January  2015, 20(1): 249-257. doi: 10.3934/dcdsb.2015.20.249

On conformal measures of parabolic meromorphic functions

1. 

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

Received  January 2014 Revised  March 2014 Published  November 2014

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.
Citation: Zuxing Xuan. On conformal measures of parabolic meromorphic functions. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 249-257. doi: 10.3934/dcdsb.2015.20.249
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. doi: 10.1090/S0002-9947-1993-1107025-2. Google Scholar

[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. doi: 10.1112/jlms/s2-43.1.107. Google Scholar

[3]

M. Denker and M. Urbański, Geometric measures for parabolic rational maps,, Ergodic. Theory Dynam. Systems, 12 (1992), 53. doi: 10.1017/S014338570000657X. Google Scholar

[4]

M. Guzmán, Real Variable Methods in Fourier Analysis,, North-Holland Math. Studies, (1981). doi: 10.2307/2152750. Google Scholar

[5]

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

[6]

J. Kotus and M. Urbański, Fractal measures and ergodic theory of transcendental meromorphic functions,, in Transcendental Dynamics and Complex Anaysis, (2008), 251. doi: 10.1017/CBO9780511735233.013. Google Scholar

[7]

B. O. Stratmann and M. Urbański, The geometry of conformal measures for parabolic rational maps,, Math. Proc. Cambridge Phil. Soc., 128 (2000), 141. doi: 10.1017/S0305004199003837. Google Scholar

[8]

M. Urbański and A. Zdunik, The parabolic map $1/e e^z$,, Indag. Math. (N.S.), 15 (2004), 419. doi: 10.1016/S0019-3577(04)80009-0. Google Scholar

[9]

J. William, Multifractal Analysis of Parabolic Rational Maps,, PHD thesis, (1998). Google Scholar

[10]

J. H. Zheng, Parabolic meromorphic functions,, Pacific Journal of Mathematics, 250 (2011), 487. doi: 10.2140/pjm.2011.250.487. Google Scholar

[11]

J. H. Zheng, Conformal and invariant measures of parabolic meromorphic functions,, Houston J. Math., 39 (2013), 1149. Google Scholar

show all references

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. doi: 10.1090/S0002-9947-1993-1107025-2. Google Scholar

[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. doi: 10.1112/jlms/s2-43.1.107. Google Scholar

[3]

M. Denker and M. Urbański, Geometric measures for parabolic rational maps,, Ergodic. Theory Dynam. Systems, 12 (1992), 53. doi: 10.1017/S014338570000657X. Google Scholar

[4]

M. Guzmán, Real Variable Methods in Fourier Analysis,, North-Holland Math. Studies, (1981). doi: 10.2307/2152750. Google Scholar

[5]

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

[6]

J. Kotus and M. Urbański, Fractal measures and ergodic theory of transcendental meromorphic functions,, in Transcendental Dynamics and Complex Anaysis, (2008), 251. doi: 10.1017/CBO9780511735233.013. Google Scholar

[7]

B. O. Stratmann and M. Urbański, The geometry of conformal measures for parabolic rational maps,, Math. Proc. Cambridge Phil. Soc., 128 (2000), 141. doi: 10.1017/S0305004199003837. Google Scholar

[8]

M. Urbański and A. Zdunik, The parabolic map $1/e e^z$,, Indag. Math. (N.S.), 15 (2004), 419. doi: 10.1016/S0019-3577(04)80009-0. Google Scholar

[9]

J. William, Multifractal Analysis of Parabolic Rational Maps,, PHD thesis, (1998). Google Scholar

[10]

J. H. Zheng, Parabolic meromorphic functions,, Pacific Journal of Mathematics, 250 (2011), 487. doi: 10.2140/pjm.2011.250.487. Google Scholar

[11]

J. H. Zheng, Conformal and invariant measures of parabolic meromorphic functions,, Houston J. Math., 39 (2013), 1149. Google Scholar

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