October  2019, 39(10): 5659-5681. doi: 10.3934/dcds.2019248

Slices of parameter spaces of generalized Nevanlinna functions

1. 

Department of Mathematics, Engineering and Computer Science, Laguardia Community College, CUNY, 31-10 Thomson Ave, Long Island City, NY 11101, USA

2. 

Mathematics Program, CUNY Graduate Center, 365 Fifth Ave, New York, NY 10016, USA

The first author is supported by PSC-CUNY

Received  May 2018 Revised  January 2019 Published  July 2019

Fund Project: The first author is supported by PSC-CUNY.

In the early 1980's, computers made it possible to observe that in complex dynamics, one often sees dynamical behavior reflected in parameter space and vice versa. This duality was first exploited by Douady, Hubbard and their students in early work on rational maps.

Here, we extend these ideas to transcendental functions.

In [16], it was shown that for the tangent family, $ \{ \lambda \tan z \} $, the hyperbolic components meet at a parameter $ \lambda^* $ such that $ f_{ \lambda^*}^n( \lambda^*i) = \infty $ for some $ n $. The behavior there reflects the dynamic behavior of $ \lambda^* \tan z $ at infinity. In Part 1. we show that this duality extends to a more general class of transcendental meromorphic functions $ \{f_{\lambda}\} $ for which infinity is not an asymptotic value. In particular, we show that in "dynamically natural" one-dimensional slices of parameter space, there are "hyperbolic-like" components $ \Omega $ with a unique distinguished boundary point such that for $ \lambda \in \Omega $, the dynamics of $ f_\lambda $ reflect the behavior of $ f_\lambda $ at infinity. Our main result is that every parameter point $ \lambda $ in such a slice for which the iterate of the asymptotic value of $ f_\lambda $ is a pole is such a distinguished boundary point.

In the second part of the paper, we apply this result to the families $ \lambda \tan^p z^q $, $ p, q \in \mathbb Z^+ $, to prove that all hyperbolic components of period greater than $ 1 $ are bounded.

Citation: Tao Chen, Linda Keen. Slices of parameter spaces of generalized Nevanlinna functions. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5659-5681. doi: 10.3934/dcds.2019248
References:
[1]

I. N. BakerJ. Kotus and Y. Lü, Iterates of meromorphic functions Ⅱ: Examples of wandering domains, J. London Math. Soc., 42 (1990), 267-278.  doi: 10.1112/jlms/s2-42.2.267.

[2]

I. N. BakerJ. Kotus and Y. Lü, Iterates of meromorphic functions Ⅰ, Ergodic Th. and Dyn. Sys., 11 (1991), 241-248.  doi: 10.1017/S014338570000612X.

[3]

I. N. BakerJ. Kotus and Y. Lü, Iterates of meromorphic functions Ⅲ: Preperiodic domains, Ergodic Th. and Dyn. Sys., 11 (1991), 603-618.  doi: 10.1017/S0143385700006386.

[4]

I. N. BakerJ. Kotus and Y. Lü, Iterates of meromorphic functions Ⅳ: Critically finite functions, Results in Mathematics, 22 (1992), 651-656.  doi: 10.1007/BF03323112.

[5]

W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc., 29 (1993), 151-188.  doi: 10.1090/S0273-0979-1993-00432-4.

[6]

W. Bergweiler and A. Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoamericana, 11 (1995), 355-373.  doi: 10.4171/RMI/176.

[7]

W. Bergweiler and J. Kotus, On the Hausdorff dimension of the escaping set of certain meromorphic functions, Trans. Amer. Math. Soc., 364 (2012), 5369-5394.  doi: 10.1090/S0002-9947-2012-05514-0.

[8]

R. L. DevaneyN. Fagella and X. Jarque, Hyperbolic components of the complex exponential family, Fundamenta Mathematicae, 174 (2002), 193-215.  doi: 10.4064/fm174-3-1.

[9]

B. Devaney and L. Keen, Dynamics of meromorphic maps: Maps with polynomial schwarzian derivative, Annales Scientifiques de l'Ecole Normale Superieure, 22 (1989), 55-79.  doi: 10.24033/asens.1575.

[10]

A. Douady and J. H. Hubbard, Étude Dynamique des Polyn es Complexes, Université de Paris-Sud, Département de Mathématiques, Orsay, 1984.

