 N. Kenmore Ave., Chicago, IL 60614-3504

Received  March 2010 Revised  September 2010 Published  January 2011

We study a two-parameter family of one-dimensional maps and related $(a,b)$-continued fractions suggested for consideration by Don Zagier. We prove that the associated natural extension maps have attractors with finite rectangular structure for the entire parameter set except for a Cantor-like set of one-dimensional Lebesgue zero measure that we completely describe. We show that the structure of these attractors can be "computed'' from the data $(a,b)$, and that for a dense open set of parameters the Reduction theory conjecture holds, i.e., every point is mapped to the attractor after finitely many iterations. We also show how this theory can be applied to the study of invariant measures and ergodic properties of the associated Gauss-like maps.
Citation: Svetlana Katok, Ilie Ugarcovici. Structure of attractors for $(a,b)$-continued fraction transformations. Journal of Modern Dynamics, 2010, 4 (4) : 637-691. doi: 10.3934/jmd.2010.4.637
References:
[1]

R. Adler and L. Flatto, The backward continued fraction map and geodesic flow,, Ergod. Th. & Dynam. Sys., 4 (1984), 487.   Google Scholar

[2]

R. Adler and L. Flatto, Geodesic flows, interval maps, and symbolic dynamics,, Bull. Amer. Math. Soc., 25 (1991), 229.  doi: 10.1090/S0273-0979-1991-16076-3.  Google Scholar

[3]

E. Artin, Ein mechanisches system mit quasiergodischen Bahnen,, Abh. Math. Sem. Univ. Hamburg, 3 (1924), 170.  doi: 10.1007/BF02954622.  Google Scholar

[4]

J. Bourdon, B. Daireaux and B. Vallée, Dynamical analysis of $\alpha$-Euclidean algorithms,, J. Algorithms, 44 (2002), 246.  doi: 10.1016/S0196-6774(02)00218-3.  Google Scholar

[5]

C. Carminati and G.Tiozzo, A canonical thickening of $\Q$ and the dynamics of continued fractions,, preprint \arXiv{1004.3790v1}., ().   Google Scholar

[6]

G. H. Hardy and E. M. Wright, "An Introduction to the Theory of Numbers,", sixth edition, (2008).   Google Scholar

[7]

A. Hurwitz, Über eine besondere Art der Kettenbruch-Entwicklung reeler Grössen,, (German), 12 (1889), 367.  doi: 10.1007/BF02592188.  Google Scholar

[8]

S. Katok, "Fuchsian Groups,", Chicago Lectures in Mathematics, (1992).   Google Scholar

[9]

S. Katok, Coding of closed geodesics after Gauss and Morse,, Geom. Dedicata, 63 (1996), 123.  doi: 10.1007/BF00148213.  Google Scholar

[10]

S. Katok and I. Ugarcovici, Arithmetic coding of geodesics on the modular surface via continued fractions,, European women in mathematics-Marseille 2003, (2005), 59.   Google Scholar

[11]

S. Katok, I. Ugarcovici, Geometrically Markov geodesics on the modular surface,, Moscow Math. J. \textbf{5} (2005), 5 (2005), 135.   Google Scholar

[12]

S. Katok and I. Ugarcovici, Symbolic dynamics for the modular surface and beyond,, Bull. Amer. Math. Soc., 44 (2007), 87.  doi: 10.1090/S0273-0979-06-01115-3.  Google Scholar

[13]

S. Katok and I. Ugarcovici, Theory of $(a,b)$-continued fraction transformations and applications,, Electron. Res. Announc. Math. Sci., 17 (2010), 20.  doi: 10.3934/era.2010.17.20.  Google Scholar

[14]

S. Katok and I. Ugarcovici, Applications of $(a,b)$-continued fraction transformations,, in preparation., ().   Google Scholar

[15]

C. Kraaikamp, A new class of continued fraction expansions,, Acta Arith., 57 (1991), 1.   Google Scholar

[16]

L. Luzzi and S. Marmi, On the entropy of Japanese continued fractions,, Discrete Cont. Dyn. Syst., 20 (2008), 673.   Google Scholar

[17]

P. Moussa, A. Cassa and S. Marmi, Continued fractions and Brjuno functions,, Continued fractions and geometric function theory (CONFUN) (Trondheim, 105 (1999), 403.  doi: 10.1016/S0377-0427(99)00029-1.  Google Scholar

[18]

H. Nakada, Metrical theory for a class of continued fraction transformations and their natural extensions,, Tokyo J. Math., 4 (1981), 399.  doi: 10.3836/tjm/1270215165.  Google Scholar

[19]

H. Nakada and R. Natsui, Some metric properties of $\alpha$-continued fractions,, Journal of Number Theory, 97 (2002), 287.  doi: 10.1016/S0022-314X(02)00008-2.  Google Scholar

[20]

H. Nakada and R. Natsui, The non-monotonicity of the entropy of $\alpha$-continued fraction transformations,, Nonlinearity, 21 (2008), 1207.  doi: 10.1088/0951-7715/21/6/003.  Google Scholar

[21]

C. Series, On coding geodesics with continued fractions,, Ergodic theory (Sem., (1981), 67.   Google Scholar

[22]

F. Schweiger, "Ergodic Theory of Fibred Systems and Metric Number Theory,", Oxford Science Publications, (1995).   Google Scholar

