# American Institute of Mathematical Sciences

June  2012, 32(6): 2027-2039. doi: 10.3934/dcds.2012.32.2027

## Density of orbits in laminations and the space of critical portraits

 1 Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294-1170, United States, United States 2 Department of Mathematics, Huntingdon College, Montgomery, AL 36106-2114, United States

Received  April 2011 Revised  November 2011 Published  February 2012

Thurston introduced $\sigma_d$-invariant laminations (where $\sigma_d(z)$ coincides with $z^d:\mathbb{S}\to \mathbb{S}$, $d\ge 2$). He defined wandering $k$-gons as sets $T\subset \mathbb{S}$ such that $\sigma_d^n(T)$ consists of $k\ge 3$ distinct points for all $n\ge 0$ and the convex hulls of all the sets $\sigma_d^n(T)$ in the plane are pairwise disjoint. Thurston proved that $\sigma_2$ has no wandering $k$-gons and posed the problem of their existence for $\sigma_d$, $d\ge 3$.
Call a lamination with wandering $k$-gons a WT-lamination. Denote the set of cubic critical portraits by $\mathcal{A}_3$. A critical portrait, compatible with a WT-lamination, is called a WT-critical portrait; let $\mathcal{WT}_3$ be the set of all of them. It was recently shown by the authors that cubic WT-laminations exist and cubic WT-critical portraits, defining polynomials with condense orbits of vertices of order three in their dendritic Julia sets, are dense and locally uncountable in $\mathcal{A}_3$ ($D\subset X$ is condense in $X$ if $D$ intersects every subcontinuum of $X$). Here we show that $\mathcal{WT}_3$ is a dense first category subset of $\mathcal{A}_3$, that critical portraits, whose laminations have a condense orbit in the topological Julia set, form a residual subset of $\mathcal{A}_3$, and that the existence of a condense orbit in the Julia set $J$ implies that $J$ is locally connected.
Citation: Alexander Blokh, Clinton Curry, Lex Oversteegen. Density of orbits in laminations and the space of critical portraits. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2027-2039. doi: 10.3934/dcds.2012.32.2027
##### References:
 [1] B. Bielefeld, Y. Fisher and J. Hubbard, The classification of critically preperiodic polynomials as dynamical systems, Journal AMS, 5 (1992), 721-762. [2] A. Blokh, C. Curry and L. Oversteegen, Cubic critical portraits and polynomials with wandering gaps,, preprint, (). [3] A. Blokh, C. Curry and L. Oversteegen, Locally connected models for Julia sets, Advances in Mathematics, 226 (2011), 1621-1661. [4] A. Blokh, R. Fokkink, J. Mayer, L. Oversteegen and E. Tymchatyn, Fixed point theorems for plane continua with applications,, preprint, (). [5] A. Blokh and G. Levin, An inequality for laminations, Julia sets and "growing trees'', Erg. Th. and Dyn. Sys., 22 (2002), 63-97. [6] A. Blokh and L. Oversteegen, Monotone images of Cremer Julia sets, Houston J. Math., 36 (2010), 469-476. [7] A. Blokh and L. Oversteegen, Wandering gaps for weakly hyperbolic cubic polynomials, in, "Complex Dynamics'' (ed. D. Schleicher), A K Peters, Wellesley, MA, (2009), 139-168. [8] A. Douady, Descriptions of compact sets in $\bbc$, in, "Topological Methods in Modern Mathematics'' (Stony Brook, NY, 1991) (eds. L. R. Goldberg and A. V. Phillips), Publish or Perish, Houston, TX, (1993), 429-465. [9] A. Douady and J. H. Hubbard, "Étude Dynamique des Polynômes Complexes," Part I Publications Mathématiques d'Orsay, 84-2, Université de Paris-Sud, Département de Mathématiques, Orsay, 1984. [10] A. Douady and J. H. Hubbard, "Étude Dynamique des Polynômes Complexes," Part II, Publications Mathématiques d'Orsay, 85-4, Université de Paris-Sud, Département de Mathématiques, Orsay, 1985. [11] Y. Fisher, "The Classification of Critically Preperiodic Polynomials,'' Ph.D thesis, Cornell University, 1989. [12] L. Goldberg and J. Milnor, Fixed points of polynomial maps. II: Fixed point portraits, Ann. Scient. École Norm. Sup. (4), 26 (1993), 51-98. [13] J. Kiwi, Wandering orbit portraits, Trans. Amer. Math. Soc., 354 (2002), 1473-1485. doi: 10.1090/S0002-9947-01-02896-3. [14] J. Kiwi, $\mathbb R$eal laminations and the topological dynamics of complex polynomials, Advances in Math., 184 (2004), 207-267. doi: 10.1016/S0001-8708(03)00144-0. [15] J. Kiwi, Combinatorial continuity in complex polynomial dynamics, Proc. London Math. Soc. (3), 91 (2005), 215-248. [16] O. Kozlovski and S. van Strien, Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials, Proc. Lond. Math. Soc. (3), 99 (2009), 275-296. doi: 10.1112/plms/pdn055. [17] G. Levin, On backward stability of holomorphic dynamical systems, Fundamenta Mathematicae, 158 (1998), 97-107. [18] J. Milnor, "Dynamics in One Complex Variable,'' 3rd edition, Annals of Mathematics Studies, 160, Princeton University Press, Princeton, 2006. [19] S. B. Nadler, Jr., "Continuum Theory. An Introduction,'' Monographs and Textbooks in Pure and Applied Mathematics, 158, Marcel Dekker, Inc., New York, 1992. [20] R. Pérez-Marco, Fixed points and circle maps, Acta Math., 179 (1997), 243-294. [21] A. Poirier, Critical portraits for postcritically finite polynomials, Fund. Math., 203 (2009), 107-163. doi: 10.4064/fm203-2-2. [22] J. Rogers, Jr., Singularities in the boundaries of local Siegel disks, Erg. Th. and Dyn. Syst., 12 (1992), 803-821. [23] P. Roesch and Y. Yin, The boundary of bounded polynomial Fatou components, C. R. Math. Acad. Sci. Paris, 346 (2008), 877-880. [24] W. Thurston, The combinatorics of iterated rational maps, in, "Complex Dynamics'' (ed. D. Schleicher), A K Peters, Wellesley, MA, (2009), 1-108.

