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May  2015, 35(5): 2273-2298. doi: 10.3934/dcds.2015.35.2273

Dynamics of hyperbolic meromorphic functions

Jian-Hua Zheng 1,

1. 

Department of Mathematical Sciences, Tsinghua University, Beijing, China

Received  October 2010 Revised  June 2014 Published  December 2014

A definition of hyperbolic meromorphic functions is given and then we discuss the dynamical behavior and the thermodynamic formalism of hyperbolic functions on their Julia sets. We prove the important expanding properties for hyperbolic functions on the complex plane or with respect to the Euclidean metric. We establish the Bowen formula for hyperbolic functions on the complex plane, that is, the Poincare exponent equals to the Hausdorff dimension of the radial Julia set and furthermore, we prove that all the results in the Walters' theory hold for hyperbolic functions on the Riemann sphere.
Keywords: Hyperbolic meromorphic functions, conformal measures, Hausdorff dimension, Bowen formula..
Mathematics Subject Classification: Primary: 37D35, 28D20; Secondary: 30D2.
Citation: Jian-Hua Zheng. Dynamics of hyperbolic meromorphic functions. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2273-2298. doi: 10.3934/dcds.2015.35.2273
References:
[1]

I. N. Baker and P. Dominguez, Boundaries of unbounded Fatou components of entire functions, Ann. Acad. Sci. Fenn., Series A., I. Math., 24 (1999), 437-464.

[2]

K. Baranski, Hausdorff dimension and measures on Julia sets of some meromorphic functions, Fund. Math., 147 (1995), 239-260.

[3]

W. Bergweiler and A. Eremenko, Meromorphic functions with two completely invariant domains, in Transcendental Dynamics and Complex Analysis, edited by P. J. Rippon & G. M. Stallard, Cambridge University Press, LMS Lecture Note Series, 348 (2008), 74-89. doi: 10.1017/CBO9780511735233.005.

[4]

W. Bergweiler, M. Haruta, H. Kriete. H. G. Meier and N. Terglane, On the limit functions of iterates in wandering domains, Ann. Acad. Sci. Fenn., Series A., I. Math., 18 (1993), 369-375.

[5]

W. Bergweiler, P. J. Rippon and G. M. Stallard, Dynamics of meromorphic functions with direct or logarithmic singularities, Proc. London Math. Soc., 97 (2008), 368-400. doi: 10.1112/plms/pdn007.

[6]

R. L. Devaney and L. Keen, Dynamics of meromorphic maps: Maps with polynomial Schwarzian derivative, Ann. Scient. Éc. Norm. Sup., $4^e$ série, t., 22 (1989), 55-79.

[7]

P. Dominguez, Dynamics of transcendental meromorphic functions, Ann. Acad. Sci. Fenn., Series A., I. Math., 23 (1998), 225-250.

[8]

K. Falconer, Fractal Geometry, John Wiley & Sons, 1999. doi: 10.1002/0470013850.

[9]

W. K. Hayman, On Iversen's Theorem for meromorphic functions with few poles, Acta Mathematica, 141 (1978), 115-145. doi: 10.1007/BF02545745.

[10]

F. Iversen, Recherches sur les Fonctions Inverses des Fonctons Méromorphes, Thése de Helsingfors, 1914.

[11]

J. Kotus and M. Urbanski, Conformal, geometric and invariant measures for transcendental expanding functions, Math. Ann., 324 (2002), 619-656. doi: 10.1007/s00208-002-0356-y.

[12]

J. Kotus and M. Urbanski, Hausdorff dimension of radial and escaping points for transcendental meromorphic functions, Illinois J. Math., 52 (2008), 1035-1044.

[13]

V. Mayer and M. Urbański, Thermodynamical formalism and multifractal analysis for meromorphic functions of finite order, Memoirs of AMS, 203 (2010), vi+107 pp. doi: 10.1090/S0065-9266-09-00577-8.

[14]

C. McMullen, Area and Hausdorff dimension of Julia sets of entire functions, Trans. Amer. Math. Soc., 300 (1987), 329-342. doi: 10.1090/S0002-9947-1987-0871679-3.

[15]

C. McMullen, In holomorphic functions and moduli I, Springer-Verlag, 10 (1988), 31-60. doi: 10.1007/978-1-4613-9602-4_3.

[16]

J. K. Moser, Stable And Random Motions In Dynamical Systems, Princeton: Princeton University Press, 1973.

[17]

T. W. Ng, J. H. Zheng and K. Choi, Residual Julia sets of meromorphic functions, Math. Proc. Cambridge Phil. Soc., 141 (2006), 113-126. doi: 10.1017/S0305004106009388.

[18]

K. Pilgrim and L. Tan, Rational maps with disconnected Julia set, Géométrie complexe et systèmes dynamiques (Orsay, 1995), 261 (2000), 349-384.

[19]

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.

