The classifiction of critically preperiodic polynomials as dynamical systems, J.Amer.Math.Soc, vol.5, pp.721-762, 1992. ,
The iteration of cubic polynomials. Part I: the global topology of parameter space, Acta Math, vol.160, pp.143-206, 1988. ,
Growing trees, laminations and the dynamics on the Julia set, Ergod. Th. and Dynam. Sys, vol.22, pp.63-97, 2002. ,
Julia sets and complex singularities in hierarchical Ising models, Commun. Math. Phys, vol.141, pp.453-474, 1991. ,
Quasisymmetries of Sierpi?ski carpet Julia sets, 2014. ,
Quasisymmetric rigidity of square Sierpi?ski carpets, Ann. of Math, vol.177, issue.2, pp.591-643, 2013. ,
Hyerbolic-parabolic deformations of rational maps ,
Sierpi?ski curve Julia sets for quadratic rational maps, Ann. Acad. Sci. Fenn. Math, vol.39, pp.3-22, 2014. ,
Topological entropy of unimodal maps: monotonicity for quadratic polynomials, Real and complex dynamical systems, vol.464, pp.65-87, 1993. ,
Étude Dynamique des Polynômes Complexes I and II, Publ. Math. Orsay, 1984. ,
The escape trichotomy for singularly perturbed rational maps, Indiana Univ. Math. J, vol.54, pp.1621-1634, 2005. ,
Fractal structure of zeros in hierarchinal models, J. Stat. Phys, vol.33, pp.559-569, 1983. ,
The Nature of Critical Points (Lectures in Theoretical Physics, VIIC) ,
Dynatomic curves and core entropy of polynomials, 2013. ,
Quasisymmetrically inequivalent hyperbolic Julia sets, Revista Math. Iberoamericana, vol.28, pp.1025-1034, 2012. ,
, Core entropy and biaccessibility of quadratic polynomials, available at
Wandering orbit portraits, Trans. Amer. Math. Soc, vol.354, pp.1473-1485, 2002. ,
Combinatorial continuity in complex polynomial dynamics, Proc. London Math. Soc, vol.91, issue.3, pp.215-248, 2005. ,
A monotonicity conjecture for the entropy of Hubbard trees, 2007. ,
Geometry and dynamics of quadratic rational maps, with an appendix by, Exper. Math, vol.2, pp.37-83, 1993. ,
On postcritically finite polynomials, part I: critical portraits, IMS preprint #5, 1993. ,
Julia sets and complex singularities of free energies, 2011. ,
On connectivity of Julia sets of Yang-Lee zeros, Commun. Math. Phys, vol.222, pp.319-326, 2001. ,
Fatou components and Julia sets of singularly perturbed rational maps with positive parameter, Acta Math. Sin. (Engl. Ser.), vol.28, issue.10, pp.1937-1954, 2012. ,
Feigenbaum Julia sets of singularities of free energy, Ergodic Th. and Dynam. Sys, vol.30, pp.1573-1591, 2010. ,
Topological entropy of quadratic polynomials and dimension of sections of the Mandelbrot set ,
Continuity of core entropy of quadratic polynomials, available at arXiv:1409.3511 ,
Quasisymmetric embeddings of metric spaces, Ann. Acad. Sci. Fenn. Ser. AI Math, 1980. ,
Lectures on quasiconformal mappings, Van Nostrand Math.Studies, issue.10, 1966. ,
, Conformal invariants: topics in geometric function theory, 1973.
