July  2012, 32(7): 2375-2402. doi: 10.3934/dcds.2012.32.2375

Measure rigidity for some transcendental meromorphic functions

1. 

Faculty of Mathematics and Information Science, Warsaw University of Technology, Pl. Politechniki 1, 00-661 Warszawa, Poland

Received  December 2009 Revised  July 2010 Published  March 2012

We consider hyperbolic meromorphic functions of the following form $f(z)=R\circ\exp(z)$, where $R$ is a non-constant rational function, satisfying so-called rapid derivative growth condition. We study several types of conjugacies in this class and prove a~measure rigidity theorem in the case when $f$ has a logarithmic tract over $\infty$ and under some additional assumptions.
Citation: 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
References:
[1]

A. Badeńska, Real analyticity of Jacobian of invariant measures for hyperbolic meromorphic functions,, Bull. Lond. Math. Soc., 40 (2008), 1017. doi: 10.1112/blms/bdn083. Google Scholar

[2]

K. Barański, B. Karpnińska and A. Zdunik, Hyperbolic dimension of Julia sets of meromorphic maps with logarithmic tracts,, Int. Math. Res. Not. IMRN, 2009 (): 615. Google Scholar

[3]

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

[4]

R. L. Devaney and L. Keen, Dynamics of meromorphic maps: Maps with polynomial Schwarzian derivative,, Ann. Sci. École Norm. Sup. (4), 22 (1989), 55. Google Scholar

[5]

E. Hille, "Analytic Function Theory,", Vol. II, (1962). Google Scholar

[6]

J. Kotus and M. Urbański, The class of pseudo non-recurrent elliptic functions; geometry and dynamics,, preprint, (2007). Google Scholar

[7]

J. Kotus and M. Urbański, Fractal measures and ergodic theory of transcendental meromorphic functions,, in, 348 (2008), 251. Google Scholar

[8]

V. Mayer, Comparing measures and invariant line fields,, Ergodic Theory Dynam. Systems, 22 (2002), 555. Google Scholar

[9]

V. Mayer and M. Urbański, Geometric thermodynamical formalism and real analyticity for meromorphic functions of finite order,, Ergodic Theory Dynam. Systems, 28 (2008), 915. Google Scholar

[10]

V. Mayer and M. Urbański, Thermodynamical formalism and multifractal analysis for meromorphic functions of finite order,, Mem. Amer. Math. Soc., 203 (2010). Google Scholar

[11]

R. Nevanlinna, "Analytic Functions,", Die Grundlehren der mathematischen Wissenschaften, (1970). Google Scholar

[12]

F. Przytycki and M. Urbański, Rigidity of tame rational functions,, Bull. Polish Acad. Sci. Math., 47 (1999), 163. Google Scholar

[13]

L. Rempe, Hyperbolic dimension and radial Julia sets of transcendental functions,, Proc. Amer. Math. Soc., 137 (2009), 1411. doi: 10.1090/S0002-9939-08-09650-0. Google Scholar

[14]

L. Rempe and S. Van Strien, Absence of line fields and Mané's theorem for nonrecurrent transcendental functions,, Trans. Amer. Math. Soc., 363 (2011), 203. doi: 10.1090/S0002-9947-2010-05125-6. Google Scholar

[15]

G. Stallard, The Hausdorff dimension of Julia sets of entire functions. II,, Math. Proc. Cambridge Philos. Soc., 119 (1996), 513. doi: 10.1017/S0305004100074387. Google Scholar

[16]

D. Sullivan, Quasiconformal homeomorphisms in dynamics, topology, and geometry,, in, (1987), 1216. Google Scholar

show all references

References:
[1]

A. Badeńska, Real analyticity of Jacobian of invariant measures for hyperbolic meromorphic functions,, Bull. Lond. Math. Soc., 40 (2008), 1017. doi: 10.1112/blms/bdn083. Google Scholar

[2]

K. Barański, B. Karpnińska and A. Zdunik, Hyperbolic dimension of Julia sets of meromorphic maps with logarithmic tracts,, Int. Math. Res. Not. IMRN, 2009 (): 615. Google Scholar

[3]

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

[4]

R. L. Devaney and L. Keen, Dynamics of meromorphic maps: Maps with polynomial Schwarzian derivative,, Ann. Sci. École Norm. Sup. (4), 22 (1989), 55. Google Scholar

[5]

E. Hille, "Analytic Function Theory,", Vol. II, (1962). Google Scholar

[6]

J. Kotus and M. Urbański, The class of pseudo non-recurrent elliptic functions; geometry and dynamics,, preprint, (2007). Google Scholar

[7]

J. Kotus and M. Urbański, Fractal measures and ergodic theory of transcendental meromorphic functions,, in, 348 (2008), 251. Google Scholar

[8]

V. Mayer, Comparing measures and invariant line fields,, Ergodic Theory Dynam. Systems, 22 (2002), 555. Google Scholar

[9]

V. Mayer and M. Urbański, Geometric thermodynamical formalism and real analyticity for meromorphic functions of finite order,, Ergodic Theory Dynam. Systems, 28 (2008), 915. Google Scholar

[10]

V. Mayer and M. Urbański, Thermodynamical formalism and multifractal analysis for meromorphic functions of finite order,, Mem. Amer. Math. Soc., 203 (2010). Google Scholar

[11]

R. Nevanlinna, "Analytic Functions,", Die Grundlehren der mathematischen Wissenschaften, (1970). Google Scholar

[12]

F. Przytycki and M. Urbański, Rigidity of tame rational functions,, Bull. Polish Acad. Sci. Math., 47 (1999), 163. Google Scholar

[13]

L. Rempe, Hyperbolic dimension and radial Julia sets of transcendental functions,, Proc. Amer. Math. Soc., 137 (2009), 1411. doi: 10.1090/S0002-9939-08-09650-0. Google Scholar

[14]

L. Rempe and S. Van Strien, Absence of line fields and Mané's theorem for nonrecurrent transcendental functions,, Trans. Amer. Math. Soc., 363 (2011), 203. doi: 10.1090/S0002-9947-2010-05125-6. Google Scholar

[15]

G. Stallard, The Hausdorff dimension of Julia sets of entire functions. II,, Math. Proc. Cambridge Philos. Soc., 119 (1996), 513. doi: 10.1017/S0305004100074387. Google Scholar

[16]

D. Sullivan, Quasiconformal homeomorphisms in dynamics, topology, and geometry,, in, (1987), 1216. Google Scholar

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