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 & Continuous Dynamical Systems - A, 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

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

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[15]

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

[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. Google Scholar

[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. Google Scholar

[18]

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

[19]

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

[20]

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

[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. Google Scholar

[22]

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

[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. Google Scholar

[24]

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

[25]

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

[26]

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

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

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

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[15]

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

[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. Google Scholar

[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. Google Scholar

[18]

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

[19]

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

[20]

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

[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. Google Scholar

[22]

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

[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. Google Scholar

[24]

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

[25]

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

[26]

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

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 $
[1]

Jian-Hua Zheng. Dynamics of hyperbolic meromorphic functions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2273-2298. doi: 10.3934/dcds.2015.35.2273

[2]

Zuxing Xuan. On conformal measures of parabolic meromorphic functions. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 249-257. doi: 10.3934/dcdsb.2015.20.249

[3]

Agnieszka Badeńska. Measure rigidity for some transcendental meromorphic functions. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2375-2402. doi: 10.3934/dcds.2012.32.2375

[4]

Shahar Nevo, Xuecheng Pang and Lawrence Zalcman. Picard-Hayman behavior of derivatives of meromorphic functions with multiple zeros. Electronic Research Announcements, 2006, 12: 37-43.

[5]

Božidar Jovanović, Vladimir Jovanović. Virtual billiards in pseudo–euclidean spaces: Discrete hamiltonian and contact integrability. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5163-5190. doi: 10.3934/dcds.2017224

[6]

Hiroki Sumi. Dynamics of postcritically bounded polynomial semigroups I: Connected components of the Julia sets. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1205-1244. doi: 10.3934/dcds.2011.29.1205

[7]

Anton Petrunin. Harmonic functions on Alexandrov spaces and their applications. Electronic Research Announcements, 2003, 9: 135-141.

[8]

Feng Luo. Geodesic length functions and Teichmuller spaces. Electronic Research Announcements, 1996, 2: 34-41.

[9]

Janina Kotus, Mariusz Urbański. The dynamics and geometry of the Fatou functions. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 291-338. doi: 10.3934/dcds.2005.13.291

[10]

Robert Azencott, Yutheeka Gadhyan. Accurate parameter estimation for coupled stochastic dynamics. Conference Publications, 2009, 2009 (Special) : 44-53. doi: 10.3934/proc.2009.2009.44

[11]

Qing Hong, Guorong Hu. Molecular decomposition and a class of Fourier multipliers for bi-parameter modulation spaces. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3103-3120. doi: 10.3934/cpaa.2019139

[12]

Rafael De La Llave, R. Obaya. Regularity of the composition operator in spaces of Hölder functions. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 157-184. doi: 10.3934/dcds.1999.5.157

[13]

Hugo Beirão da Veiga. Elliptic boundary value problems in spaces of continuous functions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 43-52. doi: 10.3934/dcdss.2016.9.43

[14]

Wolfgang Arendt, Patrick J. Rabier. Linear evolution operators on spaces of periodic functions. Communications on Pure & Applied Analysis, 2009, 8 (1) : 5-36. doi: 10.3934/cpaa.2009.8.5

[15]

Feng Rao, Carlos Castillo-Chavez, Yun Kang. Dynamics of a stochastic delayed Harrison-type predation model: Effects of delay and stochastic components. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1401-1423. doi: 10.3934/mbe.2018064

[16]

Jaeyoo Choy, Hahng-Yun Chu. On the dynamics of flows on compact metric spaces. Communications on Pure & Applied Analysis, 2010, 9 (1) : 103-108. doi: 10.3934/cpaa.2010.9.103

[17]

F. Zeyenp Sargut, H. Edwin Romeijn. Capacitated requirements planning with pricing flexibility and general cost and revenue functions. Journal of Industrial & Management Optimization, 2007, 3 (1) : 87-98. doi: 10.3934/jimo.2007.3.87

[18]

Dianmo Li, Zengxiang Gao, Zufei Ma, Baoyu Xie, Zhengjun Wang. Two general models for the simulation of insect population dynamics. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 623-628. doi: 10.3934/dcdsb.2004.4.623

[19]

Kais Hamza, Fima C. Klebaner, Olivia Mah. Volatility in options formulae for general stochastic dynamics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 435-446. doi: 10.3934/dcdsb.2014.19.435

[20]

Aylin Aydoğdu, Sean T. McQuade, Nastassia Pouradier Duteil. Opinion Dynamics on a General Compact Riemannian Manifold. Networks & Heterogeneous Media, 2017, 12 (3) : 489-523. doi: 10.3934/nhm.2017021

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (32)
  • HTML views (91)
  • Cited by (0)

Other articles
by authors

[Back to Top]