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##### References:
 [1] B. Bielefeld, Y. Fisher and J. Hubbard, The classification of critically preperiodic polynomials as dynamical systems, Journal AMS, 5 (1992), 721-762. [2] A. Blokh, C. Curry and L. Oversteegen, Cubic critical portraits and polynomials with wandering gaps,, preprint, (). [3] A. Blokh, C. Curry and L. Oversteegen, Locally connected models for Julia sets, Advances in Mathematics, 226 (2011), 1621-1661. [4] A. Blokh, R. Fokkink, J. Mayer, L. Oversteegen and E. Tymchatyn, Fixed point theorems for plane continua with applications,, preprint, (). [5] A. Blokh and G. Levin, An inequality for laminations, Julia sets and "growing trees'', Erg. Th. and Dyn. Sys., 22 (2002), 63-97. [6] A. Blokh and L. Oversteegen, Monotone images of Cremer Julia sets, Houston J. Math., 36 (2010), 469-476. [7] A. Blokh and L. Oversteegen, Wandering gaps for weakly hyperbolic cubic polynomials, in, "Complex Dynamics'' (ed. D. Schleicher), A K Peters, Wellesley, MA, (2009), 139-168. [8] A. Douady, Descriptions of compact sets in $\bbc$, in, "Topological Methods in Modern Mathematics'' (Stony Brook, NY, 1991) (eds. L. R. Goldberg and A. V. Phillips), Publish or Perish, Houston, TX, (1993), 429-465. [9] A. Douady and J. H. Hubbard, "Étude Dynamique des Polynômes Complexes," Part I Publications Mathématiques d'Orsay, 84-2, Université de Paris-Sud, Département de Mathématiques, Orsay, 1984. [10] A. Douady and J. H. Hubbard, "Étude Dynamique des Polynômes Complexes," Part II, Publications Mathématiques d'Orsay, 85-4, Université de Paris-Sud, Département de Mathématiques, Orsay, 1985. [11] Y. Fisher, "The Classification of Critically Preperiodic Polynomials,'' Ph.D thesis, Cornell University, 1989. [12] L. Goldberg and J. Milnor, Fixed points of polynomial maps. II: Fixed point portraits, Ann. Scient. École Norm. Sup. (4), 26 (1993), 51-98. [13] J. Kiwi, Wandering orbit portraits, Trans. Amer. Math. Soc., 354 (2002), 1473-1485. doi: 10.1090/S0002-9947-01-02896-3. [14] J. Kiwi, $\mathbb R$eal laminations and the topological dynamics of complex polynomials, Advances in Math., 184 (2004), 207-267. doi: 10.1016/S0001-8708(03)00144-0. [15] J. Kiwi, Combinatorial continuity in complex polynomial dynamics, Proc. London Math. Soc. (3), 91 (2005), 215-248. [16] O. Kozlovski and S. van Strien, Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials, Proc. Lond. Math. Soc. (3), 99 (2009), 275-296. doi: 10.1112/plms/pdn055. [17] G. Levin, On backward stability of holomorphic dynamical systems, Fundamenta Mathematicae, 158 (1998), 97-107. [18] J. Milnor, "Dynamics in One Complex Variable,'' 3rd edition, Annals of Mathematics Studies, 160, Princeton University Press, Princeton, 2006. [19] S. B. Nadler, Jr., "Continuum Theory. An Introduction,'' Monographs and Textbooks in Pure and Applied Mathematics, 158, Marcel Dekker, Inc., New York, 1992. [20] R. Pérez-Marco, Fixed points and circle maps, Acta Math., 179 (1997), 243-294. [21] A. Poirier, Critical portraits for postcritically finite polynomials, Fund. Math., 203 (2009), 107-163. doi: 10.4064/fm203-2-2. [22] J. Rogers, Jr., Singularities in the boundaries of local Siegel disks, Erg. Th. and Dyn. Syst., 12 (1992), 803-821. [23] P. Roesch and Y. Yin, The boundary of bounded polynomial Fatou components, C. R. Math. Acad. Sci. Paris, 346 (2008), 877-880. [24] W. Thurston, The combinatorics of iterated rational maps, in, "Complex Dynamics'' (ed. D. Schleicher), A K Peters, Wellesley, MA, (2009), 1-108.
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