Quasisymmetric rigidity of square Sierpi?ski carpets, Ann. of Math, vol.177, issue.2, pp.591-643, 2013. ,
Uniformization of Sierpi?ski carpets in the plane, Invent. Math, vol.186, pp.559-665, 2011. ,
Quasiconformal geometry of fractals, Proceedings Internat. Congress Math, pp.1349-1373, 2006. ,
Rigidity of Schottky sets, Amer. J. Math, vol.131, pp.409-443, 2009. ,
Lectures on analysis on metric spaces, 2001. ,
Quasiconformal maps in metric spaces with controlled geometry, Acta Math, vol.181, pp.1-61, 1998. ,
Hyperbolic groups with low-dimensional boundary, Ann. Sci. École Norm. Sup, issue.4, pp.647-669, 2000. ,
Quasiconformal mappings in the plane, 1973. ,
Quasisymmetric embeddings of metric spaces, Ann. Acad. Sci. Fenn. Ser. AI Math, 1980. ,
Topological characterization of the Sierpi?ski curve, Fund. Math, vol.45, pp.320-324, 1958. ,
Uniformization of Sierpi?ski carpets in the plane, Invent. Math, vol.186, pp.559-665, 2011. ,
Quasisymmetries of Sierpi?ski carpet Julia sets, 2014. ,
Rigidity of Schottky sets, Amer. J. Math, vol.131, pp.409-443, 2009. ,
Immeubles hyperboliques, dimension conforme et rigidité de Mostow, Geom. Funct. Anal, vol.7, pp.245-268, 1997. ,
Quasiconformal geometry and hyperbolic geometry, Rigidity in dynamics and geometry, pp.1-17, 2000. ,
Sierpi?ski curve Julia sets for quadratic rational maps, Ann. Acad. Sci. Fenn. Math, vol.39, pp.3-22, 2014. ,
The escape trichotomy for singularly perturbed rational maps, Indiana Univ. Math. J, vol.54, pp.1621-1634, 2005. ,
Generic hyperbolicity in the logistic family, Ann. Math, vol.146, pp.1-52, 1997. ,
Quasisymmetrically inequivalent hyperbolic Julia sets, Revista Math. Iberoamericana, vol.28, pp.1025-1034, 2012. ,
The asymptotic geometry of negatively curved spaces: uniformization, geometrization and rigidity, International Congress of Mathematicians, vol.II, pp.743-768, 2006. ,
Quasi-Additivity Law in conformal geometry, Ann. Math, vol.169, pp.561-193, 2009. ,
Quasiconformal Mappings in the Plane, 1973. ,
Dynamics of quadratic polynomials I-II, Acta Math, vol.178, pp.185-297, 1997. ,
On a lemma of Fatou, Bol. Soc. Bras. Mat, vol.24, pp.1-11, 1993. ,
Geometry and dynamics of quadratic rational maps, with an appendix by, Exper. Math, vol.2, pp.37-83, 1993. ,
Periodic orbits, externals rays and the Mandelbrot set: An expository account, Géomé Complexe et Systèmes Dynamiques, Colloque en l'honneur d'Adrien Douady, pp.277-333, 2000. ,
Dynamics in One Complex Variable, 2006. ,
, Boundary Behaviour of Conformal Maps, 1992.
Dynamics of McMullen maps, Adv. Math, vol.229, pp.2525-2577, 2012. ,
Fatou components and Julia sets of singularly perturbed rational maps with positive parameter, Acta Math. Sin. (Engl. Ser.), vol.28, issue.10, pp.1937-1954, 2012. ,
On the dynamics of the McMullen family R(z) = z m +?/z, Conform. Geom. Dyn, vol.10, pp.159-183, 2006. ,
Quasiconformal homeomorphisms and dynamics I: Solution of the Fatou-Julia problem on wandering domains, Ann. Math, vol.122, pp.401-418, 1985. ,
Topological entropy of quadratic polynomials and dimension of sections of the Manderbrot set, 2013. ,
Topological characterization of the Sierpi?ski curves, Fund. Math, vol.45, pp.320-324, 1958. ,
On the dynamics of generalized McMullen maps, Ergod. Th. & Dynam. Sys ,
On the Julia set of semi-hyperbolic rational maps, Chinese J. Contemp. Math, vol.20, issue.4, pp.469-476, 1999. ,
The classifiction of critically preperiodic polynomials as dynamical systems, J.Amer.Math.Soc, vol.5, pp.721-762, 1992. ,
Necessary conditions for the existence of wandering triangles for cubic laminations, Discrete Contin. Dyn. Syst, vol.13, issue.1, pp.13-34, 2005. ,
Growing trees, laminations and the dynamics on the Julia set, Ergod. Th. and Dynam. Sys, vol.22, pp.63-97, 2002. ,
Wandering gaps for weakly hyperbolic polynomials, pp.139-168, 2008. ,
Topological entropy of unimodal maps: monotonicity for quadratic polynomials, Real and complex dynamical systems, vol.464, pp.65-87, 1993. ,
Étude Dynamique des Polynômes Complexes I and II, Publ. Math. Orsay, 1984. ,
Dynatomic curves and core entropy of polynomials, 2013. ,
Fixed points of polynomial maps II: Fixed point portraits, Ann. Scient. École Norm. Sup, vol.4, pp.51-98, 1993. ,
, Core entropy and biaccessibility of quadratic polynomials, available at
Wandering orbit portraits, Trans. Amer. Math. Soc, vol.354, pp.1473-1485, 2002. ,
, Real laminations and the topological dynamics of complex polynomials, vol.184, pp.207-267, 2004.
Combinatorial continuity in complex polynomial dynamics, Proc. London Math. Soc, vol.91, issue.3, pp.215-248, 2005. ,
On backward stability of holomorphic dynamical systems, Fundamenta Mathematicae, vol.158, pp.97-107, 1998. ,
A monotonicity conjecture for the entropy of Hubbard trees, 2007. ,
, Complex Dynamics and Renormalization, vol.135, 1995.
