Glasnik Matematicki, Vol. 57, No. 2 (2022), 239-250. \( \)

THE HAUSDORFF DIMENSION OF DIRECTIONAL EDGE ESCAPING POINTS SET

Xiaojie Huang, Zhixiu Liu and Yuntong Li

Department of Science, Nanchang Institute of Technology, 330099 Nanchang, China
e-mail:359536229@qq.com

Department of Science, Nanchang Institute of Technology, 330099 Nanchang, China
e-mail:270144355@qq.com

Department of Basic Courses, Shaanxi Railway Institute, 714000 Weinan, China
e-mail:liyuntong2005@sohu.com

Abstract.   In this paper, we define the directional edge escaping points set of function iteration under a given plane partition and then prove that the upper bound of Hausdorff dimension of the directional edge escaping points set of \(S(z)=a e^{z}+b e^{-z}\), where \(a, b\in \mathbb{C}\) and \(|a|^{2}+|b|^{2}\neq 0\), is no more than 1.

2020 Mathematics Subject Classification.   30D05, 37F10, 37F35

Key words and phrases.   Directional edge escaping points set, plane partition, function iteration, exponential function, Hausdorff dimension

https://doi.org/10.3336/gm.57.2.05

References:

1. M. Bailesteanu, H. V. Balan and D. Schleicher, Hausdorff dimension of exponential parameter rays and their endpoints, Nonlinearity 21 (2008), 113–120.
   MathSciNet    CrossRef

2. C. J. Bishop, A transcendental Julia set of dimension 1, Invent. Math. 212 (2018), 407–460.
   MathSciNet    CrossRef

3. R. L. Devaney and M. Krych, Dynamics of \(e^{z}\), Ergodic Theory Dynam. Systems 4 (1984), 35–52.
   MathSciNet    CrossRef

4. A. E. Eremenko, On the iteration of entire functions, in Dynamical systems and ergodic theory, PWN, Warsaw, 1989, 339–345.
   MathSciNet

5. K. Falconer, Fractal geometry. Mathematical foundations and applications, Wiley Publishing, Chichester, 1990.
   MathSciNet

6. M. Förster and D. Schleiher, Parameter rays in the space of exponential maps, Ergodic Theory Dynam. Systems 29 (2009), 515–544.
   MathSciNet    CrossRef

7. X. J. Huang and W. Y. Qiu, The dimension paradox in parameter space of cosine family, Chinese Ann. Math. Ser. B 41 (2020), 645–656.
   MathSciNet    CrossRef

8. B. Karpińska, Hausdorff dimension of the hairs without endpoints for \(\lambda \exp(z)\), C. R. Acad. Sci. Paris Sér. I Math 328 (1999), 1039–1044.
   MathSciNet    CrossRef

9. B. Karpińska, Area and Hausdorff dimension of the set of accessible points of the Julia sets of \(\lambda e^z\) and \(\lambda \sin(z)\), Fund.  Math. 159 (1999), 269–287.
   MathSciNet    CrossRef

10. C. McMullen, Area and Hausdorff dimension of Julia sets of entire functions, Trans. Amer. Math. Soc. 300 (1987), 329–342.
    MathSciNet    CrossRef

11. F. C. Moon, Chaotic and fractal dynamics, Wiley Publishing, New York, 1992.
    MathSciNet    CrossRef

12. W. Y. Qiu, Hausdorff Dimension of the \(M\)-Set of \(\lambda \exp(z)\), Acta Math. Sin. (N.S.) 10 (1994), 362–368.
    MathSciNet    CrossRef

13. G. Rottenfusser and D. Schleicher, Escaping points of the cosine family, in Transcendental dynamics and complex analysis, Cambridge Univ. Press, Cambridge, 2008, 396–424.
    MathSciNet    CrossRef

14. L. Rempe, D. Schleicher and M. Förster, Classification of escaping exponential maps, Proc. Amer. Math. Soc. 136 (2008), 651–663.
    MathSciNet    CrossRef

15. D. Schleicher, The dynamical fine structure of iterated cosine maps and a dimension paradox, Duke Math. J. 136 (2007), 343–356.
    MathSciNet    CrossRef

16. D. Schleicher and J. Zimmer, Escaping points of exponential maps, J. London Math. Soc. (2) 67 (2003), 380–400.
    MathSciNet    CrossRef

17. G. M. Stallard, The Hausdorff dimension of Julia sets of entire functions. III, Math. Proc. Cambridge Philos. Soc. 122 (1997), 223–244.
    MathSciNet    CrossRef

18. G. M. Stallard, The Hausdorff dimension of Julia sets of entire functions. IV, J. London Math. Soc. (2) 61 (2000), 471–488.
    MathSciNet    CrossRef

19. T. Tian, Parameter rays for the cosine family, J. Fudan Univ. Nat. Sci. 50 (2011), 10–22.
    MathSciNet