# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2022051
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## Achievable connectivities of Fatou components for a family of singular perturbations

 1 Institut Universitari de Matemàtiques i Aplicacions de Castelló (IMAC), Universitat Jaume I, 12071 Castelló de la Plana, Spain 2 Departament de Matemàtiques i Informàtica at Universitat de Barcelona, and Barcelona Graduate School of Mathematics, 08007 Barcelona, Spain

Corresponding author: Dan Paraschiv

Received  February 2021 Early access April 2022

In this paper we study the connectivity of Fatou components for maps in a large family of singular perturbations. We prove that, for some parameters inside the family, the dynamical planes for the corresponding maps present Fatou components of arbitrarily large connectivity and we determine precisely these connectivities. In particular, these results extend the ones obtained in [5,6].

Citation: Jordi Canela, Xavier Jarque, Dan Paraschiv. Achievable connectivities of Fatou components for a family of singular perturbations. Discrete and Continuous Dynamical Systems, doi: 10.3934/dcds.2022051
##### References:
 [1] I. N. Baker, J. Kotus and Y. N. Lü, Iterates of meromorphic functions. III. Preperiodic domains, Ergodic Theory Dynam. Systems., 11 (1991), 603-618.  doi: 10.1017/S0143385700006386. [2] A. F. Beardon, Iteration of Rational Functions, Complex analytic dynamical systems. Graduate Texts in Mathematics, 132. Springer-Verlag, New York, 1991. [3] P. Blanchard, R. L. Devaney, A. Garijo and E. D. Russell, A generalized version of the McMullen domain, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), 2309-2318.  doi: 10.1142/S0218127408021725. [4] B. Branner and N. Fagella, Quasiconformal Surgery in Holomorphic Dynamics, Cambridge Studies in Advanced Mathematics, vol. 141, Cambridge University Press, 2014. [5] J. Canela, Singular perturbations of Blaschke products and connectivity of Fatou components, Discrete Contin. Dyn. Syst., 37 (2017), 3567-3585.  doi: 10.3934/dcds.2017153. [6] J. Canela, Rational maps with Fatou components of arbitrarily large connectivity, J. Math. Anal. Appl., 462 (2018), 36-56.  doi: 10.1016/j.jmaa.2018.01.061. [7] J. Canela, N. Fagella and A. Garijo, On a family of rational perturbations of the doubling map, J. Difference Equ. Appl., 21 (2015), 715-741.  doi: 10.1080/10236198.2015.1050387. [8] L. Carleson and T. W. Gamelin, Complex Dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4364-9. [9] R. L. Devaney, D. M. Look and D. Uminsky, The escape trichotomy for singularly perturbed rational maps, Indiana Univ. Math. J., 54 (2005), 1621-1634.  doi: 10.1512/iumj.2005.54.2615. [10] 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. [11] A. Garijo, S. M. Marotta and E. D. Russell, Singular perturbations in the quadratic family with multiple poles, J. Difference Equ. Appl., 19 (2013), 124-145.  doi: 10.1080/10236198.2011.630668. [12] I. Hiroyuki, Personal communication. [13] C. McMullen, Automorphisms of rational maps, Holomorphic Functions and Moduli, Vol. I (Berkeley, CA, 1986), Math. Sci. Res. Inst. Publ., Springer, New York, 10 (1988), 31–60. doi: 10.1007/978-1-4613-9602-4_3. [14] J. Milnor, Dynamics in One Complex Variable, 3rd edition, Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006. [15] ——, Cubic polynomial maps with periodic critical orbit. I, Complex Dynamics, A. K Peters, Wellesley, MA, (2009), 333–411. [16] J. Y. Qiao and J. Y. Gao, The connectivity numbers of Fatou components of rational mappings, Acta Math. Sinica (Chin. Ser.), 47 (2004), 625-628. [17] P. Roesch, Hyperbolic components of polynomials with a fixed critical point of maximal order, Ann. Sci. École Norm. Sup., 40 (2007), 901-949.  doi: 10.1016/j.ansens.2007.10.001. [18] N. Steinmetz, The formula of Riemann-Hurwitz and iteration of rational functions, Complex Variables Theory Appl., 22 (1993), 203-206.  doi: 10.1080/17476939308814660. [19] D. Sullivan, Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou-Julia problem on wandering domains, Ann. of Math., 122 (1985), 401-418.  doi: 10.2307/1971308.