[20]

D. Ruelle, Repellers for real analytic maps, Ergodic Th. Dyn. Sys., 2 (1982), 99-107. doi: 10.1017/S0143385700009603.

[21]

G. M. Stallard, The Hausdorff dimension of Julia sets of meromorphic functions, J. London Math. soc., 49 (1994), 281-295. doi: 10.1112/jlms/49.2.281.

[22]

G. M. Stallard, The Hausdorff dimension of Julia sets of meromorphic functions II, J. London Math. soc., 60 (1999), 847-859. doi: 10.1112/S0024610799008029.

[23]

G. M. Stallard, Dimension of Julia sets of hyperbolic meromorphic functions, Math. Proc. Camb. Phil. Soc., 127 (1999), 271-288. doi: 10.1017/S0305004199003813.

[24]

G. M. Stallard, The Hausdorff dimension of Julia sets of hyperbolic meromorphic functions, Ergodic Theory Dynam. Systems, 20 (2000), 895-910. doi: 10.1017/S0143385700000481.

[25]

G. M. Stallard, Meromorphic functions whose Julia sets contain a free Jordan arc, Ann. Acad. Sci. Fenn., Series A., I. Math., 18 (1993), 273-298.

[26]

N. Steinmetz, Rational Iterations, Berlin: Walter de Gruyter, 1993. doi: 10.1515/9783110889314.

[27]

D. Sullivan, Conformal dynamical systems: In Geometric dynamics, Lecture Notes in Math., Springer, 1007 (1983), 725-752. doi: 10.1007/BFb0061443.

[28]

O. Teichmüller, Eine Umkehrung des zweiten Hauptsatzes der Wertverteilungstheorie, Deutsche Math., 2 (1937), 96-107.

[29]

L. A. Ter-Israelyan, Meromorphic functions of zero order with non-asymptotic deficient values, Math. Zam., 13 (1973), 195-206.

[30]

P. Walters, Invariant measures and equilibrium states for some mappings which expand distances, Trans. Amer. Math. Soc., 236 (1978), 121-153. doi: 10.1090/S0002-9947-1978-0466493-1.

[31]

M. Urbanski and A. Zdunik, The finer geometry and dynamics of exponential family, Michingan Math. J., 51 (2003), 227-250. doi: 10.1307/mmj/1060013195.

[32]

M. Urbanski and A. Zdunik, Real analyticity of Hausdorff dimension of finer Julia sets of exponential family, Ergodic Th. and Dynam. Sys., 24 (2004), 279-315. doi: 10.1017/S0143385703000208.

[33]

J. H. Zheng, Singularties and limit functions in iteration of meromorphic functions, J. London Math. Soc., 67 (2003), 195-207. doi: 10.1112/S0024610702003800.

[34]

J. H. Zheng, On transcendental meromorphic functions which are geometrically finite, J. Austral. Math. Soc., 72 (2002), 93-107. doi: 10.1017/S144678870000361X.

[35]

J. H. Zheng, Dynamics Of Meromorphic Functions, Monograph of Tsinghua University, Tsinghua University Press, 2006 (in Chinese).

show all references

References:
[1]

I. N. Baker and P. Dominguez, Boundaries of unbounded Fatou components of entire functions, Ann. Acad. Sci. Fenn., Series A., I. Math., 24 (1999), 437-464.

[2]

K. Baranski, Hausdorff dimension and measures on Julia sets of some meromorphic functions, Fund. Math., 147 (1995), 239-260.

[3]

W. Bergweiler and A. Eremenko, Meromorphic functions with two completely invariant domains, in Transcendental Dynamics and Complex Analysis, edited by P. J. Rippon & G. M. Stallard, Cambridge University Press, LMS Lecture Note Series, 348 (2008), 74-89. doi: 10.1017/CBO9780511735233.005.

[4]

W. Bergweiler, M. Haruta, H. Kriete. H. G. Meier and N. Terglane, On the limit functions of iterates in wandering domains, Ann. Acad. Sci. Fenn., Series A., I. Math., 18 (1993), 369-375.

[5]

W. Bergweiler, P. J. Rippon and G. M. Stallard, Dynamics of meromorphic functions with direct or logarithmic singularities, Proc. London Math. Soc., 97 (2008), 368-400. doi: 10.1112/plms/pdn007.

[6]

R. L. Devaney and L. Keen, Dynamics of meromorphic maps: Maps with polynomial Schwarzian derivative, Ann. Scient. Éc. Norm. Sup., $4^e$ série, t., 22 (1989), 55-79.

[7]

P. Dominguez, Dynamics of transcendental meromorphic functions, Ann. Acad. Sci. Fenn., Series A., I. Math., 23 (1998), 225-250.

[8]

K. Falconer, Fractal Geometry, John Wiley & Sons, 1999. doi: 10.1002/0470013850.

[9]

W. K. Hayman, On Iversen's Theorem for meromorphic functions with few poles, Acta Mathematica, 141 (1978), 115-145. doi: 10.1007/BF02545745.

