August  2017, 37(7): 3567-3585. doi: 10.3934/dcds.2017153

Singular perturbations of Blaschke products and connectivity of Fatou components

1. 

Inst. Univ. de Matemàtiques i Aplicacions de Castelló (IMAC), Universitat Jaume I, Av. de Vicent Sos Baynat, s/n, 12071 Castelló de la Plana, Spain

2. 

Inst. of Math. Polish Academy of Sciences (IMPAN), ul. Śniadeckich 8, 00-656 Warszawa, Poland

Received  May 2016 Revised  February 2017 Published  April 2017

Fund Project: The author was partially supported by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund, by RedIUM and MINECO (Spain) through the research network MTM2014-55580-REDT, and by the mathematics institute IMAC.

The goal of this paper is to study the family of singular perturbations of Blaschke products given by $B_{a, λ}(z)=z^3\frac{z-a}{1-\overline{a}z}+\frac{λ}{z^2}$. We focus on the study of these rational maps for parameters $a$ in the punctured disk $\mathbb{D}^*$ and $|λ|$ small. We prove that, under certain conditions, all Fatou components of a singularly perturbed Blaschke product $B_{a, λ}$ have finite connectivity but there are components of arbitrarily large connectivity within its dynamical plane. Under the same conditions we prove that the Julia set is the union of countably many Cantor sets of quasicircles and uncountably many point components.

Citation: Canela Jordi. Singular perturbations of Blaschke products and connectivity of Fatou components. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3567-3585. doi: 10.3934/dcds.2017153
References:
[1]

I. N. BakerJ. 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, vol. 132 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1991, Complex analytic dynamical systems.

[3]

P. BlanchardR. L. DevaneyA. 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, vol. 141 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2014.

[5]

J. CanelaN. 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.

[6]

J. CanelaN. Fagella and A. Garijo, Tongues in degree 4 Blaschke products, Nonlinearity, 29 (2016), 3464-3495.  doi: 10.1088/0951-7715/29/11/3464.

[7]

R. L. DevaneyD. 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.

[8]

J. Y. Gao and G. Liu, On connectivity of Fatou components concerning a family of rational maps, Abstr. Appl. Anal. , (2014), Art. ID 621312, 7pp. doi: 10.1155/2014/621312.

[9]

A. GarijoS. 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.

[10]

C. McMullen, Automorphisms of rational maps, in Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986), vol. 10 of Math. Sci. Res. Inst. Publ., Springer, New York, 1988, 31–60. doi: 10.1007/978-1-4613-9602-4_3.

[11]

J. Milnor, Dynamics in One Complex Variable, vol. 160 of Annals of Mathematics Studies, 3rd edition, Princeton University Press, Princeton, NJ, 2006.

[12]

M. Morabito and R. L. Devaney, Limiting Julia sets for singularly perturbed rational maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), 3175-3181.  doi: 10.1142/S0218127408022342.

[13]

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. 

[14]

W. Y. QiuP. RoeschX. G. Wang and Y. C. Yin, Hyperbolic components of McMullen maps, Ann. Sci. Ec. Norm. Supér. (4), 48 (2015), 703-737. 

[15]

W. Y. QiuX. G. Wang and Y. C. Yin, Dynamics of McMullen maps, Adv. Math., 229 (2012), 2525-2577.  doi: 10.1016/j.aim.2011.12.026.

[16]

N. Steinmetz, The formula of Riemann-Hurwitz and iteration of rational functions, Complex Variables Theory Appl., 22 (1993), 203-206. 

[17]

M. Stiemer, Rational maps with Fatou components of arbitrary connectivity number, Comput. Methods Funct. Theory, 7 (2007), 415-427.  doi: 10.1007/BF03321654.

[18]

D. Sullivan, Quasiconformal homeomorphisms and dynamics. Ⅰ. Solution of the Fatou-Julia problem on wandering domains, Ann. of Math. (2), 122 (1985), 401-418.  doi: 10.2307/1971308.

show all references

References:
[1]

I. N. BakerJ. 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, vol. 132 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1991, Complex analytic dynamical systems.

[3]

P. BlanchardR. L. DevaneyA. 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, vol. 141 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2014.

[5]

J. CanelaN. 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.

[6]

J. CanelaN. Fagella and A. Garijo, Tongues in degree 4 Blaschke products, Nonlinearity, 29 (2016), 3464-3495.  doi: 10.1088/0951-7715/29/11/3464.

[7]

R. L. DevaneyD. 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.

[8]

J. Y. Gao and G. Liu, On connectivity of Fatou components concerning a family of rational maps, Abstr. Appl. Anal. , (2014), Art. ID 621312, 7pp. doi: 10.1155/2014/621312.

[9]

A. GarijoS. 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.

[10]

C. McMullen, Automorphisms of rational maps, in Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986), vol. 10 of Math. Sci. Res. Inst. Publ., Springer, New York, 1988, 31–60. doi: 10.1007/978-1-4613-9602-4_3.

[11]

J. Milnor, Dynamics in One Complex Variable, vol. 160 of Annals of Mathematics Studies, 3rd edition, Princeton University Press, Princeton, NJ, 2006.

