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, 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.  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), 151-188. 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), 55-79.  Google Scholar

[5]

E. Hille, "Analytic Function Theory," Vol. II, Introductions to Higher Mathematics, Ginn and Co., Boston, Mass.-New York-Toronto, Ont., 1962.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[8]

V. Mayer, Comparing measures and invariant line fields, Ergodic Theory Dynam. Systems, 22 (2002), 555-570.  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), 915-946.  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), vi+107 pp.  Google Scholar

[11]

R. Nevanlinna, "Analytic Functions," Die Grundlehren der mathematischen Wissenschaften, Band 162, Springer Verlag, New York-Berlin, 1970.  Google Scholar

[12]

F. Przytycki and M. Urbański, Rigidity of tame rational functions, Bull. Polish Acad. Sci. Math., 47 (1999), 163-182.  Google Scholar

[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.  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), 203-228. 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), 513-536. doi: 10.1017/S0305004100074387.  Google Scholar

[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.  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), 1017-1024. 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), 151-188. 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), 55-79.  Google Scholar

[5]

E. Hille, "Analytic Function Theory," Vol. II, Introductions to Higher Mathematics, Ginn and Co., Boston, Mass.-New York-Toronto, Ont., 1962.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[8]

V. Mayer, Comparing measures and invariant line fields, Ergodic Theory Dynam. Systems, 22 (2002), 555-570.  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), 915-946.  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), vi+107 pp.  Google Scholar

[11]

R. Nevanlinna, "Analytic Functions," Die Grundlehren der mathematischen Wissenschaften, Band 162, Springer Verlag, New York-Berlin, 1970.  Google Scholar

[12]

F. Przytycki and M. Urbański, Rigidity of tame rational functions, Bull. Polish Acad. Sci. Math., 47 (1999), 163-182.  Google Scholar

[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.  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), 203-228. 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), 513-536. doi: 10.1017/S0305004100074387.  Google Scholar

[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.  Google Scholar

[1]

Kingshook Biswas. Complete conjugacy invariants of nonlinearizable holomorphic dynamics. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 847-856. doi: 10.3934/dcds.2010.26.847

[2]

Carlos Cabrera, Peter Makienko, Peter Plaumann. Semigroup representations in holomorphic dynamics. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1333-1349. doi: 10.3934/dcds.2013.33.1333

[3]

Anatole Katok, Federico Rodriguez Hertz. Measure and cocycle rigidity for certain nonuniformly hyperbolic actions of higher-rank abelian groups. Journal of Modern Dynamics, 2010, 4 (3) : 487-515. doi: 10.3934/jmd.2010.4.487

[4]

Tomasz Downarowicz, Yonatan Gutman, Dawid Huczek. Rank as a function of measure. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2741-2750. doi: 10.3934/dcds.2014.34.2741

[5]

Tomasz Szarek, Mariusz Urbański, Anna Zdunik. Continuity of Hausdorff measure for conformal dynamical systems. Discrete & Continuous Dynamical Systems, 2013, 33 (10) : 4647-4692. doi: 10.3934/dcds.2013.33.4647

[6]

David Constantine. 2-Frame flow dynamics and hyperbolic rank-rigidity in nonpositive curvature. Journal of Modern Dynamics, 2008, 2 (4) : 719-740. doi: 10.3934/jmd.2008.2.719

[7]

Boris Kalinin, Anatole Katok and Federico Rodriguez Hertz. New progress in nonuniform measure and cocycle rigidity. Electronic Research Announcements, 2008, 15: 79-92. doi: 10.3934/era.2008.15.79

[8]

Zhenqi Jenny Wang. Local rigidity of partially hyperbolic actions. Journal of Modern Dynamics, 2010, 4 (2) : 271-327. doi: 10.3934/jmd.2010.4.271

[9]

Zhenqi Jenny Wang. Local rigidity of partially hyperbolic actions. Electronic Research Announcements, 2010, 17: 68-79. doi: 10.3934/era.2010.17.68

[10]

Nuno Luzia. On the uniqueness of an ergodic measure of full dimension for non-conformal repellers. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5763-5780. doi: 10.3934/dcds.2017250

[11]

Ralf Spatzier. On the work of Rodriguez Hertz on rigidity in dynamics. Journal of Modern Dynamics, 2016, 10: 191-207. doi: 10.3934/jmd.2016.10.191

[12]

Andrei Török. Rigidity of partially hyperbolic actions of property (T) groups. Discrete & Continuous Dynamical Systems, 2003, 9 (1) : 193-208. doi: 10.3934/dcds.2003.9.193

[13]

Misha Bialy. Hopf rigidity for convex billiards on the hemisphere and hyperbolic plane. Discrete & Continuous Dynamical Systems, 2013, 33 (9) : 3903-3913. doi: 10.3934/dcds.2013.33.3903

[14]

Boris Kalinin, Anatole Katok. Measure rigidity beyond uniform hyperbolicity: invariant measures for cartan actions on tori. Journal of Modern Dynamics, 2007, 1 (1) : 123-146. doi: 10.3934/jmd.2007.1.123

[15]

Ugo Bessi. The stochastic value function in metric measure spaces. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 1819-1839. doi: 10.3934/dcds.2017076

[16]

Kanji Inui, Hikaru Okada, Hiroki Sumi. The Hausdorff dimension function of the family of conformal iterated function systems of generalized complex continued fractions. Discrete & Continuous Dynamical Systems, 2020, 40 (2) : 753-766. doi: 10.3934/dcds.2020060

[17]

Juan Wang, Jing Wang, Yongluo Cao, Yun Zhao. Dimensions of $ C^1- $average conformal hyperbolic sets. Discrete & Continuous Dynamical Systems, 2020, 40 (2) : 883-905. doi: 10.3934/dcds.2020065

[18]

Eugen Mihailescu. Applications of thermodynamic formalism in complex dynamics on $\mathbb{P}^2$. Discrete & Continuous Dynamical Systems, 2001, 7 (4) : 821-836. doi: 10.3934/dcds.2001.7.821

[19]

Anja Randecker, Giulio Tiozzo. Cusp excursion in hyperbolic manifolds and singularity of harmonic measure. Journal of Modern Dynamics, 2021, 17: 183-211. doi: 10.3934/jmd.2021006

[20]

Richard Sharp. Conformal Markov systems, Patterson-Sullivan measure on limit sets and spectral triples. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2711-2727. doi: 10.3934/dcds.2016.36.2711

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (53)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]