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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
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show all references

References:
[1]

Acta Math., 195 (2005), 1-20. doi: 10.1007/BF02588048.  Google Scholar

[2]

Thèse d'État, Université Bordeaux I, 1983. Google Scholar

[3]

C. R. Acad. Sci. Paris Sér. I, Math., 296 (1983), 159-162.  Google Scholar

[4]

Acta Math. Hungar., 91 (2001), 325-332. doi: 10.1023/A:1010667918943.  Google Scholar

[5]

Amer. Math. Monthly, 107 (2000), 448-449.  Google Scholar

[6]

Coll. Math., 116 (2009), 165-189.  Google Scholar

[7]

Ergodic Theory Dynam. Systems, 32 (2012), 1249-1269. doi: 10.1017/S0143385711000447.  Google Scholar

[8]

Nonlinearity, 23 (2010), 2429-2456. doi: 10.1088/0951-7715/23/10/005.  Google Scholar

[9]

Theoret. Comput. Sci., 218 (1999), 3-12.  Google Scholar

[10]

Springer-Verlag, Berlin, Heidelberg, 1993.  Google Scholar

[11]

Acta Math. Hungar, 119, (2008), 57-62. doi: 10.1007/s10474-007-6252-x.  Google Scholar

[12]

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]

Acta Math. Hungar, 58 (1991), 129-132. doi: 10.1109/LICS.1991.151636.  Google Scholar

[14]

Bull. Soc. Math. France, 118 (1990), 377-390.  Google Scholar

[15]

$2^{nd}$ edition, John Wiley and Sons, Chichester UK, 2003.  Google Scholar

[16]

Ann. of Math., 146 (1997), 1-52.  Google Scholar

[17]

J. Number Theory, 40 (1992), 336-358. doi: 10.1016/0022-314X(92)90006-B.  Google Scholar

[18]

Far East J. Dyn. Syst., 6 (2004), 79-96.  Google Scholar

[19]

J. Stat. Phys., 61 (1990), 263-291. doi: 10.1007/BF01013965.  Google Scholar

[20]

Amer. Math. Monthly, 105 (1998), 936-939.  Google Scholar

[21]

J. Number Theory, 122 (2007), 157-183.  Google Scholar

[22]

Nonlinearity, 25 (2012), 2207-2243. doi: 10.1088/0951-7715/25/8/2207.  Google Scholar

[23]

Discrete Contin. Dyn. Syst., 20 (2008), 673-711.  Google Scholar

[24]

Acta Math., 178 (1997), 185-247, 247-297. doi: 10.1007/BF02392694.  Google Scholar

[25]

Math. Annalen, 101 (1929), 342-366. Corrigendum, 103 (1930), 532.  Google Scholar

[26]

Ark. Mat., 25 (1987), 41-89. doi: 10.1007/BF02384436.  Google Scholar

[27]

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]

Tokyo J. Math., 4 (1981), 399-426. doi: 10.3836/tjm/1270215165.  Google Scholar

[30]

Nonlinearity, 21 (2008), 1207-1225. doi: 10.1088/0951-7715/21/6/003.  Google Scholar

[31]

Trans. Amer. Math. Soc., 53 (1943), 427-439.  Google Scholar

[32]

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]

Ergodic Theory Dynam. Systems, 23 (2003), 637-660.  Google Scholar

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