[12]

M. Morabito and R. L. Devaney, Limiting Julia sets for singularly perturbed rational maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), 3175-3181.  doi: 10.1142/S0218127408022342.

[13]

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. 

[14]

W. Y. QiuP. RoeschX. G. Wang and Y. C. Yin, Hyperbolic components of McMullen maps, Ann. Sci. Ec. Norm. Supér. (4), 48 (2015), 703-737. 

[15]

W. Y. QiuX. G. Wang and Y. C. Yin, Dynamics of McMullen maps, Adv. Math., 229 (2012), 2525-2577.  doi: 10.1016/j.aim.2011.12.026.

[16]

N. Steinmetz, The formula of Riemann-Hurwitz and iteration of rational functions, Complex Variables Theory Appl., 22 (1993), 203-206. 

[17]

M. Stiemer, Rational maps with Fatou components of arbitrary connectivity number, Comput. Methods Funct. Theory, 7 (2007), 415-427.  doi: 10.1007/BF03321654.

[18]

D. Sullivan, Quasiconformal homeomorphisms and dynamics. Ⅰ. Solution of the Fatou-Julia problem on wandering domains, Ann. of Math. (2), 122 (1985), 401-418.  doi: 10.2307/1971308.

Figure 1.  Figure (a) corresponds to the dynamical plane of $ Q_{λ, 3, 2}(z) = z^3+λ/z^2 $ for $ λ = 10^{-4} $ . Figure (b) corresponds to the dynamical plane of $ p_{2, -1}(z) = z^2-1 $ , known as the Basilica. The map $ p_{2, -1} $ has a period 2 superattracting cycle at $ \{0, -1\} $ . Figure (c) corresponds to the dynamical plane of $ f(z) = z^2-1+λ/(z^7(z+1) ^5) $ for $ λ = 10^{-22} $ . This map is a singular perturbation of the polynomial $ p_{2, -1} $ which adds a pole at each point of the superattracting cycle. Figure (d) is a magnification of (c) around the point $ z = 0 $ . The colours are as follows. We use a scaling from yellow to red to plot the basin of attraction of $ z = \infty $ . In Figure (b) we plot the basin of attraction of the cycle $ \{0, -1\} $ in black. In the other figures we may observe an approximation of the Julia set in yellow
Figure 2.  Figures (a) and (b) represent the dynamical plane of the map $ B_{a, λ} $ where $ a = 0.5 $ and $ λ = 3.022× 10^{-5} $ . Figures (c) and (d) represent the dynamical plane of the map $ B_{a, λ} $ where $ a = 0.5 $ and $ λ = 2.8×10^{-5}+8.4× 10^{-7}i $ . These maps correspond to singularly perturbed Blaschke products for which statements a) and b) of Theorem A hold. The colours are as follows. We use a scaling from yellow to red to plot the basin of attraction of $ z = \infty $ . An approximation of the Julia set may be observed in yellow
Figure 3.  The left figure corresponds to the dynamical plane of the map $ B_{a, λ} $ where $ a = 0.5 $ and $ λ = 10^{-5} $ . The right figure is a magnification of the left one. Statement c) of Theorem A holds for this map. Colours are as in Figure 2
Figure 4.  The left figure corresponds to the parameter space of the family $ B_{a, λ} $ for $ a = 0.5 $ , $ Re(λ)\in(-8.7× 10^{-5}, 7.3×10^{-5}) $ and $ Im(λ)\in(-8× 10^{-5}, 8×10^{-5}) $ . The right figure is a magnification of the left one. The colours are as follows. We use a scaling from yellow to red to plot parameters such that $ c_-\in A(\infty) $ and green otherwise
Figure 5.  Scheme of the sets described in the proof of Proposition 2.2
Figure 6.  Scheme of the dynamics described in Theorem 2.5. We draw in red the preimages of zero and in black the critical points
Figure 7.  Summary of the dynamics of $ B_{a, λ} $ described in Proposition 3.1. The triply connected region $ \mathcal{U}_c $ is mapped with degree 4 onto the annulus $ B_{a, λ}(\mathcal{U}_c) $ . The green annular region $ \mathcal{V}_4 $ is mapped with degree 4 to the green annular region $ \mathcal{W}_4 $ . The blue annular region $ \mathcal{V}_3 $ is mapped with degree 3 to the blue annular region $ \mathcal{W}_3 $ . The pallid blue disk $ \mathcal{V}_1 $ is mapped with degree 1 to the region bounded by $ B_{a, λ}(\mathcal{U}_c) $ . The red region $ \mathcal{V}_2 $ is mapped with degree 2 to the full annular region bounded by $ A^*(\infty) $ and $ T_0 $ . Since $ \mathcal{U}_c\subset \mathcal{W}_4 $ , $ \mathcal{V}_4\subset \mathcal{W}_4 $ and either $ B_{a, λ}(\mathcal{U}_c) = A_0 $ or $ B_{a, λ}(\mathcal{U}_c)\subset \mathcal{V}_3\cup \mathcal{V}_2 $
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