# Periodic points and normal families concerning multiplicity

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## Abstract

In 1992, Yang Lo posed the following problem: let *F* be a family of entire functions, let *D* be a domain in ℂ, and let *k* ⩾ 2 be a positive integer. If, for every *f* ∈ *F*, both *f* and its iteration *f*^{k} have no fixed points in *D*, is *F* normal in *D*? This problem was solved by Essén and Wu in 1998, and then solved for meromorphic functions by Chang and Fang in 2005. In this paper, we study the problem in which *f* and *f*^{k} have fixed points. We give positive answers for holomorphic and meromorphic functions.

- (I)
Let

*F*be a family of holomorphic functions in a domain*D*and let*k*⩾ 2 be a positive integer. If, for each*f*∈*F*, all zeros of*f*(*z*) −*z*are multiple and*f*^{k}has at most*k*distinct fixed points in*D*, then*F*is normal in*D*. Examples show that the conditions “all zeros of*f*(*z*) −*z*are multiple” and “*f*^{k}having at most*k*distinct fixed points in*D*” are the best possible. - (II)
Let

*F*be a family of meromorphic functions in a domain*D*, and let*k*⩾ 2;*l*be two positive integers satisfying*l*⩾ 4 for*k*= 2 and*l*⩾ 3 for*k*⩾ 3. If, for each*f*∈*F*, all zeros of*f*(*z*) −*z*have a multiplicity at least*l*and*f*^{k}has at most one fixed point in*D*, then*F*is normal in*D*. Examples show that the conditions “*l*⩾ 3 for*k*⩾ 3” and “*f*^{k}having at most one fixed point in*D*” are the best possible.

## Keywords

normality iteration periodic points## MSC(2010)

35D45## Preview

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## Notes

### Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11371149 and 11231009 ) and the Graduate Student Overseas Study Program from South China Agricultural University (Grant No. 2017LHPY003). The authors thank the referees for valuable suggestions.

## References

- 1.Baker I N. Fix-points of polynomials and rational functions. J London Math Soc (2), 1964, 39: 615–622CrossRefzbMATHGoogle Scholar
- 2.Bergweiler W. Iteration of meromorphic functions. Bull Amer Math Soc (NS), 1993, 29: 151–188MathSciNetCrossRefzbMATHGoogle Scholar
- 3.Chang J M, Fang M L. Normal families and fixed points. J Anal Math, 2005, 95: 389–395MathSciNetCrossRefzbMATHGoogle Scholar
- 4.Chang J M, Fang M L. Repelling periodic points of given periods of rational functions. Sci China Ser A, 2006, 49: 1165–1174MathSciNetCrossRefzbMATHGoogle Scholar
- 5.Chen H H, Gu Y X. Improvement of Marty’s criterion and its application. Sci China Ser A, 1993, 36: 674–681MathSciNetzbMATHGoogle Scholar
- 6.Essén M, Wu S J. Fix-points and a normal families of analytic functions. Complex Variables, 1998, 37: 171–178zbMATHGoogle Scholar
- 7.Essén M, Wu S J. Repulsive fixpoints of analytic functions with application to complex dynamics. J Lond Math Soc (2), 2000, 62: 139–148MathSciNetCrossRefzbMATHGoogle Scholar
- 8.Hayman W K. Meromorphic Functions. Oxford: Clarendon Press, 1964zbMATHGoogle Scholar
- 9.Pang X C, Zalcman L. Normal families and shared values. Bull Lond Math Soc, 2000, 32: 325–331MathSciNetCrossRefzbMATHGoogle Scholar
- 10.Yang L. Some recent results and problems in the theory of value distribution. In: Proceedings of the Symposium on Value Distribution Theory in Several Complex Variables. Notre Dame: University of Notre Dame Press, 1992: 157–171Google Scholar
- 11.Yang L. Value Distribution Theory. Berlin: Springer-Verlag, 1993zbMATHGoogle Scholar
- 12.Zalcman L. Normal families: New perspectives. Bull Amer Math Soc (NS), 1998, 35: 215–230MathSciNetCrossRefzbMATHGoogle Scholar