[11]

A. Eremenko and A. Gabrielov, Analytic continuation of eigenvalues of a quartic oscillator, Comm. Math. Phys., 287 (2009), 431-457.  doi: 10.1007/s00220-008-0663-6.

[12]

N. Fagella and A. Garijo, The parameter planes of $\lambda z^me^z$ for $m\geq 2$, Commun. Math. Phys., 273 (2007), 755-783.  doi: 10.1007/s00220-007-0265-8.

[13]

N. Fagella and L. Keen, Stable components in the parameter plane of meromorphic functions of finite type, Preprint, http://arXiv.org/abs/1702.06563. Submitted.

[14]

L. Goldberg and L. Keen, A finiteness theorem for a dynamical class of entire functions, Ergodic Theory Dynam. Systems, 6 (1986), 183-192.  doi: 10.1017/S0143385700003394.

[15]

E. Hille, Ordinary Differential Equations in the Complex Domain, John Wiley and Sons, New York, 1976.

[16]

L. Keen and J. Kotus, Dynamics of the family of $\lambda \tan z$., Conformal Geometry and Dynamics, 1 (1997), 28-57.  doi: 10.1090/S1088-4173-97-00017-9.

[17]

A. I. Markushevich, Theory of Functions of a Complex Variable, Vol. Ⅰ, Ⅱ, Ⅲ, Revised English edition, translated and edited by Richard A. Silverman, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967.

[18]

C. T. McMullen, Complex Dynamics and Renormalization, Annals of Mathematics Studies, Vol. 135, Princeton University Press, 1994.

[19]

J. Milnor, Dynamics in One Complex Variable, Third Edition, AM(160), Princeton University Press, 2006.

[20]

R. Nevanlinna, Über Riemannsche Flächen mit endlich vielen Windungspunkten, Acta Math., 58 (1932), 295-373.  doi: 10.1007/BF02547780.

[21]

R. Nevanlinna, Analytic Functions, Translated from the second edition by Philip Emig. Die Grundlehren der mathematischen Wissenschaften, Band 162, Springer, Berlin, Heidelberg, and New York, 1970.

[22]

L. Rempe-Gillen, Dynamics of Exponential Maps, Ph.D. thesis, Christian-Albrechts-Universität Kiel, 2003.

[23]

P. J. Rippon and G. M. Stallard, Iteration of a class of hyperbolic meromorphic functions, Proc. Amer. Math. Soc., 127 (1999), 3251-3258.  doi: 10.1090/S0002-9939-99-04942-4.

[24]

D. Schleicher, Attracting dynamics of exponential maps, Ann. Acad. Sci. Fenn. Math., 28 (2003), 3-34. 

[25]

L. Tan, Similarity between the Mandelbrot set and Julia sets, Comm. Math. Phys., 134 (1990), 587-617.  doi: 10.1007/BF02098448.

[26]

J. Zheng, Dynamics of hyperbolic meromorphic functions, Discrete Contin. Dyn. Syst., 35 (2015), 2273-2298.  doi: 10.3934/dcds.2015.35.2273.

show all references

References:
[1]

I. N. BakerJ. Kotus and Y. Lü, Iterates of meromorphic functions Ⅱ: Examples of wandering domains, J. London Math. Soc., 42 (1990), 267-278.  doi: 10.1112/jlms/s2-42.2.267.

[2]

I. N. BakerJ. Kotus and Y. Lü, Iterates of meromorphic functions Ⅰ, Ergodic Th. and Dyn. Sys., 11 (1991), 241-248.  doi: 10.1017/S014338570000612X.

[3]

I. N. BakerJ. Kotus and Y. Lü, Iterates of meromorphic functions Ⅲ: Preperiodic domains, Ergodic Th. and Dyn. Sys., 11 (1991), 603-618.  doi: 10.1017/S0143385700006386.

[4]

I. N. BakerJ. Kotus and Y. Lü, Iterates of meromorphic functions Ⅳ: Critically finite functions, Results in Mathematics, 22 (1992), 651-656.  doi: 10.1007/BF03323112.

[5]

W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc., 29 (1993), 151-188.  doi: 10.1090/S0273-0979-1993-00432-4.

[6]

W. Bergweiler and A. Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoamericana, 11 (1995), 355-373.  doi: 10.4171/RMI/176.

