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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 & Continuous Dynamical Systems - A, 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., 24 (1999), 437.   Google Scholar

[2]

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

[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, 348 (2008), 74.  doi: 10.1017/CBO9780511735233.005.  Google Scholar

[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., 18 (1993), 369.   Google Scholar

[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.  doi: 10.1112/plms/pdn007.  Google Scholar

[6]

R. L. Devaney and L. Keen, Dynamics of meromorphic maps: Maps with polynomial Schwarzian derivative,, Ann. Scient. Éc. Norm. Sup., 22 (1989), 55.   Google Scholar

[7]

P. Dominguez, Dynamics of transcendental meromorphic functions,, Ann. Acad. Sci. Fenn., 23 (1998), 225.   Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

G. M. Stallard, Meromorphic functions whose Julia sets contain a free Jordan arc,, Ann. Acad. Sci. Fenn., 18 (1993), 273.   Google Scholar

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[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.  doi: 10.1017/S0143385703000208.  Google Scholar

[33]

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

[34]

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

[35]

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

show all references

References:
[1]

I. N. Baker and P. Dominguez, Boundaries of unbounded Fatou components of entire functions,, Ann. Acad. Sci. Fenn., 24 (1999), 437.   Google Scholar

[2]

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

[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, 348 (2008), 74.  doi: 10.1017/CBO9780511735233.005.  Google Scholar

[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., 18 (1993), 369.   Google Scholar

[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.  doi: 10.1112/plms/pdn007.  Google Scholar

[6]

R. L. Devaney and L. Keen, Dynamics of meromorphic maps: Maps with polynomial Schwarzian derivative,, Ann. Scient. Éc. Norm. Sup., 22 (1989), 55.   Google Scholar

[7]

P. Dominguez, Dynamics of transcendental meromorphic functions,, Ann. Acad. Sci. Fenn., 23 (1998), 225.   Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

G. M. Stallard, Meromorphic functions whose Julia sets contain a free Jordan arc,, Ann. Acad. Sci. Fenn., 18 (1993), 273.   Google Scholar

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[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.  doi: 10.1017/S0143385703000208.  Google Scholar

[33]

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

[34]

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

[35]

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

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