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

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

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

Élise Janvresse ^1, , Benoît Rittaud ^2, and Thierry de la Rue ^3,

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

Abstract
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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.

Keywords: symbolic dynamics, homographic functions, maps of the interval., Continued fraction, beta-expansion.

Mathematics Subject Classification: 511K16, 11J70, 37A45, 37B1.

Citation: Élise Janvresse, Benoît Rittaud, Thierry de la Rue. Dynamics of $\lambda$-continued fractions and $\beta$-shifts. Discrete and 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.
[2]	Karma Dajani and Martijn de Vries, Measures of maximal entropy for random $\beta$-expansions, J. Eur. Math. Soc. (JEMS), 7 (2005), 51-68.
[3]	Karma Dajani and Martijn de Vries, Invariant densities for random $\beta$-expansions, J. Eur. Math. Soc. (JEMS), 9 (2007), 157-176.
[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.
[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.
[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.
[7]	W. Parry, On the $\beta $-expansions of real numbers, Acta Math. Acad. Sci. Hungar, 11 (1960), 401-416. doi: 10.1007/BF02020954.
[8]	A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar, 8 (1957), 477-493. doi: 10.1007/BF02020331.
[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.

show all references

References:

[1]	F. Blanchard, $\beta$-expansions and symbolic dynamics, Theoret. Comput. Sci., 65 (1989), 131-141.
[2]	Karma Dajani and Martijn de Vries, Measures of maximal entropy for random $\beta$-expansions, J. Eur. Math. Soc. (JEMS), 7 (2005), 51-68.
[3]	Karma Dajani and Martijn de Vries, Invariant densities for random $\beta$-expansions, J. Eur. Math. Soc. (JEMS), 9 (2007), 157-176.
[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.
[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.
[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.
[7]	W. Parry, On the $\beta $-expansions of real numbers, Acta Math. Acad. Sci. Hungar, 11 (1960), 401-416. doi: 10.1007/BF02020954.
[8]	A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar, 8 (1957), 477-493. doi: 10.1007/BF02020331.
[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.

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