[7]

W. Bergweiler and J. Kotus, On the Hausdorff dimension of the escaping set of certain meromorphic functions, Trans. Amer. Math. Soc., 364 (2012), 5369-5394.  doi: 10.1090/S0002-9947-2012-05514-0.

[8]

R. L. DevaneyN. Fagella and X. Jarque, Hyperbolic components of the complex exponential family, Fundamenta Mathematicae, 174 (2002), 193-215.  doi: 10.4064/fm174-3-1.

[9]

B. Devaney and L. Keen, Dynamics of meromorphic maps: Maps with polynomial schwarzian derivative, Annales Scientifiques de l'Ecole Normale Superieure, 22 (1989), 55-79.  doi: 10.24033/asens.1575.

[10]

A. Douady and J. H. Hubbard, Étude Dynamique des Polyn es Complexes, Université de Paris-Sud, Département de Mathématiques, Orsay, 1984.

[11]

A. Eremenko and A. Gabrielov, Analytic continuation of eigenvalues of a quartic oscillator, Comm. Math. Phys., 287 (2009), 431-457.  doi: 10.1007/s00220-008-0663-6.

[12]

N. Fagella and A. Garijo, The parameter planes of $\lambda z^me^z$ for $m\geq 2$, Commun. Math. Phys., 273 (2007), 755-783.  doi: 10.1007/s00220-007-0265-8.

[13]

N. Fagella and L. Keen, Stable components in the parameter plane of meromorphic functions of finite type, Preprint, http://arXiv.org/abs/1702.06563. Submitted.

[14]

L. Goldberg and L. Keen, A finiteness theorem for a dynamical class of entire functions, Ergodic Theory Dynam. Systems, 6 (1986), 183-192.  doi: 10.1017/S0143385700003394.

[15]

E. Hille, Ordinary Differential Equations in the Complex Domain, John Wiley and Sons, New York, 1976.

[16]

L. Keen and J. Kotus, Dynamics of the family of $\lambda \tan z$., Conformal Geometry and Dynamics, 1 (1997), 28-57.  doi: 10.1090/S1088-4173-97-00017-9.

[17]

A. I. Markushevich, Theory of Functions of a Complex Variable, Vol. Ⅰ, Ⅱ, Ⅲ, Revised English edition, translated and edited by Richard A. Silverman, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967.

[18]

C. T. McMullen, Complex Dynamics and Renormalization, Annals of Mathematics Studies, Vol. 135, Princeton University Press, 1994.

[19]

J. Milnor, Dynamics in One Complex Variable, Third Edition, AM(160), Princeton University Press, 2006.

[20]

R. Nevanlinna, Über Riemannsche Flächen mit endlich vielen Windungspunkten, Acta Math., 58 (1932), 295-373.  doi: 10.1007/BF02547780.

[21]

R. Nevanlinna, Analytic Functions, Translated from the second edition by Philip Emig. Die Grundlehren der mathematischen Wissenschaften, Band 162, Springer, Berlin, Heidelberg, and New York, 1970.

[22]

L. Rempe-Gillen, Dynamics of Exponential Maps, Ph.D. thesis, Christian-Albrechts-Universität Kiel, 2003.

[23]

P. J. Rippon and G. M. Stallard, Iteration of a class of hyperbolic meromorphic functions, Proc. Amer. Math. Soc., 127 (1999), 3251-3258.  doi: 10.1090/S0002-9939-99-04942-4.

[24]

D. Schleicher, Attracting dynamics of exponential maps, Ann. Acad. Sci. Fenn. Math., 28 (2003), 3-34. 

[25]

L. Tan, Similarity between the Mandelbrot set and Julia sets, Comm. Math. Phys., 134 (1990), 587-617.  doi: 10.1007/BF02098448.

[26]

J. Zheng, Dynamics of hyperbolic meromorphic functions, Discrete Contin. Dyn. Syst., 35 (2015), 2273-2298.  doi: 10.3934/dcds.2015.35.2273.

Figure 1.  The map $ g_{ \lambda} $ on parameter space. $ S $ is a sector inside all the asymptotic tracts $ A_{ \lambda} $, $ \lambda \in V $. Note that $ g( \lambda^*) = \infty $
Figure 2.  The dynamic plane for $ f_{ \lambda} $. The region $ f_\lambda^{n}( {\mathcal T}) $ is contained inside $ {\mathcal T} $
Figure 3.  The parameter plane for $ \lambda \tan^2 z^3 $
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