# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 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), 1017-1024. doi: 10.1112/blms/bdn083. [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. [3] W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc., 29 (1993), 151-188. doi: 10.1090/S0273-0979-1993-00432-4. [4] R. L. Devaney and L. Keen, Dynamics of meromorphic maps: Maps with polynomial Schwarzian derivative, Ann. Sci. École Norm. Sup. (4), 22 (1989), 55-79. [5] E. Hille, "Analytic Function Theory," Vol. II, Introductions to Higher Mathematics, Ginn and Co., Boston, Mass.-New York-Toronto, Ont., 1962. [6] J. Kotus and M. Urbański, The class of pseudo non-recurrent elliptic functions; geometry and dynamics, preprint, 2007. Available from: http://www.math.unt.edu/~urbanski. [7] J. Kotus and M. Urbański, Fractal measures and ergodic theory of transcendental meromorphic functions, in "Transcendental Dynamics and Complex Analysis," London Math. Soc. Lecture Note Ser., 348, Cambridge Univ. Press, Cambridge, (2008), 251-316. [8] V. Mayer, Comparing measures and invariant line fields, Ergodic Theory Dynam. Systems, 22 (2002), 555-570. [9] V. Mayer and M. Urbański, Geometric thermodynamical formalism and real analyticity for meromorphic functions of finite order, Ergodic Theory Dynam. Systems, 28 (2008), 915-946. [10] V. Mayer and M. Urbański, Thermodynamical formalism and multifractal analysis for meromorphic functions of finite order, Mem. Amer. Math. Soc., 203 (2010), vi+107 pp. [11] R. Nevanlinna, "Analytic Functions," Die Grundlehren der mathematischen Wissenschaften, Band 162, Springer Verlag, New York-Berlin, 1970. [12] F. Przytycki and M. Urbański, Rigidity of tame rational functions, Bull. Polish Acad. Sci. Math., 47 (1999), 163-182. [13] L. Rempe, Hyperbolic dimension and radial Julia sets of transcendental functions, Proc. Amer. Math. Soc., 137 (2009), 1411-1420. doi: 10.1090/S0002-9939-08-09650-0. [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), 203-228. doi: 10.1090/S0002-9947-2010-05125-6. [15] G. Stallard, The Hausdorff dimension of Julia sets of entire functions. II, Math. Proc. Cambridge Philos. Soc., 119 (1996), 513-536. doi: 10.1017/S0305004100074387. [16] D. Sullivan, Quasiconformal homeomorphisms in dynamics, topology, and geometry, in "Proc. Internat. Congress of Math." (Berkeley, Calif., 1986), Vol. 1, 2, Amer. Math. Soc., Providence, RI, (1987), 1216-1228.

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##### References:
 [1] A. Badeńska, Real analyticity of Jacobian of invariant measures for hyperbolic meromorphic functions, Bull. Lond. Math. Soc., 40 (2008), 1017-1024. doi: 10.1112/blms/bdn083. [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. [3] W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc., 29 (1993), 151-188. doi: 10.1090/S0273-0979-1993-00432-4. [4] R. L. Devaney and L. Keen, Dynamics of meromorphic maps: Maps with polynomial Schwarzian derivative, Ann. Sci. École Norm. Sup. (4), 22 (1989), 55-79. [5] E. Hille, "Analytic Function Theory," Vol. II, Introductions to Higher Mathematics, Ginn and Co., Boston, Mass.-New York-Toronto, Ont., 1962. [6] J. Kotus and M. Urbański, The class of pseudo non-recurrent elliptic functions; geometry and dynamics, preprint, 2007. Available from: http://www.math.unt.edu/~urbanski. [7] J. Kotus and M. Urbański, Fractal measures and ergodic theory of transcendental meromorphic functions, in "Transcendental Dynamics and Complex Analysis," London Math. Soc. Lecture Note Ser., 348, Cambridge Univ. Press, Cambridge, (2008), 251-316. [8] V. Mayer, Comparing measures and invariant line fields, Ergodic Theory Dynam. Systems, 22 (2002), 555-570. [9] V. Mayer and M. Urbański, Geometric thermodynamical formalism and real analyticity for meromorphic functions of finite order, Ergodic Theory Dynam. Systems, 28 (2008), 915-946. [10] V. Mayer and M. Urbański, Thermodynamical formalism and multifractal analysis for meromorphic functions of finite order, Mem. Amer. Math. Soc., 203 (2010), vi+107 pp. [11] R. Nevanlinna, "Analytic Functions," Die Grundlehren der mathematischen Wissenschaften, Band 162, Springer Verlag, New York-Berlin, 1970. [12] F. Przytycki and M. Urbański, Rigidity of tame rational functions, Bull. Polish Acad. Sci. Math., 47 (1999), 163-182. [13] L. Rempe, Hyperbolic dimension and radial Julia sets of transcendental functions, Proc. Amer. Math. Soc., 137 (2009), 1411-1420. doi: 10.1090/S0002-9939-08-09650-0. [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), 203-228. doi: 10.1090/S0002-9947-2010-05125-6. [15] G. Stallard, The Hausdorff dimension of Julia sets of entire functions. II, Math. Proc. Cambridge Philos. Soc., 119 (1996), 513-536. doi: 10.1017/S0305004100074387. [16] D. Sullivan, Quasiconformal homeomorphisms in dynamics, topology, and geometry, in "Proc. Internat. Congress of Math." (Berkeley, Calif., 1986), Vol. 1, 2, Amer. Math. Soc., Providence, RI, (1987), 1216-1228.
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