Periodic orbits, externals rays and the Mandelbrot set: An expository account, Géomé Complexe et Systèmes Dynamiques, Colloque en l'honneur d'Adrien Douady, pp.277-333, 2000. ,
, Dynamics in one complex variable, 2006.
DOI : 10.1007/978-3-663-08092-3
Continuum theory. An introduction, Monographs and Textbooks in Pure and Applied Mathematics, vol.158, 1992. ,
On postcritically finite polynomials, part I: critical portraits, IMS preprint #5, 1993. ,
On post critically finite polynomials, part II: Hubbard trees, IMS preprint #7, 1993. ,
, Real and Complex Analysis, 1987.
Conformal Dynamical Systems, Springer Lecture Notes in Mathematics, vol.1007, pp.725-752, 1983. ,
Complex Dynamics: families and Friends, The combinatorics of iterated rational maps, pp.3-137, 1985. ,
Entropy in dimension one, Frontiers in Complex Dynamics: In Celebration of John Milonor's 80th Birthday, 2014. ,
Topological entropy of quadratic polynomials and dimension of sections of the Mandelbrot set ,
Continuity of core entropy of quadratic polynomials, available at arXiv:1409.3511 ,
A simple proof for monotonicity of entropy in the quadratic family, Ergod. Th. and Dynam. Sys, vol.20, issue.3, pp.925-933, 2000. ,
Biaccessibility in quadratic Julia sets, Ergod. Th. and Dynam. Sys, vol.20, issue.6, pp.1859-1883, 2000. ,
On biaccessible points in Julia sets of polynomials, Fund. Math, vol.163, issue.3, pp.277-286, 2000. ,
, Conformal Invariants, Topics in Geometric Function Theory, 1973.
The classifiction of critically preperiodic polynomials as dynamical systems, J.Amer.Math.Soc, vol.5, pp.721-762, 1992. ,
The iteration of cubic polynomials. Part I: the global topology of parameter space, Acta Math, vol.160, pp.143-206, 1988. ,
Conjugacies between rational maps and extremal quasiconformal maps, Proc. Amer. Math. Soc, vol.129, issue.7, pp.1949-1953, 2001. ,
Geometrically finite and semi-rational branched coverings of the two-sphere, Trans. Amer. Math. Soc, vol.363, pp.2701-2714, 2011. ,
On a theorem of Rees-Shishikura, Annales de la Facult'es des Sciences de Toulouse, Sr, vol.6, issue.5, pp.981-993, 2012. ,
, Distortion control of conjugacies between quadratic polynomials, vol.53, pp.625-634, 2010.
A characterization of hyperbolic rational maps, Invent. Math, vol.183, pp.451-516, 2011. ,
URL : https://hal.archives-ouvertes.fr/hal-00166325
Hyerbolic-parabolic deformations of rational maps ,
Étude Dynamique des Polynômes Complexes I and II, Publ. Math. Orsay, 1984. ,
On the multiplier of a repelling fixed point, Invent. Math, vol.118, pp.85-108, 1994. ,
Algebraic topology, 2012. ,
Combinatorial continuity in complex polynomial dynamics, Proc. London Math. Soc, vol.91, issue.3, pp.215-248, 2005. ,
Quasi-Additivity Law in conformal geometry, Ann. Math, vol.169, pp.561-193, 2009. ,
, Univalent Functions and Teichmüller Spaces, 1987.
On a lemma of Fatou, Bol.Soc.Bras.Mat, vol.24, pp.1-11, 1993. ,
Periodic orbits, externals rays and the Mandelbrot set: An expository account, Géomé Complexe et Systèmes Dynamiques, Colloque en l'honneur d'Adrien Douady, pp.277-333, 2000. ,
, Dynamics in one complex variable, 2006.
Convergence of rational rays in parameter spaces, The Mandelbrot set, theme and variations, London Mathematical Society Lecture Note Series, vol.274, pp.161-172, 2000. ,
Dynamics of McMullen maps, Adv.Math, vol.229, pp.2525-2577, 2012. ,
Introduction to Riemann Surfaces, 1981. ,
, Classification Theory of Riemann Surfaces, 1970.
