# American Institute of Mathematical Sciences

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), 487-492. [2] R. Adler and L. Flatto, Geodesic flows, interval maps, and symbolic dynamics, Bull. Amer. Math. Soc., 25 (1991), 229-334. doi: 10.1090/S0273-0979-1991-16076-3. [3] E. Artin, Ein mechanisches system mit quasiergodischen Bahnen, Abh. Math. Sem. Univ. Hamburg, 3 (1924), 170-175. doi: 10.1007/BF02954622. [4] J. Bourdon, B. Daireaux and B. Vallée, Dynamical analysis of $\alpha$-Euclidean algorithms, J. Algorithms, 44 (2002), 246-285. doi: 10.1016/S0196-6774(02)00218-3. [5] C. Carminati and G.Tiozzo, A canonical thickening of $\Q$ and the dynamics of continued fractions,, preprint \arXiv{1004.3790v1}., (). [6] G. H. Hardy and E. M. Wright, "An Introduction to the Theory of Numbers," sixth edition, Revised by D. R. Heath-Brown and J. H. Silverman, Oxford University Press, Oxford, 2008. [7] A. Hurwitz, Über eine besondere Art der Kettenbruch-Entwicklung reeler Grössen, (German), Acta Math., 12 (1889), 367-405. doi: 10.1007/BF02592188. [8] S. Katok, "Fuchsian Groups," Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1992. [9] S. Katok, Coding of closed geodesics after Gauss and Morse, Geom. Dedicata, 63 (1996), 123-145. doi: 10.1007/BF00148213. [10] S. Katok and I. Ugarcovici, Arithmetic coding of geodesics on the modular surface via continued fractions, European women in mathematics-Marseille 2003, 59-77, CWI Tract, 135, Centrum Wisk. Inform., Amsterdam, (2005). [11] S. Katok, I. Ugarcovici, Geometrically Markov geodesics on the modular surface, Moscow Math. J. 5 (2005), 135-151. [12] S. Katok and I. Ugarcovici, Symbolic dynamics for the modular surface and beyond, Bull. Amer. Math. Soc., 44 (2007), 87-132. doi: 10.1090/S0273-0979-06-01115-3. [13] S. Katok and I. Ugarcovici, Theory of $(a,b)$-continued fraction transformations and applications, Electron. Res. Announc. Math. Sci., 17 (2010), 20-33. doi: 10.3934/era.2010.17.20. [14] S. Katok and I. Ugarcovici, Applications of $(a,b)$-continued fraction transformations,, in preparation., (). [15] C. Kraaikamp, A new class of continued fraction expansions, Acta Arith., 57 (1991), 1-39. [16] L. Luzzi and S. Marmi, On the entropy of Japanese continued fractions, Discrete Cont. Dyn. Syst., 20 (2008), 673-711. [17] P. Moussa, A. Cassa and S. Marmi, Continued fractions and Brjuno functions, Continued fractions and geometric function theory (CONFUN) (Trondheim, 1997), J. Comput. Appl. Math., 105 (1999), 403-415. doi: 10.1016/S0377-0427(99)00029-1. [18] H. Nakada, Metrical theory for a class of continued fraction transformations and their natural extensions, Tokyo J. Math., 4 (1981), 399-426. doi: 10.3836/tjm/1270215165. [19] H. Nakada and R. Natsui, Some metric properties of $\alpha$-continued fractions, Journal of Number Theory, 97 (2002), 287-300. doi: 10.1016/S0022-314X(02)00008-2. [20] H. Nakada and R. Natsui, The non-monotonicity of the entropy of $\alpha$-continued fraction transformations, Nonlinearity, 21 (2008) 1207-1225. doi: 10.1088/0951-7715/21/6/003. [21] C. Series, On coding geodesics with continued fractions, Ergodic theory (Sem., Les Plans-sur-Bex, 1980) (French), 67-76, Monograph. Enseign. Math., 29, Univ. Genéve, Geneva, (1981). [22] F. Schweiger, "Ergodic Theory of Fibred Systems and Metric Number Theory," Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995. [23] D. Zagier, "Zetafunkionen und Quadratische Körper: Eine Einführung in die Höhere Zahlentheorie," Springer-Verlag, 1981. [24] R. Zweimüller, Ergodic structure and invariant densities of non-Markovian interval maps with indifferent fixed points, Nonlinearity, 11 (1998), 1263-1276. doi: 10.1088/0951-7715/11/5/005.

