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July  2020, 40(7): 4179-4196. doi: 10.3934/dcds.2020177

A new type of non-landing exponential rays

Jianxun Fu 1, and  Song Zhang 2,,

1. 

Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan 430074, China
2. 

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

* Corresponding author: Song Zhang

Received  April 2019 Revised  January 2020 Published  April 2020

Fund Project: This work is supported by the Fundamental Research Funds for the Central Universities (Grant No. 2019kfyXJJS135) and the National Natural Science Foundation of China (Grant No. 11901219)

In this paper, we will construct a new type of non-landing exponential rays, each of whose accumulation sets is bounded, disjoint from the ray and homeomorphic to the closed topologist's sine curve.

Keywords: Exponential map, itinerary, non-landing ray, accumulation set, topologist's sine curve.
Mathematics Subject Classification: Primary: 37F45; Secondary: 37F10.
Citation: Jianxun Fu, Song Zhang. A new type of non-landing exponential rays. Discrete & Continuous Dynamical Systems, 2020, 40 (7) : 4179-4196. doi: 10.3934/dcds.2020177
References:
[1]

C. Bodelón, R. L. Devaney, L. Goldberg, M. Hayes, J. Hubbard and G. Roberts, Hairs for the complex exponential family, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 9 (1999), 1517-1534.  doi: 10.1142/S0218127499001061.  Google Scholar

[2]

R. L. Devaney, Complex exponential dynamics, in Handbook of Dynamical Systems, Elsevier, North-Holland, 2010. doi: 10.1016/S1874-575X(10)00312-7.  Google Scholar

[3]

R. L. Devaney, Knaster-like continua and complex dynamics, Ergodic Theory Dynam. Systems, 13 (1993), 627-634.  doi: 10.1017/S0143385700007586.  Google Scholar

[4]

R. L. Devaney and X. Jarque, Indecomposable continua in exponential dynamics, Conform. Geom. Dyn., 6 (2002), 1-12.  doi: 10.1090/S1088-4173-02-00080-2.  Google Scholar

[5]

R. L. Devaney, X. Jarque and M. M. Rocha, Indecomposable continua and Misiurewicz points in exponential dynamics, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 3281-3293.  doi: 10.1142/S0218127405013885.  Google Scholar

[6]

J. Fu and G. Zhang, On the accumulation sets of exponential rays, Ergodic Theory Dynam. Systems, 39 (2019), 370-391.  doi: 10.1017/etds.2017.33.  Google Scholar

[7]

J. Milnor, Dynamics in One Complex Variable, Annals of Mathematics Studies, 160, Princeton University Press, Princeton, NJ, 2006.  Google Scholar

[8]

J. R. Munkres, Topology, Prentice Hall, Inc., Upper Saddle River, NJ, 2000.  Google Scholar

[9]

S. B. Nadler Jr., Continuum Theory. An Introduction, Monographs and Textbooks in Pure and Applied Mathematics, 158, Marcel Dekker, Inc., New York, 1992.  Google Scholar

[10] M. H. A. Newman, Elements of the Topology of Plane Sets of Points, Cambridge University Press, New York, 1961.   Google Scholar
[11]

L. Rempe, A landing theorem for periodic rays of exponential maps, Proc. Amer. Math. Soc., 134 (2006), 2639-2648.  doi: 10.1090/S0002-9939-06-08287-6.  Google Scholar

[12]

L. Rempe, Arc-like continua, Julia sets of entire functions, and Eremenko's conjecture, preprint, arXiv: 1610.06278v3. Google Scholar

[13]

L. Rempe, Dynamics of Exponential Maps, Ph.D thesis, Christian-Albrechts-Universität Kiel, 2003. Google Scholar

[14]

L. Rempe, On nonlanding dynamic rays of exponential maps, Ann. Acad. Sci. Fenn. Math., 32 (2007), 353-369.   Google Scholar

[15]

G. Rottenfusser, J. Rückert, L. Rempe and D. Schleicher, Dynamic rays of bounded-type entire functions, Ann. of Math. (2), 173 (2011), 77-125.  doi: 10.4007/annals.2011.173.1.3.  Google Scholar

[16]

D. Schleicher and J. Zimmer, Escaping points of exponential maps, J. London. Math. Soc. (2), 67 (2003), 380-400.  doi: 10.1112/S0024610702003897.  Google Scholar

