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January  2019, 39(1): 115-130. doi: 10.3934/dcds.2019005

Core entropy of polynomials with a critical point of maximal order

Domingo González , and  Gamaliel Blé

División Académica de Ciencias Básicas, Universidad Juárez Autónoma de Tabasco, Carr. Cunduacán-Jalpa Km 1, Cunduacán Tabasco, México

* Corresponding author

Received  September 2017 Revised  July 2018 Published  October 2018

Fund Project: The first author is supported by CONACY CB-2012/181247 and 488419/278289.

This paper discusses some properties of the topological entropy of the systems generated by polynomials of degree $ d$ with two critical points. A partial order in the parameter space is defined. The monotonicity of the topological entropy of postcritically finite polynomials of degree $ d$ acting on Hubbard tree is generalized.

Keywords: Entropy, chaos, Hubbard tree, external rays.
Mathematics Subject Classification: Primary: 37F10, 37B40; Secondary: 28D20, 37F45.
Citation: Domingo González, Gamaliel Blé. Core entropy of polynomials with a critical point of maximal order. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 115-130. doi: 10.3934/dcds.2019005
References:
[1]

L. S. Block and W. A. Coppel, Dynamics in One Dimension, Lecture Notes in Mathematics, 1513. Springer-Verlag, Berlin, 1992. doi: 10.1007/BFb0084762.  Google Scholar

[2]

L. Carleson and T. W. Gamelin, Complex Dynamics, (English summary), Universitext: Tracts in Mathematics. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4364-9.  Google Scholar

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[4]

A. Douady, Topological entropy of unimodal maps: Monotonicity for quadratic polynomials. Real and Complex Dynamical Systems (Hiller$ \oslash $d, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 464, Kluwer Acad. Publ., Dordrecht, (1995), 65-87.  Google Scholar

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A. Douady and J. H. Hubbard, Étude Dynamique des Polynômes Complexes, Publications Mathématiques d'Orsay [Mathematical Publications of Orsay], 84-2. Université de Paris-Sud, Département de Mathématiques, Orsay, 1984.  Google Scholar

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A. Douady and J. H. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup., 18 (1985), 287-343.  doi: 10.24033/asens.1491.  Google Scholar

[7]

A. Kaffl, Trees and Kneading Sequences for Unicritical and Cubic Polynomials, Ph.D. thesis, International University Bremen, 2007. Google Scholar

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T. Li, A monotonicity conjecture for the entropy of Hubbard trees, Thesis (Ph.D.)-State University of New York at Stony Brook. ProQuest LLC, Ann Arbor, MI, 2007.  Google Scholar

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

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J. Milnor, Cubic polynomial maps with periodic critical orbit. Ⅰ. Complex dynamics, A K Peters, Wellesley, MA, (2009), 333-411. doi: 10.1201/b10617-13.  Google Scholar

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J. Milnor, Periodic orbits, external rays and the mandelbrot set, Geometrie Complexe et Systemes Dynamiques, Astérisque, 261 (2000), 277-333 [Stony Brook IMS Preprint 1999 3].  Google Scholar

[12]

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. doi: 10.1007/BFb0082847.  Google Scholar

[13]

J. Milnor and Ch. Tresser, On entropy and monotonicity for real cubic maps, With an appendix by Adrien Douady and Pierrette Sentenac, Comm. Math. Phys., 209 (2000), 123-178.  doi: 10.1007/s002200050018.  Google Scholar

[14]

A. Poirier, Hubbard trees, Fundamenta Mathematicae, 208 (2010), 193-248.  doi: 10.4064/fm208-3-1.  Google Scholar

[15]

A. Radulescu, The connected isentropes conjecture in a space of quartic polynomials, Discrete Contin. Dyn. Syst., 19 (2007), 139-175.  doi: 10.3934/dcds.2007.19.139.  Google Scholar

[16]

P. Roesch, Hyperbolic components of polynomials with a fixed critical point of maximal order, (English, French summary) Ann. Sci. École Norm. Sup., 40 (2007), 901-949. doi: 10.1016/j.ansens.2007.10.001.  Google Scholar

[17]

W. P. Thurston, On geometry and dynamics of iterated Rational Maps. Edited by Dierk Schleicher and Nikita Selinger and with an appendix by Schleicher. Complex Dynamics, A K Peters, Wellesley, MA, (2009), 3-137. doi: 10.1201/b10617-3.  Google Scholar

[18]

G. Tiozzo, Entropy, dimension and combinatorial moduli for one-dimensional dynamical systems, Thesis (Ph.D.)-Harvard University. ProQuest LLC, Ann Arbor, MI, 2013.  Google Scholar

[19]