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##### References:
 [1] R. Adler and L. Flatto, The backward continued fraction map and geodesic flow, Ergod. Th. & Dynam. Sys., 4 (1984), 487-492. [2] R. Adler and L. Flatto, Geodesic flows, interval maps, and symbolic dynamics, Bull. Amer. Math. Soc., 25 (1991), 229-334. doi: 10.1090/S0273-0979-1991-16076-3. [3] E. Artin, Ein mechanisches system mit quasiergodischen Bahnen, Abh. Math. Sem. Univ. Hamburg, 3 (1924), 170-175. doi: 10.1007/BF02954622. [4] J. Bourdon, B. Daireaux and B. Vallée, Dynamical analysis of $\alpha$-Euclidean algorithms, J. Algorithms, 44 (2002), 246-285. doi: 10.1016/S0196-6774(02)00218-3. [5] C. Carminati and G.Tiozzo, A canonical thickening of $\Q$ and the dynamics of continued fractions,, preprint \arXiv{1004.3790v1}., (). [6] G. H. Hardy and E. M. Wright, "An Introduction to the Theory of Numbers," sixth edition, Revised by D. R. Heath-Brown and J. H. Silverman, Oxford University Press, Oxford, 2008. [7] A. Hurwitz, Über eine besondere Art der Kettenbruch-Entwicklung reeler Grössen, (German), Acta Math., 12 (1889), 367-405. doi: 10.1007/BF02592188. [8] S. Katok, "Fuchsian Groups," Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1992. [9] S. Katok, Coding of closed geodesics after Gauss and Morse, Geom. Dedicata, 63 (1996), 123-145. doi: 10.1007/BF00148213. [10] S. Katok and I. Ugarcovici, Arithmetic coding of geodesics on the modular surface via continued fractions, European women in mathematics-Marseille 2003, 59-77, CWI Tract, 135, Centrum Wisk. Inform., Amsterdam, (2005). [11] S. Katok, I. Ugarcovici, Geometrically Markov geodesics on the modular surface, Moscow Math. J. 5 (2005), 135-151. [12] S. Katok and I. Ugarcovici, Symbolic dynamics for the modular surface and beyond, Bull. Amer. Math. Soc., 44 (2007), 87-132. doi: 10.1090/S0273-0979-06-01115-3. [13] S. Katok and I. Ugarcovici, Theory of $(a,b)$-continued fraction transformations and applications, Electron. Res. Announc. Math. Sci., 17 (2010), 20-33. doi: 10.3934/era.2010.17.20. [14] S. Katok and I. Ugarcovici, Applications of $(a,b)$-continued fraction transformations,, in preparation., (). [15] C. Kraaikamp, A new class of continued fraction expansions, Acta Arith., 57 (1991), 1-39. [16] L. Luzzi and S. Marmi, On the entropy of Japanese continued fractions, Discrete Cont. Dyn. Syst., 20 (2008), 673-711. [17] P. Moussa, A. Cassa and S. Marmi, Continued fractions and Brjuno functions, Continued fractions and geometric function theory (CONFUN) (Trondheim, 1997), J. Comput. Appl. Math., 105 (1999), 403-415. doi: 10.1016/S0377-0427(99)00029-1. [18] H. Nakada, Metrical theory for a class of continued fraction transformations and their natural extensions, Tokyo J. Math., 4 (1981), 399-426. doi: 10.3836/tjm/1270215165. [19] H. Nakada and R. Natsui, Some metric properties of $\alpha$-continued fractions, Journal of Number Theory, 97 (2002), 287-300. doi: 10.1016/S0022-314X(02)00008-2. [20] H. Nakada and R. Natsui, The non-monotonicity of the entropy of $\alpha$-continued fraction transformations, Nonlinearity, 21 (2008) 1207-1225. doi: 10.1088/0951-7715/21/6/003. [21] C. Series, On coding geodesics with continued fractions, Ergodic theory (Sem., Les Plans-sur-Bex, 1980) (French), 67-76, Monograph. Enseign. Math., 29, Univ. Genéve, Geneva, (1981). [22] F. Schweiger, "Ergodic Theory of Fibred Systems and Metric Number Theory," Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995. [23] D. Zagier, "Zetafunkionen und Quadratische Körper: Eine Einführung in die Höhere Zahlentheorie," Springer-Verlag, 1981. [24] R. Zweimüller, Ergodic structure and invariant densities of non-Markovian interval maps with indifferent fixed points, Nonlinearity, 11 (1998), 1263-1276. doi: 10.1088/0951-7715/11/5/005.
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