[17]

D. Schleicher and J. Zimmer, Periodic points and dynamic rays of exponential maps, Ann. Acad. Sci. Fenn. Math., 28 (2003), 327-354.   Google Scholar

show all references

References:
[1]

C. Bodelón, R. L. Devaney, L. Goldberg, M. Hayes, J. Hubbard and G. Roberts, Hairs for the complex exponential family, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 9 (1999), 1517-1534.  doi: 10.1142/S0218127499001061.  Google Scholar

[2]

R. L. Devaney, Complex exponential dynamics, in Handbook of Dynamical Systems, Elsevier, North-Holland, 2010. doi: 10.1016/S1874-575X(10)00312-7.  Google Scholar

[3]

R. L. Devaney, Knaster-like continua and complex dynamics, Ergodic Theory Dynam. Systems, 13 (1993), 627-634.  doi: 10.1017/S0143385700007586.  Google Scholar

[4]

R. L. Devaney and X. Jarque, Indecomposable continua in exponential dynamics, Conform. Geom. Dyn., 6 (2002), 1-12.  doi: 10.1090/S1088-4173-02-00080-2.  Google Scholar

[5]

R. L. Devaney, X. Jarque and M. M. Rocha, Indecomposable continua and Misiurewicz points in exponential dynamics, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 3281-3293.  doi: 10.1142/S0218127405013885.  Google Scholar

[6]

J. Fu and G. Zhang, On the accumulation sets of exponential rays, Ergodic Theory Dynam. Systems, 39 (2019), 370-391.  doi: 10.1017/etds.2017.33.  Google Scholar

[7]

J. Milnor, Dynamics in One Complex Variable, Annals of Mathematics Studies, 160, Princeton University Press, Princeton, NJ, 2006.  Google Scholar

[8]

J. R. Munkres, Topology, Prentice Hall, Inc., Upper Saddle River, NJ, 2000.  Google Scholar

[9]

S. B. Nadler Jr., Continuum Theory. An Introduction, Monographs and Textbooks in Pure and Applied Mathematics, 158, Marcel Dekker, Inc., New York, 1992.  Google Scholar

[10] M. H. A. Newman, Elements of the Topology of Plane Sets of Points, Cambridge University Press, New York, 1961.   Google Scholar
[11]

L. Rempe, A landing theorem for periodic rays of exponential maps, Proc. Amer. Math. Soc., 134 (2006), 2639-2648.  doi: 10.1090/S0002-9939-06-08287-6.  Google Scholar

[12]

L. Rempe, Arc-like continua, Julia sets of entire functions, and Eremenko's conjecture, preprint, arXiv: 1610.06278v3. Google Scholar

[13]

L. Rempe, Dynamics of Exponential Maps, Ph.D thesis, Christian-Albrechts-Universität Kiel, 2003. Google Scholar

[14]

L. Rempe, On nonlanding dynamic rays of exponential maps, Ann. Acad. Sci. Fenn. Math., 32 (2007), 353-369.   Google Scholar

[15]

G. Rottenfusser, J. Rückert, L. Rempe and D. Schleicher, Dynamic rays of bounded-type entire functions, Ann. of Math. (2), 173 (2011), 77-125.  doi: 10.4007/annals.2011.173.1.3.  Google Scholar

[16]

D. Schleicher and J. Zimmer, Escaping points of exponential maps, J. London. Math. Soc. (2), 67 (2003), 380-400.  doi: 10.1112/S0024610702003897.  Google Scholar

[17]

D. Schleicher and J. Zimmer, Periodic points and dynamic rays of exponential maps, Ann. Acad. Sci. Fenn. Math., 28 (2003), 327-354.   Google Scholar

Figure 1.  The folding process of $ \gamma_{6}^0 $ by $ g_{1}^- \circ g_{2}^+ \circ \cdots\circ g_{5}^-\circ g_{6}^+ $ for the above choice of $ \xi_i $ with $ i = 0,1,\cdots,5 $
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Figure 2.  Here is the sketch of $ \gamma_0^{-12} $. The black polylines with different-sized bold show the idea of how the limit curve $ \eta $ is produced in the accumulation set of $ \gamma_0 $
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