[23]

D. Zagier, "Zetafunkionen und Quadratische Körper: Eine Einführung in die Höhere Zahlentheorie,", Springer-Verlag, (1981).   Google Scholar

[24]

R. Zweimüller, Ergodic structure and invariant densities of non-Markovian interval maps with indifferent fixed points,, Nonlinearity, 11 (1998), 1263.  doi: 10.1088/0951-7715/11/5/005.  Google Scholar

show all references

References:
[1]

R. Adler and L. Flatto, The backward continued fraction map and geodesic flow,, Ergod. Th. & Dynam. Sys., 4 (1984), 487.   Google Scholar

[2]

R. Adler and L. Flatto, Geodesic flows, interval maps, and symbolic dynamics,, Bull. Amer. Math. Soc., 25 (1991), 229.  doi: 10.1090/S0273-0979-1991-16076-3.  Google Scholar

[3]

E. Artin, Ein mechanisches system mit quasiergodischen Bahnen,, Abh. Math. Sem. Univ. Hamburg, 3 (1924), 170.  doi: 10.1007/BF02954622.  Google Scholar

[4]

J. Bourdon, B. Daireaux and B. Vallée, Dynamical analysis of $\alpha$-Euclidean algorithms,, J. Algorithms, 44 (2002), 246.  doi: 10.1016/S0196-6774(02)00218-3.  Google Scholar

[5]

C. Carminati and G.Tiozzo, A canonical thickening of $\Q$ and the dynamics of continued fractions,, preprint \arXiv{1004.3790v1}., ().   Google Scholar

[6]

G. H. Hardy and E. M. Wright, "An Introduction to the Theory of Numbers,", sixth edition, (2008).   Google Scholar

[7]

A. Hurwitz, Über eine besondere Art der Kettenbruch-Entwicklung reeler Grössen,, (German), 12 (1889), 367.  doi: 10.1007/BF02592188.  Google Scholar

[8]

S. Katok, "Fuchsian Groups,", Chicago Lectures in Mathematics, (1992).   Google Scholar

[9]

S. Katok, Coding of closed geodesics after Gauss and Morse,, Geom. Dedicata, 63 (1996), 123.  doi: 10.1007/BF00148213.  Google Scholar

[10]

S. Katok and I. Ugarcovici, Arithmetic coding of geodesics on the modular surface via continued fractions,, European women in mathematics-Marseille 2003, (2005), 59.   Google Scholar

[11]

S. Katok, I. Ugarcovici, Geometrically Markov geodesics on the modular surface,, Moscow Math. J. \textbf{5} (2005), 5 (2005), 135.   Google Scholar

[12]

S. Katok and I. Ugarcovici, Symbolic dynamics for the modular surface and beyond,, Bull. Amer. Math. Soc., 44 (2007), 87.  doi: 10.1090/S0273-0979-06-01115-3.  Google Scholar

[13]

S. Katok and I. Ugarcovici, Theory of $(a,b)$-continued fraction transformations and applications,, Electron. Res. Announc. Math. Sci., 17 (2010), 20.  doi: 10.3934/era.2010.17.20.  Google Scholar

[14]

S. Katok and I. Ugarcovici, Applications of $(a,b)$-continued fraction transformations,, in preparation., ().   Google Scholar

[15]

C. Kraaikamp, A new class of continued fraction expansions,, Acta Arith., 57 (1991), 1.   Google Scholar

[16]

L. Luzzi and S. Marmi, On the entropy of Japanese continued fractions,, Discrete Cont. Dyn. Syst., 20 (2008), 673.   Google Scholar

[17]

P. Moussa, A. Cassa and S. Marmi, Continued fractions and Brjuno functions,, Continued fractions and geometric function theory (CONFUN) (Trondheim, 105 (1999), 403.  doi: 10.1016/S0377-0427(99)00029-1.  Google Scholar

[18]

H. Nakada, Metrical theory for a class of continued fraction transformations and their natural extensions,, Tokyo J. Math., 4 (1981), 399.  doi: 10.3836/tjm/1270215165.  Google Scholar

[19]

H. Nakada and R. Natsui, Some metric properties of $\alpha$-continued fractions,, Journal of Number Theory, 97 (2002), 287.  doi: 10.1016/S0022-314X(02)00008-2.  Google Scholar

[20]

H. Nakada and R. Natsui, The non-monotonicity of the entropy of $\alpha$-continued fraction transformations,, Nonlinearity, 21 (2008), 1207.  doi: 10.1088/0951-7715/21/6/003.  Google Scholar

[21]

C. Series, On coding geodesics with continued fractions,, Ergodic theory (Sem., (1981), 67.   Google Scholar

[22]

F. Schweiger, "Ergodic Theory of Fibred Systems and Metric Number Theory,", Oxford Science Publications, (1995).   Google Scholar

[23]

D. Zagier, "Zetafunkionen und Quadratische Körper: Eine Einführung in die Höhere Zahlentheorie,", Springer-Verlag, (1981).   Google Scholar

[24]

R. Zweimüller, Ergodic structure and invariant densities of non-Markovian interval maps with indifferent fixed points,, Nonlinearity, 11 (1998), 1263.  doi: 10.1088/0951-7715/11/5/005.  Google Scholar

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