Dynamics of continued fractions and kneading sequences of unimodal maps

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April 2013, 33(4): 1313-1332. doi: 10.3934/dcds.2013.33.1313

Dynamics of continued fractions and kneading sequences of unimodal maps

Claudio Bonanno ^1, , Carlo Carminati ^1, , Stefano Isola ^2, and Giulio Tiozzo ^3,

1.	Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, I-56127, Italy, Italy
2.	Dipartimento di Matematica e Informatica, Università di Camerino, via Madonna delle Carceri, I-62032 Camerino
3.	Department of Mathematics, Harvard University, One Oxford Street Cambridge, MA 02138, United States

Received September 2011 Revised January 2012 Published October 2012

Abstract
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In this paper we construct a correspondence between the parameter spaces of two families of one-dimensional dynamical systems, the $\alpha$-continued fraction transformations $T_\alpha$ and unimodal maps. This correspondence identifies bifurcation parameters in the two families, and allows one to transfer topological and metric properties from one setting to the other. As an application, we recover results about the real slice of the Mandelbrot set, and the set of univoque numbers.

Keywords: univoque numbers., period-doubling bifurcations, kneading theory, Continued fractions.

Mathematics Subject Classification: Primary: 11A55, 37A10; Secondary: 11K10, 11J06, 37E05, 37E20, 37F4.

Citation: Claudio Bonanno, Carlo Carminati, Stefano Isola, Giulio Tiozzo. Dynamics of continued fractions and kneading sequences of unimodal maps. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1313-1332. doi: 10.3934/dcds.2013.33.1313

References:

[1]	B. Adamczewski and Y. Bugeaud, On the complexity of algebraic numbers. II. Continued fractions, Acta Math., 195 (2005), 1-20. doi: 10.1007/BF02588048.
[2]	J.-P. Allouche, "Théorie des Nombres et Automates,'' Thèse d'État, Université Bordeaux I, 1983.
[3]	J.-P. Allouche and M. Cosnard, Itérations de fonctions unimodales et suites engendrées par automates, C. R. Acad. Sci. Paris Sér. I, Math., 296 (1983), 159-162.
[4]	J.-P. Allouche and M. Cosnard, Non-integer bases, iteration of continuous real maps, and an arithmetic self-similar set, Acta Math. Hungar., 91 (2001), 325-332. doi: 10.1023/A:1010667918943.
[5]	J.-P. Allouche and M. Cosnard, The Komornik-Loreti constant is transcendental, Amer. Math. Monthly, 107 (2000), 448-449.
[6]	C. Bonanno and S. Isola, Orderings of the rationals and dynamical systems, Coll. Math., 116 (2009), 165-189.
[7]	C. Carminati and G. Tiozzo, A canonical thickening of $\Q$ and the entropy of $\alpha$-continued fraction transformations, Ergodic Theory Dynam. Systems, 32 (2012), 1249-1269. doi: 10.1017/S0143385711000447.
[8]	C. Carminati, S. Marmi, A. Profeti and G. Tiozzo, The entropy of $\alpha$-continued fractions: numerical results, Nonlinearity, 23 (2010), 2429-2456. doi: 10.1088/0951-7715/23/10/005.
[9]	J. Cassaigne, Limit values of the recurrence quotient of Sturmian sequences, Theoret. Comput. Sci., 218 (1999), 3-12.
[10]	W. de Melo and S. van Strien, "One-dimensional Dynamics,'' Springer-Verlag, Berlin, Heidelberg, 1993.
[11]	M. de Vries, A property of algebraic univoque numbers, Acta Math. Hungar, 119, (2008), 57-62. doi: 10.1007/s10474-007-6252-x.
[12]	A. Douady, Topological entropy of unimodal maps: monotonicity for quadratic polynomials, in "Real and Complex Dynamical Systems (Hillerød, 1993)'', NATO Adv. Sci. Inst. Ser. C, Math. Phys. Sci., 464, Kluwer, Dordrecht (1995), 65-87.
[13]	P. Erdös, M. Horváth and I. Joó, On the uniqueness of the expansions $1=\sum q^{-n_i}$, Acta Math. Hungar, 58 (1991), 129-132. doi: 10.1109/LICS.1991.151636.
[14]	P. Erdös, I. Joó and V. Komornik, Characterization of the unique expansions $1=\sum q^{-n_i}$ and related problems, Bull. Soc. Math. France, 118 (1990), 377-390.
[15]	K. Falconer, "Fractal Geometry - Mathematical Foundations and Applications,'' $2^{nd}$ edition, John Wiley and Sons, Chichester UK, 2003.
[16]	J. Graczyk and G. Światek, Generic hyperbolicity in the logistic family, Ann. of Math., 146 (1997), 1-52.
[17]	D. Hensley, Continued fractions Cantor sets, Hausdorff dimension, and functional analysis, J. Number Theory, 40 (1992), 336-358. doi: 10.1016/0022-314X(92)90006-B.
[18]	S. Isola, On a set of numbers arising in the dynamics of unimodal maps, Far East J. Dyn. Syst., 6 (2004), 79-96.
[19]	S. Isola and A. Politi, Universal encoding for unimodal maps, J. Stat. Phys., 61 (1990), 263-291. doi: 10.1007/BF01013965.
[20]	V. Komornik and P. Loreti, Unique developments in non-integer bases, Amer. Math. Monthly, 105 (1998), 936-939.
[21]	V. Komornik and P. Loreti, On the topological structure of univoque sets, J. Number Theory, 122 (2007), 157-183.
[22]	C. Kraaikamp, T. A. Schmidt and W. Steiner, Natural extensions and entropy of $\alpha$-continued fractions, Nonlinearity, 25 (2012), 2207-2243. doi: 10.1088/0951-7715/25/8/2207.
[23]	L. Luzzi and S. Marmi, On the entropy of Japanese continued fractions, Discrete Contin. Dyn. Syst., 20 (2008), 673-711.
[24]	M. Lyubich, Dynamics of quadratic polynomials. I, II, Acta Math., 178 (1997), 185-247, 247-297. doi: 10.1007/BF02392694.
[25]	K. Mahler, Aritmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen, Math. Annalen, 101 (1929), 342-366. Corrigendum, 103 (1930), 532.
[26]	N. G. Makarov, Conformal mapping and Hausdorff measures, Ark. Mat., 25 (1987), 41-89. doi: 10.1007/BF02384436.
[27]	J. Milnor and W. Thurston, On iterated maps of the interval, in "Dynamical Systems (College Park, MD, 1986-87)'', Lecture Notes in Math., 1342, Springer, Berlin (1988), 465-563.
[28]	C. G. Moreira, Geometric properties of the Markov and Lagrange spectra, available from: http://w3.impa.br/~gugu/Geometric_Properties.pdf.
[29]	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.
[30]	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.
[31]	R. Salem, On some singular monotone functions which are strictly increasing, Trans. Amer. Math. Soc., 53 (1943), 427-439.
[32]	W. Thurston, On the geometry and dynamics of iterated rational maps, in "Complex Dynamics: Families and Friends'' (eds. D. Schleicher and N. Selinger), A K Peters, Wellesley, MA, (2009), 3-137. doi: 10.1016/j.phycom.2009.02.007.
[33]	G. Tiozzo, The entropy of Nakada's $\alpha$-continued fractions: analytical results, to appear on Ann. Sc. Norm. Super. Pisa Cl. Sci., arXiv:0912.2379.
[34]	S. Zakeri, External rays and the real slice of the Mandelbrot set, Ergodic Theory Dynam. Systems, 23 (2003), 637-660.

show all references

References:

[1]	B. Adamczewski and Y. Bugeaud, On the complexity of algebraic numbers. II. Continued fractions, Acta Math., 195 (2005), 1-20. doi: 10.1007/BF02588048.
[2]	J.-P. Allouche, "Théorie des Nombres et Automates,'' Thèse d'État, Université Bordeaux I, 1983.
[3]	J.-P. Allouche and M. Cosnard, Itérations de fonctions unimodales et suites engendrées par automates, C. R. Acad. Sci. Paris Sér. I, Math., 296 (1983), 159-162.
[4]	J.-P. Allouche and M. Cosnard, Non-integer bases, iteration of continuous real maps, and an arithmetic self-similar set, Acta Math. Hungar., 91 (2001), 325-332. doi: 10.1023/A:1010667918943.
[5]	J.-P. Allouche and M. Cosnard, The Komornik-Loreti constant is transcendental, Amer. Math. Monthly, 107 (2000), 448-449.
[6]	C. Bonanno and S. Isola, Orderings of the rationals and dynamical systems, Coll. Math., 116 (2009), 165-189.
[7]	C. Carminati and G. Tiozzo, A canonical thickening of $\Q$ and the entropy of $\alpha$-continued fraction transformations, Ergodic Theory Dynam. Systems, 32 (2012), 1249-1269. doi: 10.1017/S0143385711000447.
[8]	C. Carminati, S. Marmi, A. Profeti and G. Tiozzo, The entropy of $\alpha$-continued fractions: numerical results, Nonlinearity, 23 (2010), 2429-2456. doi: 10.1088/0951-7715/23/10/005.
[9]	J. Cassaigne, Limit values of the recurrence quotient of Sturmian sequences, Theoret. Comput. Sci., 218 (1999), 3-12.
[10]	W. de Melo and S. van Strien, "One-dimensional Dynamics,'' Springer-Verlag, Berlin, Heidelberg, 1993.
[11]	M. de Vries, A property of algebraic univoque numbers, Acta Math. Hungar, 119, (2008), 57-62. doi: 10.1007/s10474-007-6252-x.
[12]	A. Douady, Topological entropy of unimodal maps: monotonicity for quadratic polynomials, in "Real and Complex Dynamical Systems (Hillerød, 1993)'', NATO Adv. Sci. Inst. Ser. C, Math. Phys. Sci., 464, Kluwer, Dordrecht (1995), 65-87.
[13]	P. Erdös, M. Horváth and I. Joó, On the uniqueness of the expansions $1=\sum q^{-n_i}$, Acta Math. Hungar, 58 (1991), 129-132. doi: 10.1109/LICS.1991.151636.
[14]	P. Erdös, I. Joó and V. Komornik, Characterization of the unique expansions $1=\sum q^{-n_i}$ and related problems, Bull. Soc. Math. France, 118 (1990), 377-390.
[15]	K. Falconer, "Fractal Geometry - Mathematical Foundations and Applications,'' $2^{nd}$ edition, John Wiley and Sons, Chichester UK, 2003.
[16]	J. Graczyk and G. Światek, Generic hyperbolicity in the logistic family, Ann. of Math., 146 (1997), 1-52.
[17]	D. Hensley, Continued fractions Cantor sets, Hausdorff dimension, and functional analysis, J. Number Theory, 40 (1992), 336-358. doi: 10.1016/0022-314X(92)90006-B.
[18]	S. Isola, On a set of numbers arising in the dynamics of unimodal maps, Far East J. Dyn. Syst., 6 (2004), 79-96.
[19]	S. Isola and A. Politi, Universal encoding for unimodal maps, J. Stat. Phys., 61 (1990), 263-291. doi: 10.1007/BF01013965.
[20]	V. Komornik and P. Loreti, Unique developments in non-integer bases, Amer. Math. Monthly, 105 (1998), 936-939.
[21]	V. Komornik and P. Loreti, On the topological structure of univoque sets, J. Number Theory, 122 (2007), 157-183.
[22]	C. Kraaikamp, T. A. Schmidt and W. Steiner, Natural extensions and entropy of $\alpha$-continued fractions, Nonlinearity, 25 (2012), 2207-2243. doi: 10.1088/0951-7715/25/8/2207.
[23]	L. Luzzi and S. Marmi, On the entropy of Japanese continued fractions, Discrete Contin. Dyn. Syst., 20 (2008), 673-711.
[24]	M. Lyubich, Dynamics of quadratic polynomials. I, II, Acta Math., 178 (1997), 185-247, 247-297. doi: 10.1007/BF02392694.
[25]	K. Mahler, Aritmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen, Math. Annalen, 101 (1929), 342-366. Corrigendum, 103 (1930), 532.
[26]	N. G. Makarov, Conformal mapping and Hausdorff measures, Ark. Mat., 25 (1987), 41-89. doi: 10.1007/BF02384436.
[27]	J. Milnor and W. Thurston, On iterated maps of the interval, in "Dynamical Systems (College Park, MD, 1986-87)'', Lecture Notes in Math., 1342, Springer, Berlin (1988), 465-563.
[28]	C. G. Moreira, Geometric properties of the Markov and Lagrange spectra, available from: http://w3.impa.br/~gugu/Geometric_Properties.pdf.
[29]	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.
[30]	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.
[31]	R. Salem, On some singular monotone functions which are strictly increasing, Trans. Amer. Math. Soc., 53 (1943), 427-439.
[32]	W. Thurston, On the geometry and dynamics of iterated rational maps, in "Complex Dynamics: Families and Friends'' (eds. D. Schleicher and N. Selinger), A K Peters, Wellesley, MA, (2009), 3-137. doi: 10.1016/j.phycom.2009.02.007.
[33]	G. Tiozzo, The entropy of Nakada's $\alpha$-continued fractions: analytical results, to appear on Ann. Sc. Norm. Super. Pisa Cl. Sci., arXiv:0912.2379.
[34]	S. Zakeri, External rays and the real slice of the Mandelbrot set, Ergodic Theory Dynam. Systems, 23 (2003), 637-660.

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