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.

[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.

[3]

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

[4]

M. Guzmán, Real Variable Methods in Fourier Analysis,, North-Holland Math. Studies, (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. 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, (2008), 251. 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. doi: 10.1017/S0305004199003837.

[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.

[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. 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.

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.

[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.

[3]

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

[4]

M. Guzmán, Real Variable Methods in Fourier Analysis,, North-Holland Math. Studies, (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. 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, (2008), 251. 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. doi: 10.1017/S0305004199003837.

[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.

[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. 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.

[1]

Thomas Jordan, Mark Pollicott. The Hausdorff dimension of measures for iterated function systems which contract on average. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 235-246. doi: 10.3934/dcds.2008.22.235

[2]

Krzysztof Barański, Michał Wardal. On the Hausdorff dimension of the Sierpiński Julia sets. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3293-3313. doi: 10.3934/dcds.2015.35.3293

[3]

Jon Chaika. Hausdorff dimension for ergodic measures of interval exchange transformations. Journal of Modern Dynamics, 2008, 2 (3) : 457-464. doi: 10.3934/jmd.2008.2.457

[4]

Paul Wright. Differentiability of Hausdorff dimension of the non-wandering set in a planar open billiard. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3993-4014. doi: 10.3934/dcds.2016.36.3993

[5]

Luke G. Rogers, Alexander Teplyaev. Laplacians on the basilica Julia set. Communications on Pure & Applied Analysis, 2010, 9 (1) : 211-231. doi: 10.3934/cpaa.2010.9.211

[6]

Yan Huang. On Hausdorff dimension of the set of non-ergodic directions of two-genus double cover of tori. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2395-2409. doi: 10.3934/dcds.2018099

[7]

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

[8]

Sabyasachi Mukherjee. Parabolic arcs of the multicorns: Real-analyticity of Hausdorff dimension, and singularities of $\mathrm{Per}_n(1)$ curves. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2565-2588. doi: 10.3934/dcds.2017110

[9]

Juan Wang, Yongluo Cao, Yun Zhao. Dimension estimates in non-conformal setting. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3847-3873. doi: 10.3934/dcds.2014.34.3847

[10]

Artem Dudko. Computability of the Julia set. Nonrecurrent critical orbits. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2751-2778. doi: 10.3934/dcds.2014.34.2751

[11]

Hiroki Sumi, Mariusz Urbański. Bowen parameter and Hausdorff dimension for expanding rational semigroups. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2591-2606. doi: 10.3934/dcds.2012.32.2591

[12]

Shmuel Friedland, Gunter Ochs. Hausdorff dimension, strong hyperbolicity and complex dynamics. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 405-430. doi: 10.3934/dcds.1998.4.405

[13]

Sara Munday. On Hausdorff dimension and cusp excursions for Fuchsian groups. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2503-2520. doi: 10.3934/dcds.2012.32.2503

[14]

Luis Barreira and Jorg Schmeling. Invariant sets with zero measure and full Hausdorff dimension. Electronic Research Announcements, 1997, 3: 114-118.

[15]

Michihiro Hirayama. Periodic probability measures are dense in the set of invariant measures. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1185-1192. doi: 10.3934/dcds.2003.9.1185

[16]

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

[17]

Koh Katagata. On a certain kind of polynomials of degree 4 with disconnected Julia set. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 975-987. doi: 10.3934/dcds.2008.20.975

[18]

Volodymyr Nekrashevych. The Julia set of a post-critically finite endomorphism of $\mathbb{PC}^2$. Journal of Modern Dynamics, 2012, 6 (3) : 327-375. doi: 10.3934/jmd.2012.6.327

[19]

Rich Stankewitz. Density of repelling fixed points in the Julia set of a rational or entire semigroup, II. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2583-2589. doi: 10.3934/dcds.2012.32.2583

[20]

Hiroki Sumi, Mariusz Urbański. Measures and dimensions of Julia sets of semi-hyperbolic rational semigroups. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 313-363. doi: 10.3934/dcds.2011.30.313

2017 Impact Factor: 0.972

Metrics

  • PDF downloads (4)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]