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August  2013, 33(8): 3321-3327. doi: 10.3934/dcds.2013.33.3321

No entire function with real multipliers in class $\mathcal{S}$

Agnieszka Badeńska 1,

1. 

Faculty of Mathematics and Information Science, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warszawa, Poland

Received  July 2012 Revised  August 2012 Published  January 2013

We prove that there is no entire transcendental function in class $\mathcal{S}$ with real multipliers of all repelling periodic orbits.
Keywords: Entire transcendental functions, multiplier, periodic orbit, invariant line field..
Mathematics Subject Classification: Primary: 30D05; Secondary: 37F1.
Citation: Agnieszka Badeńska. No entire function with real multipliers in class $\mathcal{S}$. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3321-3327. doi: 10.3934/dcds.2013.33.3321
References:
[1]

A. Badeńska, Measure rigidity for some transcendental meromorphic functions, Discrete Contin. Dyn. Syst., 32 (2012), 2375-2402. doi: 10.3934/dcds.2012.32.2375.  Google Scholar

[2]

W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc., 29 (1993), 151-188. doi: 10.1090/S0273-0979-1993-00432-4.  Google Scholar

[3]

A. Eremenko and S. van Strien, Rational maps with real multipliers, Trans. Amer. Math. Soc., 363 (2011), 6453-6463. doi: 10.1090/S0002-9947-2011-05308-0.  Google Scholar

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P. Fatou, Sur les équations fonctionnelles, Bull. Soc. Math. France, 48 (1920), 208-314.  Google Scholar

[5]

V. Mayer, Comparing measures and invariant line fields, Ergodic Theory Dynam. Systems, 22 (2002), 555-570. doi: 10.1017/S0143385702000275.  Google Scholar

[6]

L. Rempe and S. van Strien, Absence of line fields and Mañé's theorem for nonrecurrent transcendental functions, Trans. Amer. Math. Soc., 363 (2011), 203-228. doi: 10.1090/S0002-9947-2010-05125-6.  Google Scholar

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G. M. 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

show all references

References:
[1]

A. Badeńska, Measure rigidity for some transcendental meromorphic functions, Discrete Contin. Dyn. Syst., 32 (2012), 2375-2402. doi: 10.3934/dcds.2012.32.2375.  Google Scholar

[2]

W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc., 29 (1993), 151-188. doi: 10.1090/S0273-0979-1993-00432-4.  Google Scholar

[3]

A. Eremenko and S. van Strien, Rational maps with real multipliers, Trans. Amer. Math. Soc., 363 (2011), 6453-6463. doi: 10.1090/S0002-9947-2011-05308-0.  Google Scholar

[4]

P. Fatou, Sur les équations fonctionnelles, Bull. Soc. Math. France, 48 (1920), 208-314.  Google Scholar

[5]

V. Mayer, Comparing measures and invariant line fields, Ergodic Theory Dynam. Systems, 22 (2002), 555-570. doi: 10.1017/S0143385702000275.  Google Scholar

[6]

L. Rempe and S. van Strien, Absence of line fields and Mañé's theorem for nonrecurrent transcendental functions, Trans. Amer. Math. Soc., 363 (2011), 203-228. doi: 10.1090/S0002-9947-2010-05125-6.  Google Scholar

[7]

G. M. 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

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