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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 |
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.
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, vol. 132 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1991, Complex analytic dynamical systems. |
[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, vol. 141 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2014. |
[5] |
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. |
[6] |
J. Canela, N. 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. 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. |
[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. 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. |
[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. Qiu, P. Roesch, X. G. Wang and Y. C. Yin,
Hyperbolic components of McMullen maps, Ann. Sci. Ec. Norm. Supér. (4), 48 (2015), 703-737.
|
[15] |
W. Y. Qiu, X. 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. 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, vol. 132 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1991, Complex analytic dynamical systems. |
[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, vol. 141 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2014. |
[5] |
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. |
[6] |
J. Canela, N. 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. 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. |
[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. 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. |
[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. Qiu, P. Roesch, X. G. Wang and Y. C. Yin,
Hyperbolic components of McMullen maps, Ann. Sci. Ec. Norm. Supér. (4), 48 (2015), 703-737.
|
[15] |
W. Y. Qiu, X. 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. |






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