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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
División Académica de Ciencias Básicas, Universidad Juárez Autónoma de Tabasco, Carr. Cunduacán-Jalpa Km 1, Cunduacán Tabasco, México |
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.
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 and Continuous Dynamical Systems,
2019, 39
(1)
: 115-130.
doi: 10.3934/dcds.2019005
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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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [7] |
A. Kaffl,
Trees and Kneading Sequences for Unicritical and Cubic Polynomials, Ph.D. thesis, International University Bremen, 2007. |
| [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. |
| [9] |
M. Lyubich,
Dynamics of quadratic polynomials Ⅰ-Ⅱ, Acta Math, 178 (1997), 185-297.
doi: 10.1007/BF02392694. |
| [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. |
| [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]. |
| [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. |
| [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. |
| [14] |
A. Poirier,
Hubbard trees, Fundamenta Mathematicae, 208 (2010), 193-248.
doi: 10.4064/fm208-3-1. |
| [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. |
| [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. |
| [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. |
| [18] |
G. Tiozzo, Entropy, dimension and combinatorial moduli for one-dimensional dynamical systems, Thesis (Ph.D.)-Harvard University. ProQuest LLC, Ann Arbor, MI, 2013. |
| [19] |
G. Tiozzo,
Continuity of core entropy of quadratic polynomials, Invent. Math., 203 (2016), 891-921.
doi: 10.1007/s00222-015-0605-9. |
| [20] |
S. Zakeri,
Biaccessibility in quadratic Julia Sets, Ergodic Theory Dynam. Systems, 20 (2000), 1859-1883.
doi: 10.1017/S0143385700001024. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [7] |
A. Kaffl,
Trees and Kneading Sequences for Unicritical and Cubic Polynomials, Ph.D. thesis, International University Bremen, 2007. |
| [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. |
| [9] |
M. Lyubich,
Dynamics of quadratic polynomials Ⅰ-Ⅱ, Acta Math, 178 (1997), 185-297.
doi: 10.1007/BF02392694. |
| [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. |
| [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]. |
| [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. |
| [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. |
| [14] |
A. Poirier,
Hubbard trees, Fundamenta Mathematicae, 208 (2010), 193-248.
doi: 10.4064/fm208-3-1. |
| [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. |
| [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. |
| [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. |
| [18] |
G. Tiozzo, Entropy, dimension and combinatorial moduli for one-dimensional dynamical systems, Thesis (Ph.D.)-Harvard University. ProQuest LLC, Ann Arbor, MI, 2013. |
| [19] |
G. Tiozzo,
Continuity of core entropy of quadratic polynomials, Invent. Math., 203 (2016), 891-921.
doi: 10.1007/s00222-015-0605-9. |
| [20] |
S. Zakeri,
Biaccessibility in quadratic Julia Sets, Ergodic Theory Dynam. Systems, 20 (2000), 1859-1883.
doi: 10.1017/S0143385700001024. |
Figure 2.
Filled Julia set and intervals of angles of $P_a$
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