[10]

F. Iversen, Recherches sur les Fonctions Inverses des Fonctons Méromorphes, Thése de Helsingfors, 1914.

[11]

J. Kotus and M. Urbanski, Conformal, geometric and invariant measures for transcendental expanding functions, Math. Ann., 324 (2002), 619-656. doi: 10.1007/s00208-002-0356-y.

[12]

J. Kotus and M. Urbanski, Hausdorff dimension of radial and escaping points for transcendental meromorphic functions, Illinois J. Math., 52 (2008), 1035-1044.

[13]

V. Mayer and M. Urbański, Thermodynamical formalism and multifractal analysis for meromorphic functions of finite order, Memoirs of AMS, 203 (2010), vi+107 pp. doi: 10.1090/S0065-9266-09-00577-8.

[14]

C. McMullen, Area and Hausdorff dimension of Julia sets of entire functions, Trans. Amer. Math. Soc., 300 (1987), 329-342. doi: 10.1090/S0002-9947-1987-0871679-3.

[15]

C. McMullen, In holomorphic functions and moduli I, Springer-Verlag, 10 (1988), 31-60. doi: 10.1007/978-1-4613-9602-4_3.

[16]

J. K. Moser, Stable And Random Motions In Dynamical Systems, Princeton: Princeton University Press, 1973.

[17]

T. W. Ng, J. H. Zheng and K. Choi, Residual Julia sets of meromorphic functions, Math. Proc. Cambridge Phil. Soc., 141 (2006), 113-126. doi: 10.1017/S0305004106009388.

[18]

K. Pilgrim and L. Tan, Rational maps with disconnected Julia set, Géométrie complexe et systèmes dynamiques (Orsay, 1995), 261 (2000), 349-384.

[19]

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.

[20]

D. Ruelle, Repellers for real analytic maps, Ergodic Th. Dyn. Sys., 2 (1982), 99-107. doi: 10.1017/S0143385700009603.

[21]

G. M. Stallard, The Hausdorff dimension of Julia sets of meromorphic functions, J. London Math. soc., 49 (1994), 281-295. doi: 10.1112/jlms/49.2.281.

[22]

G. M. Stallard, The Hausdorff dimension of Julia sets of meromorphic functions II, J. London Math. soc., 60 (1999), 847-859. doi: 10.1112/S0024610799008029.

[23]

G. M. Stallard, Dimension of Julia sets of hyperbolic meromorphic functions, Math. Proc. Camb. Phil. Soc., 127 (1999), 271-288. doi: 10.1017/S0305004199003813.

[24]

G. M. Stallard, The Hausdorff dimension of Julia sets of hyperbolic meromorphic functions, Ergodic Theory Dynam. Systems, 20 (2000), 895-910. doi: 10.1017/S0143385700000481.

[25]

G. M. Stallard, Meromorphic functions whose Julia sets contain a free Jordan arc, Ann. Acad. Sci. Fenn., Series A., I. Math., 18 (1993), 273-298.

[26]

N. Steinmetz, Rational Iterations, Berlin: Walter de Gruyter, 1993. doi: 10.1515/9783110889314.

[27]

D. Sullivan, Conformal dynamical systems: In Geometric dynamics, Lecture Notes in Math., Springer, 1007 (1983), 725-752. doi: 10.1007/BFb0061443.

[28]

O. Teichmüller, Eine Umkehrung des zweiten Hauptsatzes der Wertverteilungstheorie, Deutsche Math., 2 (1937), 96-107.

[29]

L. A. Ter-Israelyan, Meromorphic functions of zero order with non-asymptotic deficient values, Math. Zam., 13 (1973), 195-206.

[30]

P. Walters, Invariant measures and equilibrium states for some mappings which expand distances, Trans. Amer. Math. Soc., 236 (1978), 121-153. doi: 10.1090/S0002-9947-1978-0466493-1.

[31]

M. Urbanski and A. Zdunik, The finer geometry and dynamics of exponential family, Michingan Math. J., 51 (2003), 227-250. doi: 10.1307/mmj/1060013195.

[32]

M. Urbanski and A. Zdunik, Real analyticity of Hausdorff dimension of finer Julia sets of exponential family, Ergodic Th. and Dynam. Sys., 24 (2004), 279-315. doi: 10.1017/S0143385703000208.

[33]

J. H. Zheng, Singularties and limit functions in iteration of meromorphic functions, J. London Math. Soc., 67 (2003), 195-207. doi: 10.1112/S0024610702003800.

[34]

J. H. Zheng, On transcendental meromorphic functions which are geometrically finite, J. Austral. Math. Soc., 72 (2002), 93-107. doi: 10.1017/S144678870000361X.

[35]

J. H. Zheng, Dynamics Of Meromorphic Functions, Monograph of Tsinghua University, Tsinghua University Press, 2006 (in Chinese).

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