On conformal measures of parabolic meromorphic functions

Zuxing Xuan

doi:10.3934/dcdsb.2015.20.249

Article Contents

2015, Volume 20, Issue 1: 249-257. Doi: 10.3934/dcdsb.2015.20.249

This issue Previous Article The stability of bifurcating steady states of several classes of chemotaxis systems Next Article Graph-theoretic approach to stability of multi-group models with dispersal

On conformal measures of parabolic meromorphic functions

Zuxing Xuan^1,

1.
Beijing Key Laboratory of Information Service Engineering, Department of General Education, Beijing Union University, Beijing, 100101

Received: December 31, 2013

Revised: February 28, 2014

Published: December 2014

Abstract Related Papers Cited by

Abstract

We prove the absolute continuity of the Hausdorff measure with respect to any conformal measure. These results extend Denker and Urbanski's work on parabolic rational functions.

Keywords:

Mathematics Subject Classification: Primary: 30B20, 30D05, 30D10; Secondary: 34M05, 37F10.

Citation:

Related Papers

Cited by

References

[1]	J. Aaronson, M. Denker and M. Urbański, Ergodic theory for Markov fibred systems and parabolic rational maps, Trans. Amer. Math. Soc., 337 (1993), 495-548.doi: 10.1090/S0002-9947-1993-1107025-2.
[2]	M. Denker and M. Urbański, Hausdorff and conformal measures on Julia sets with a rationally indifferent periodic point, J. London Math. Soc., 43 (1991), 107-118.doi: 10.1112/jlms/s2-43.1.107.
[3]	M. Denker and M. Urbański, Geometric measures for parabolic rational maps, Ergodic. Theory Dynam. Systems, 12 (1992), 53-66.doi: 10.1017/S014338570000657X.
[4]	M. Guzmán, Real Variable Methods in Fourier Analysis, North-Holland Math. Studies, {46}, Amsterdam, New York, 1981.doi: 10.2307/2152750.
[5]	J. Kotus and M. Urbański, Conformal, geometric and invariant measures for transcendental expanding functions, Math. Ann., 324 (2002), 619-656.doi: 10.1007/s00208-002-0356-y.
[6]	J. Kotus and M. Urbański, Fractal measures and ergodic theory of transcendental meromorphic functions, in Transcendental Dynamics and Complex Anaysis, London Math. Soc. Lecture Note Ser., {348}, Cambridge Univ. Press, Combridge, 2008, 251-316.doi: 10.1017/CBO9780511735233.013.
[7]	B. O. Stratmann and M. Urbański, The geometry of conformal measures for parabolic rational maps, Math. Proc. Cambridge Phil. Soc., 128 (2000), 141-156.doi: 10.1017/S0305004199003837.
[8]	M. Urbański and A. Zdunik, The parabolic map $1/e e^z$, Indag. Math. (N.S.), 15 (2004), 419-433.doi: 10.1016/S0019-3577(04)80009-0.
[9]	J. William, Multifractal Analysis of Parabolic Rational Maps, PHD thesis, 1998.
[10]	J. H. Zheng, Parabolic meromorphic functions, Pacific Journal of Mathematics, 250 (2011), 487-509.doi: 10.2140/pjm.2011.250.487.
[11]	J. H. Zheng, Conformal and invariant measures of parabolic meromorphic functions, Houston J. Math., 39 (2013), 1149-1159.