American Institute of Mathematical Sciences

Advanced
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 $
Figure Options

Download full-size image

Download as PowerPoint slide

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 $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

James W. Cannon, Mark H. Meilstrup, Andreas Zastrow. The period set of a map from the Cantor set to itself. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 2667-2679. doi: 10.3934/dcds.2013.33.2667
[2]

Marie-Claude Arnaud. A nondifferentiable essential irrational invariant curve for a $C^1$ symplectic twist map. Journal of Modern Dynamics, 2011, 5 (3) : 583-591. doi: 10.3934/jmd.2011.5.583
[3]

Xu Xu, Xin Zhao. Exponential upper bounds on the spectral gaps and homogeneous spectrum for the non-critical extended Harper's model. Discrete & Continuous Dynamical Systems, 2020, 40 (8) : 4777-4800. doi: 10.3934/dcds.2020201
[4]

Marc Deschamps, Olivier Poncelet. Complex ray in anisotropic solids: Extended Fermat's principle. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1623-1633. doi: 10.3934/dcdss.2019110
[5]

Hsuan-Wen Su. Finding invariant tori with Poincare's map. Communications on Pure & Applied Analysis, 2008, 7 (2) : 433-443. doi: 10.3934/cpaa.2008.7.433
[6]

Amadeu Delshams, Josep J. Masdemont, Pablo Roldán. Computing the scattering map in the spatial Hill's problem. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 455-483. doi: 10.3934/dcdsb.2008.10.455
[7]

Michela Eleuteri, Jana Kopfov, Pavel Krej?. Fatigue accumulation in an oscillating plate. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 909-923. doi: 10.3934/dcdss.2013.6.909
[8]

Héctor A. Tabares-Ospina, Mauricio Osorio. Methodology for the characterization of the electrical power demand curve, by means of fractal orbit diagrams on the complex plane of Mandelbrot set. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1895-1905. doi: 10.3934/dcdsb.2020008
[9]

C. R. Chen, S. J. Li. Semicontinuity of the solution set map to a set-valued weak vector variational inequality. Journal of Industrial & Management Optimization, 2007, 3 (3) : 519-528. doi: 10.3934/jimo.2007.3.519
[10]

Yangrong Li, Shuang Yang. Backward compact and periodic random attractors for non-autonomous sine-Gordon equations with multiplicative noise. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1155-1175. doi: 10.3934/cpaa.2019056
[11]

V. V. Chepyzhov, M. I. Vishik, W. L. Wendland. On non-autonomous sine-Gordon type equations with a simple global attractor and some averaging. Discrete & Continuous Dynamical Systems, 2005, 12 (1) : 27-38. doi: 10.3934/dcds.2005.12.27
[12]

Shuang Yang, Yangrong Li. Forward controllability of a random attractor for the non-autonomous stochastic sine-Gordon equation on an unbounded domain. Evolution Equations & Control Theory, 2020, 9 (3) : 581-604. doi: 10.3934/eect.2020025
[13]

James Benn. Fredholm properties of the $L^{2}$ exponential map on the symplectomorphism group. Journal of Geometric Mechanics, 2016, 8 (1) : 1-12. doi: 10.3934/jgm.2016.8.1
[14]

Fernando Lenarduzzi. Recoding the classical Hénon-Devaney map. Discrete & Continuous Dynamical Systems, 2020, 40 (7) : 4073-4092. doi: 10.3934/dcds.2020172
[15]

Xiaoxiao Li, Yingjing Shi, Rui Li, Shida Cao. Energy management method for an unpowered landing. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020180
[16]

Gabriel Soler López. Accumulation points of flows on the Klein bottle. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 497-503. doi: 10.3934/dcds.2003.9.497
[17]

Yu-Hao Liang, Wan-Rou Wu, Jonq Juang. Fastest synchronized network and synchrony on the Julia set of complex-valued coupled map lattices. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 173-184. doi: 10.3934/dcdsb.2016.21.173
[18]

Chihiro Matsuoka, Koichi Hiraide. Special functions created by Borel-Laplace transform of Hénon map. Electronic Research Announcements, 2011, 18: 1-11. doi: 10.3934/era.2011.18.1
[19]

Ming Zhao, Cuiping Li, Jinliang Wang, Zhaosheng Feng. Bifurcation analysis of the three-dimensional Hénon map. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 625-645. doi: 10.3934/dcdss.2017031
[20]

Feng-Yu Wang. Exponential convergence of non-linear monotone SPDEs. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5239-5253. doi: 10.3934/dcds.2015.35.5239

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (128)
  • HTML views (94)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences