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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

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 & 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.  Google Scholar

[2]

J.-P. Allouche, "Théorie des Nombres et Automates,'' Thèse d'État, Université Bordeaux I, 1983. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

J.-P. Allouche and M. Cosnard, The Komornik-Loreti constant is transcendental, Amer. Math. Monthly, 107 (2000), 448-449.  Google Scholar

[6]

C. Bonanno and S. Isola, Orderings of the rationals and dynamical systems, Coll. Math., 116 (2009), 165-189.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

J. Cassaigne, Limit values of the recurrence quotient of Sturmian sequences, Theoret. Comput. Sci., 218 (1999), 3-12.  Google Scholar

[10]

W. de Melo and S. van Strien, "One-dimensional Dynamics,'' Springer-Verlag, Berlin, Heidelberg, 1993.  Google Scholar

[11]

M. de Vries, A property of algebraic univoque numbers, Acta Math. Hungar, 119, (2008), 57-62. doi: 10.1007/s10474-007-6252-x.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

K. Falconer, "Fractal Geometry - Mathematical Foundations and Applications,'' $2^{nd}$ edition, John Wiley and Sons, Chichester UK, 2003.  Google Scholar

[16]

J. Graczyk and G. Światek, Generic hyperbolicity in the logistic family, Ann. of Math., 146 (1997), 1-52.  Google Scholar

[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.  Google Scholar

[18]

S. Isola, On a set of numbers arising in the dynamics of unimodal maps, Far East J. Dyn. Syst., 6 (2004), 79-96.  Google Scholar

[19]

S. Isola and A. Politi, Universal encoding for unimodal maps, J. Stat. Phys., 61 (1990), 263-291. doi: 10.1007/BF01013965.  Google Scholar

[20]

V. Komornik and P. Loreti, Unique developments in non-integer bases, Amer. Math. Monthly, 105 (1998), 936-939.  Google Scholar

[21]

V. Komornik and P. Loreti, On the topological structure of univoque sets, J. Number Theory, 122 (2007), 157-183.  Google Scholar

[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.  Google Scholar

[23]

L. Luzzi and S. Marmi, On the entropy of Japanese continued fractions, Discrete Contin. Dyn. Syst., 20 (2008), 673-711.  Google Scholar

[24]

M. Lyubich, Dynamics of quadratic polynomials. I, II, Acta Math., 178 (1997), 185-247, 247-297. doi: 10.1007/BF02392694.  Google Scholar

[25]

K. Mahler, Aritmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen, Math. Annalen, 101 (1929), 342-366. Corrigendum, 103 (1930), 532.  Google Scholar

[26]

N. G. Makarov, Conformal mapping and Hausdorff measures, Ark. Mat., 25 (1987), 41-89. doi: 10.1007/BF02384436.  Google Scholar

[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.  Google Scholar

[28]

C. G. Moreira, Geometric properties of the Markov and Lagrange spectra,, available from: , ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

R. Salem, On some singular monotone functions which are strictly increasing, Trans. Amer. Math. Soc., 53 (1943), 427-439.  Google Scholar

[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.  Google Scholar

[33]

G. Tiozzo, The entropy of Nakada's $\alpha$-continued fractions: analytical results,, to appear on Ann. Sc. Norm. Super. Pisa Cl. Sci., ().   Google Scholar

[34]

S. Zakeri, External rays and the real slice of the Mandelbrot set, Ergodic Theory Dynam. Systems, 23 (2003), 637-660.  Google Scholar

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.  Google Scholar

[2]

J.-P. Allouche, "Théorie des Nombres et Automates,'' Thèse d'État, Université Bordeaux I, 1983. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

J.-P. Allouche and M. Cosnard, The Komornik-Loreti constant is transcendental, Amer. Math. Monthly, 107 (2000), 448-449.  Google Scholar

[6]

C. Bonanno and S. Isola, Orderings of the rationals and dynamical systems, Coll. Math., 116 (2009), 165-189.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

J. Cassaigne, Limit values of the recurrence quotient of Sturmian sequences, Theoret. Comput. Sci., 218 (1999), 3-12.  Google Scholar

[10]

W. de Melo and S. van Strien, "One-dimensional Dynamics,'' Springer-Verlag, Berlin, Heidelberg, 1993.  Google Scholar

[11]

M. de Vries, A property of algebraic univoque numbers, Acta Math. Hungar, 119, (2008), 57-62. doi: 10.1007/s10474-007-6252-x.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

K. Falconer, "Fractal Geometry - Mathematical Foundations and Applications,'' $2^{nd}$ edition, John Wiley and Sons, Chichester UK, 2003.  Google Scholar

[16]

J. Graczyk and G. Światek, Generic hyperbolicity in the logistic family, Ann. of Math., 146 (1997), 1-52.  Google Scholar

[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.  Google Scholar

[18]

S. Isola, On a set of numbers arising in the dynamics of unimodal maps, Far East J. Dyn. Syst., 6 (2004), 79-96.  Google Scholar

[19]

S. Isola and A. Politi, Universal encoding for unimodal maps, J. Stat. Phys., 61 (1990), 263-291. doi: 10.1007/BF01013965.  Google Scholar

[20]

V. Komornik and P. Loreti, Unique developments in non-integer bases, Amer. Math. Monthly, 105 (1998), 936-939.  Google Scholar

[21]

V. Komornik and P. Loreti, On the topological structure of univoque sets, J. Number Theory, 122 (2007), 157-183.  Google Scholar

[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.  Google Scholar

[23]

L. Luzzi and S. Marmi, On the entropy of Japanese continued fractions, Discrete Contin. Dyn. Syst., 20 (2008), 673-711.  Google Scholar

[24]

M. Lyubich, Dynamics of quadratic polynomials. I, II, Acta Math., 178 (1997), 185-247, 247-297. doi: 10.1007/BF02392694.  Google Scholar

[25]

K. Mahler, Aritmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen, Math. Annalen, 101 (1929), 342-366. Corrigendum, 103 (1930), 532.  Google Scholar

[26]

N. G. Makarov, Conformal mapping and Hausdorff measures, Ark. Mat., 25 (1987), 41-89. doi: 10.1007/BF02384436.  Google Scholar

[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.  Google Scholar

[28]

C. G. Moreira, Geometric properties of the Markov and Lagrange spectra,, available from: , ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

R. Salem, On some singular monotone functions which are strictly increasing, Trans. Amer. Math. Soc., 53 (1943), 427-439.  Google Scholar

[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.  Google Scholar

[33]

G. Tiozzo, The entropy of Nakada's $\alpha$-continued fractions: analytical results,, to appear on Ann. Sc. Norm. Super. Pisa Cl. Sci., ().   Google Scholar

[34]

S. Zakeri, External rays and the real slice of the Mandelbrot set, Ergodic Theory Dynam. Systems, 23 (2003), 637-660.  Google Scholar

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