An alternative proof of Mañé's theorem on nonexpanding Julia sets, The Mandelbrot set, theme and variations, London Mathematical Society Lecture Note Series, vol.274, pp.265-279, 2000. ,
On the Julia set of semi-hyperbolic rational maps, Chinese J. Contemp. Math, vol.20, issue.4, pp.469-476, 1999. ,
, U d? (x) = 0 if and only if x = 1. By a straightforward calculation, we have 0 < U d? (0) < 1. Now we divide the arguments into two cases
, A d? (1) ? A d? (?) = J d? = ?. If there exists at least one fixed point of U d? in (0, 1), we denote all of them by 0 < x 1 <, particular
, For every d ? 2 and ? ? R, the Julia set J d? is not a Sierpi?sk carpet
, But this contradicts Lemma 5.3. The proofs of Theorems 5.5 and ?? are finished. By computer experiments, it is shown that A d? (1) ? A d? (?) = {z 0 } for ? ? C, Note that if J d? is a Sierpi?ski carpet
, In this section, we divide the parameter space of T d? into the non-escaping locus M d union countably many capture domains. Recall that A d? (1) and A d? (?) are the immediate superattracting basins of 1 and ? respectively
, For each ? ? C * , the following conditions are equivalent: (1) The Julia set J d? of T d? is a quasicircle
In particular, ? k ? A d? (1) if and only if ? l ? A d? (1) ,
, If J d? is a quasicircle, the Fatou set of T d? consists of two simply connected Fatou components A d? (1) and A d? (?) whose common boundary is J d? . Since T d? permutes 1 and ?, by (5.9), it follows that (2) holds and {? 1 , · · · , ? d } lies in a single Fatou component. Applying the Riemann-Hurwitz formula to U d? : A d? (?) ? A d? (?), We first prove
, we have (3) ? (4) ? (5)
Note that T d? : A d? (1) ? A d? (?) is d to 1. We claim that ? k ? A d? (1) for every 0 ? k ? d ? 1. In fact, if not, then 1 ? ? has Specifically ,
, Let D ? := dim H (J ? ) be the Hausdorff dimension of J ? . One can write the corresponding
Mating non-renormalizable quadratic polynomials, Commun. Math. Phys, vol.287, pp.1-40, 2009. ,
Julia sets and complex singularities in hierarchical Ising models, Commun. Math. Phys, vol.141, pp.453-474, 1991. ,
Fractal structure of zeros in hierarchinal models, J. Stat. Phys, vol.33, pp.559-569, 1983. ,
, Systèmes dynamiques holomorphes, pp.39-63, 1983.
Itération des polynômes quadratiques complexes, C. R. Acad. Sci, vol.294, pp.123-126, 1982. ,
, Fractal geometry: Mathematical Foundations and Applications, Second Edition, 2003.
The Nature of Critical Points (Lectures in Theoretical Physics, VIIC) ,
Julia sets, Hausdorff dimension and phase transition, Chaos, Solitons & Fractals, vol.44, pp.871-877, 2011. ,
Yang-Lee zero, Julia sets and their singularity spectra, Phys. Rev, vol.39, pp.4789-4796, 1989. ,
On the dynamics of rational maps, Ann. Sci. École Norm. Sup, issue.4, pp.193-217, 1983. ,
Complex Dynamics and Renormalization, Ann. of Math. Studies, vol.135, 1994. ,
Quasiconformal homeomorphisms and dynamics. II-I. The Teichmüller space of a holomorphic dynamical system, Adv. Math, vol.135, issue.2, pp.351-395, 1998. ,
Geometry and dynamics of quadratic rational maps, With an appendix by the author and Tan Lei, vol.2, pp.37-83, 1993. ,
1/s-expansion for generalized dimensions in a hierarchical s-state Potts model, J. Phys., A: Math. Gen, vol.28, pp.5951-5962, 1995. ,
, Hyperbolic components of McMullen maps. ArXiv: math.DS/1207.0266v1, 2012.
URL : https://hal.archives-ouvertes.fr/hal-01312943
Julia sets and complex singularities of free energies, 2011. ,
On connectivity of Julia sets of Yang-Lee zeros, Commun. Math. Phys, vol.222, pp.319-326, 2001. ,
Feigenbaum Julia sets of singularities of free energy, Ergodic Th. and Dynam. Sys, vol.30, pp.1573-1591, 2010. ,
Topologie locale des méthodes de Newton cubiques, 1997. ,
On capture zones for the family f (z) = z 2 + ?/z 2 , in "Dynamics on the Riemann Sphere: A Bodil Branner Festschrift, European Mathematical Society, pp.121-129, 2006. ,
Repellers for real analytic maps, vol.2, pp.99-107, 1982. ,
Connectivity of the Mandelbrot set for the family of renormalization transformations, Science in China, vol.53, pp.849-862, 2010. ,
Strange objects in the complex plane, Journal of Statistical Physics, pp.443-454, 1983. ,
Renormalization group and critical phenomena I and II, Phys. Rev. B, vol.4, pp.3184-3205, 1971. ,
Statistical theory of equations of state and phase transitions. I. Theory of condensation, Phys. Rev, vol.87, pp.404-419, 1952. ,
Rational maps without Herman rings ,
The Hausdorff dimension of the boundary of the immediate basin of infinity of McMullen maps, 2012. ,
Dynamics of cubic Siegel polynomials, Commun. Math. Phys, vol.206, pp.185-233, 1999. ,