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##### References:
 [1] I. N. Baker, J. Kotus and Y. N. Lü, Iterates of meromorphic functions. III. Preperiodic domains, Ergodic Theory Dynam. Systems., 11 (1991), 603-618.  doi: 10.1017/S0143385700006386. [2] A. F. Beardon, Iteration of Rational Functions, Complex analytic dynamical systems. Graduate Texts in Mathematics, 132. Springer-Verlag, New York, 1991. [3] P. Blanchard, R. L. Devaney, A. Garijo and E. D. Russell, A generalized version of the McMullen domain, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), 2309-2318.  doi: 10.1142/S0218127408021725. [4] B. Branner and N. Fagella, Quasiconformal Surgery in Holomorphic Dynamics, Cambridge Studies in Advanced Mathematics, vol. 141, Cambridge University Press, 2014. [5] J. Canela, Singular perturbations of Blaschke products and connectivity of Fatou components, Discrete Contin. Dyn. Syst., 37 (2017), 3567-3585.  doi: 10.3934/dcds.2017153. [6] J. Canela, Rational maps with Fatou components of arbitrarily large connectivity, J. Math. Anal. Appl., 462 (2018), 36-56.  doi: 10.1016/j.jmaa.2018.01.061. [7] J. Canela, N. Fagella and A. Garijo, On a family of rational perturbations of the doubling map, J. Difference Equ. Appl., 21 (2015), 715-741.  doi: 10.1080/10236198.2015.1050387. [8] L. Carleson and T. W. Gamelin, Complex Dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4364-9. [9] R. L. Devaney, D. M. Look and D. Uminsky, The escape trichotomy for singularly perturbed rational maps, Indiana Univ. Math. J., 54 (2005), 1621-1634.  doi: 10.1512/iumj.2005.54.2615. [10] 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. [11] A. Garijo, S. M. Marotta and E. D. Russell, Singular perturbations in the quadratic family with multiple poles, J. Difference Equ. Appl., 19 (2013), 124-145.  doi: 10.1080/10236198.2011.630668. [12] I. Hiroyuki, Personal communication. [13] C. McMullen, Automorphisms of rational maps, Holomorphic Functions and Moduli, Vol. I (Berkeley, CA, 1986), Math. Sci. Res. Inst. Publ., Springer, New York, 10 (1988), 31–60. doi: 10.1007/978-1-4613-9602-4_3. [14] J. Milnor, Dynamics in One Complex Variable, 3rd edition, Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006. [15] ——, Cubic polynomial maps with periodic critical orbit. I, Complex Dynamics, A. K Peters, Wellesley, MA, (2009), 333–411. [16] J. Y. Qiao and J. Y. Gao, The connectivity numbers of Fatou components of rational mappings, Acta Math. Sinica (Chin. Ser.), 47 (2004), 625-628. [17] P. Roesch, Hyperbolic components of polynomials with a fixed critical point of maximal order, Ann. Sci. École Norm. Sup., 40 (2007), 901-949.  doi: 10.1016/j.ansens.2007.10.001. [18] N. Steinmetz, The formula of Riemann-Hurwitz and iteration of rational functions, Complex Variables Theory Appl., 22 (1993), 203-206.  doi: 10.1080/17476939308814660. [19] D. Sullivan, Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou-Julia problem on wandering domains, Ann. of Math., 122 (1985), 401-418.  doi: 10.2307/1971308.
Dynamical planes of the family $B_{n, d, a, \lambda}(z)$ for $d = 3$. The top-left figure corresponds to $n = 2$ and $\lambda = 2\cdot10^{-8}$; the top-right corresponds to $n = 3$ and $\lambda = -5\cdot10^{-8}$; the bottom-left figure corresponds to $n = 4$ and $\lambda = -6.3\cdot10^{-9}$; and bottom-right corresponds to $n = 5$ and $\lambda = -1.2\cdot10^{-10}$. In all cases we can see the triply connected regions (where the critical point $\nu_\lambda$ lies) and their eventual preimages, which are Fatou components with increasing connectivity
Left figure illustrates the dynamical planes of $M_{n, a}$ for $n = 2$ and $a = (0.9+0.6i)$. Right picture illustrates the dynamical plane of the (perturbed) family $\mathcal{S}_{n, d, \lambda}$ when the unperturbed map is precisely $M_{2, a}$, and the pertubation corresponds to $d = 3$ and $\lambda = -10^{-7}$. We can see in the right figure the triply connected Fatou component which contains $\nu_{ \lambda}$ and its eventual preimages with higher connectivity
Partition of the dynamical plane with respect to $\mathcal{A}^{*}_{ \lambda}(\infty)$, $A_{ \lambda}$, $T_{ \lambda}$, and $D_{ \lambda}$, described in Proposition 2. Blue and purple points denote zeros and critical points, respectively
Partitions of the dynamical plane introduced in Theorem 3.5
Description of the situation in the proof of Lemma 4.2, where $k = 2$ and $W_2 \subset \mathcal{U}_d$. In this case, $B_2 = B_{2}^{ \text{out}} \cup \overline{ A_{ \lambda}} \cup B_{2}^{ \text{in}}$
The top figures correspond to the possible cases of $\nu_{ \lambda}$ lying in a neighbourhood of $\partial^ \text{Int} A_{m, \lambda}$. The top figures correspond to the possible cases of $\nu_{ \lambda}$ lying in a neighbourhood of $\partial^ \text{Ext} A_{m, \lambda}$
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