G. Tiozzo, Continuity of core entropy of quadratic polynomials, Invent. Math., 203 (2016), 891-921.  doi: 10.1007/s00222-015-0605-9.  Google Scholar

[20]

S. Zakeri, Biaccessibility in quadratic Julia Sets, Ergodic Theory Dynam. Systems, 20 (2000), 1859-1883.  doi: 10.1017/S0143385700001024.  Google Scholar

show all references

References:
[1]

L. S. Block and W. A. Coppel, Dynamics in One Dimension, Lecture Notes in Mathematics, 1513. Springer-Verlag, Berlin, 1992. doi: 10.1007/BFb0084762.  Google Scholar

[2]

L. Carleson and T. W. Gamelin, Complex Dynamics, (English summary), Universitext: Tracts in Mathematics. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4364-9.  Google Scholar

[3]

A. Douady, Algorithms for computing angles in the Mandelbort set, Chaotic dynamics and fractals (Atlanta, Ga. 1985), Notes Rep. Math. Sci. Engrg. 2, Academic Press, Orlando, FL, (1986), 155-168.  Google Scholar

[4]

A. Douady, Topological entropy of unimodal maps: Monotonicity for quadratic polynomials. Real and Complex Dynamical Systems (Hiller$ \oslash $d, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 464, Kluwer Acad. Publ., Dordrecht, (1995), 65-87.  Google Scholar

[5]

A. Douady and J. H. Hubbard, Étude Dynamique des Polynômes Complexes, Publications Mathématiques d'Orsay [Mathematical Publications of Orsay], 84-2. Université de Paris-Sud, Département de Mathématiques, Orsay, 1984.  Google Scholar

[6]

A. Douady and J. H. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup., 18 (1985), 287-343.  doi: 10.24033/asens.1491.  Google Scholar

[7]

A. Kaffl, Trees and Kneading Sequences for Unicritical and Cubic Polynomials, Ph.D. thesis, International University Bremen, 2007. Google Scholar

[8]

T. Li, A monotonicity conjecture for the entropy of Hubbard trees, Thesis (Ph.D.)-State University of New York at Stony Brook. ProQuest LLC, Ann Arbor, MI, 2007.  Google Scholar

[9]

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

[10]

J. Milnor, Cubic polynomial maps with periodic critical orbit. Ⅰ. Complex dynamics, A K Peters, Wellesley, MA, (2009), 333-411. doi: 10.1201/b10617-13.  Google Scholar

[11]

J. Milnor, Periodic orbits, external rays and the mandelbrot set, Geometrie Complexe et Systemes Dynamiques, Astérisque, 261 (2000), 277-333 [Stony Brook IMS Preprint 1999 3].  Google Scholar

[12]

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. doi: 10.1007/BFb0082847.  Google Scholar

[13]

J. Milnor and Ch. Tresser, On entropy and monotonicity for real cubic maps, With an appendix by Adrien Douady and Pierrette Sentenac, Comm. Math. Phys., 209 (2000), 123-178.  doi: 10.1007/s002200050018.  Google Scholar

[14]

A. Poirier, Hubbard trees, Fundamenta Mathematicae, 208 (2010), 193-248.  doi: 10.4064/fm208-3-1.  Google Scholar

[15]

A. Radulescu, The connected isentropes conjecture in a space of quartic polynomials, Discrete Contin. Dyn. Syst., 19 (2007), 139-175.  doi: 10.3934/dcds.2007.19.139.  Google Scholar

[16]

P. Roesch, Hyperbolic components of polynomials with a fixed critical point of maximal order, (English, French summary) Ann. Sci. École Norm. Sup., 40 (2007), 901-949. doi: 10.1016/j.ansens.2007.10.001.  Google Scholar

[17]

W. P. Thurston, On geometry and dynamics of iterated Rational Maps. Edited by Dierk Schleicher and Nikita Selinger and with an appendix by Schleicher. Complex Dynamics, A K Peters, Wellesley, MA, (2009), 3-137. doi: 10.1201/b10617-3.  Google Scholar

[18]

G. Tiozzo, Entropy, dimension and combinatorial moduli for one-dimensional dynamical systems, Thesis (Ph.D.)-Harvard University. ProQuest LLC, Ann Arbor, MI, 2013.  Google Scholar

[19]

G. Tiozzo, Continuity of core entropy of quadratic polynomials, Invent. Math., 203 (2016), 891-921.  doi: 10.1007/s00222-015-0605-9.  Google Scholar

[20]

S. Zakeri, Biaccessibility in quadratic Julia Sets, Ergodic Theory Dynam. Systems, 20 (2000), 1859-1883.  doi: 10.1017/S0143385700001024.  Google Scholar

Figure 1.  Connectedness locus for $d = 4$ and $d = 5$
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Figure 2.  Filled Julia set and intervals of angles of $P_a$
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