# American Institute of Mathematical Sciences

April  2013, 33(4): 1477-1498. doi: 10.3934/dcds.2013.33.1477

## Dynamics of $\lambda$-continued fractions and $\beta$-shifts

 1 Laboratoire de Mathématiques Raphaël Salem, Université de Rouen, CNRS, Avenue de l’Université, 76801 Saint Étienne du Rouvray, France 2 Laboratoire Analyse, Géométrie et Applications, Université Paris 13 Institut Galilée, CNRS, 99 avenue Jean-Baptiste Clément, 93430 Villetaneuse, France 3 Laboratoire de Mathématiques Raphaël Salem, Université de Rouen, CNRS, Avenue de l'Université, Avenue de l'Université, 76801 Saint Étienne du Rouvray, France

Received  October 2011 Revised  May 2012 Published  October 2012

For a real number $0<\lambda<2$, we introduce a transformation $T_\lambda$ naturally associated to expansion in $\lambda$-continued fraction, for which we also give a geometrical interpretation. The symbolic coding of the orbits of $T_\lambda$ provides an algorithm to expand any positive real number in $\lambda$-continued fraction. We prove the conjugacy between $T_\lambda$ and some $\beta$-shift, $\beta>1$. Some properties of the map $\lambda\mapsto\beta(\lambda)$ are established: It is increasing and continuous from $]0, 2[$ onto $]1,\infty[$ but non-analytic.
Citation: Élise Janvresse, Benoît Rittaud, Thierry de la Rue. Dynamics of $\lambda$-continued fractions and $\beta$-shifts. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1477-1498. doi: 10.3934/dcds.2013.33.1477
##### References:
 [1] F. Blanchard, $\beta$-expansions and symbolic dynamics, Theoret. Comput. Sci., 65 (1989), 131-141.  Google Scholar [2] Karma Dajani and Martijn de Vries, Measures of maximal entropy for random $\beta$-expansions, J. Eur. Math. Soc. (JEMS), 7 (2005), 51-68.  Google Scholar [3] Karma Dajani and Martijn de Vries, Invariant densities for random $\beta$-expansions, J. Eur. Math. Soc. (JEMS), 9 (2007), 157-176.  Google Scholar [4] Shunji Ito and Yōichirō Takahashi, Markov subshifts and realization of $\beta$-expansions, J. Math. Soc. Japan, 26 (1974), 33-55. doi: 10.2969/jmsj/02610033.  Google Scholar [5] Élise Janvresse, Benoît Rittaud and Thierry de la Rue, How do random Fibonacci sequences grow?, Prob. Th. Rel. Fields, 142 (2008), 619-648. doi: 10.1007/s00440-007-0117-7.  Google Scholar [6] . Janvresse, B. Rittaud and T. de la Rue, Almost-sure growth rate of generalized random Fibonacci sequences, Ann. IHP, Probab. Stat., 46 (2010), 135-158. doi: 10.1214/09-AIHP312.  Google Scholar [7] W. Parry, On the $\beta$-expansions of real numbers, Acta Math. Acad. Sci. Hungar, 11 (1960), 401-416. doi: 10.1007/BF02020954.  Google Scholar [8] A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar, 8 (1957), 477-493. doi: 10.1007/BF02020331.  Google Scholar [9] David Rosen, A class of continued fractions associated with certain properly discontinuous groups, Duke Math. J., 21 (1954), 549-563. doi: 10.1215/S0012-7094-54-02154-7.  Google Scholar

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##### References:
 [1] F. Blanchard, $\beta$-expansions and symbolic dynamics, Theoret. Comput. Sci., 65 (1989), 131-141.  Google Scholar [2] Karma Dajani and Martijn de Vries, Measures of maximal entropy for random $\beta$-expansions, J. Eur. Math. Soc. (JEMS), 7 (2005), 51-68.  Google Scholar [3] Karma Dajani and Martijn de Vries, Invariant densities for random $\beta$-expansions, J. Eur. Math. Soc. (JEMS), 9 (2007), 157-176.  Google Scholar [4] Shunji Ito and Yōichirō Takahashi, Markov subshifts and realization of $\beta$-expansions, J. Math. Soc. Japan, 26 (1974), 33-55. doi: 10.2969/jmsj/02610033.  Google Scholar [5] Élise Janvresse, Benoît Rittaud and Thierry de la Rue, How do random Fibonacci sequences grow?, Prob. Th. Rel. Fields, 142 (2008), 619-648. doi: 10.1007/s00440-007-0117-7.  Google Scholar [6] . Janvresse, B. Rittaud and T. de la Rue, Almost-sure growth rate of generalized random Fibonacci sequences, Ann. IHP, Probab. Stat., 46 (2010), 135-158. doi: 10.1214/09-AIHP312.  Google Scholar [7] W. Parry, On the $\beta$-expansions of real numbers, Acta Math. Acad. Sci. Hungar, 11 (1960), 401-416. doi: 10.1007/BF02020954.  Google Scholar [8] A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar, 8 (1957), 477-493. doi: 10.1007/BF02020331.  Google Scholar [9] David Rosen, A class of continued fractions associated with certain properly discontinuous groups, Duke Math. J., 21 (1954), 549-563. doi: 10.1215/S0012-7094-54-02154-7.  Google Scholar
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