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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July
2020, 40(7): 4179-4196.
doi: 10.3934/dcds.2020177
A new type of non-landing exponential rays
| 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 |
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.
Mathematics Subject Classification:
Primary: 37F45; Secondary: 37F10.
Citation:
Jianxun Fu, Song Zhang. A new type of non-landing exponential rays. Discrete and 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. |
| [2] |
R. L. Devaney, Complex exponential dynamics, in Handbook of Dynamical Systems, Elsevier, North-Holland, 2010.
doi: 10.1016/S1874-575X(10)00312-7. |
| [3] |
R. L. Devaney,
Knaster-like continua and complex dynamics, Ergodic Theory Dynam. Systems, 13 (1993), 627-634.
doi: 10.1017/S0143385700007586. |
| [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. |
| [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. |
| [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. |
| [7] |
J. Milnor, Dynamics in One Complex Variable, Annals of Mathematics Studies, 160, Princeton University Press, Princeton, NJ, 2006. |
| [8] |
J. R. Munkres, Topology, Prentice Hall, Inc., Upper Saddle River, NJ, 2000. |
| [9] |
S. B. Nadler Jr., Continuum Theory. An Introduction, Monographs and Textbooks in Pure and Applied Mathematics, 158, Marcel Dekker, Inc., New York, 1992. |
| [10] |
M. H. A. Newman, Elements of the Topology of Plane Sets of Points, Cambridge University Press, New York, 1961.
|
| [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. |
| [12] |
L. Rempe, Arc-like continua, Julia sets of entire functions, and Eremenko's conjecture, preprint, arXiv: 1610.06278v3. |
| [13] |
L. Rempe, Dynamics of Exponential Maps, Ph.D thesis, Christian-Albrechts-Universität Kiel, 2003. |
| [14] |
L. Rempe,
On nonlanding dynamic rays of exponential maps, Ann. Acad. Sci. Fenn. Math., 32 (2007), 353-369.
|
| [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. |
| [16] |
D. Schleicher and J. Zimmer,
Escaping points of exponential maps, J. London. Math. Soc. (2), 67 (2003), 380-400.
doi: 10.1112/S0024610702003897. |
| [17] |
D. Schleicher and J. Zimmer,
Periodic points and dynamic rays of exponential maps, Ann. Acad. Sci. Fenn. Math., 28 (2003), 327-354.
|
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. |
| [2] |
R. L. Devaney, Complex exponential dynamics, in Handbook of Dynamical Systems, Elsevier, North-Holland, 2010.
doi: 10.1016/S1874-575X(10)00312-7. |
| [3] |
R. L. Devaney,
Knaster-like continua and complex dynamics, Ergodic Theory Dynam. Systems, 13 (1993), 627-634.
doi: 10.1017/S0143385700007586. |
| [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. |
| [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. |
| [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. |
| [7] |
J. Milnor, Dynamics in One Complex Variable, Annals of Mathematics Studies, 160, Princeton University Press, Princeton, NJ, 2006. |
| [8] |
J. R. Munkres, Topology, Prentice Hall, Inc., Upper Saddle River, NJ, 2000. |
| [9] |
S. B. Nadler Jr., Continuum Theory. An Introduction, Monographs and Textbooks in Pure and Applied Mathematics, 158, Marcel Dekker, Inc., New York, 1992. |
| [10] |
M. H. A. Newman, Elements of the Topology of Plane Sets of Points, Cambridge University Press, New York, 1961.
|
| [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. |
| [12] |
L. Rempe, Arc-like continua, Julia sets of entire functions, and Eremenko's conjecture, preprint, arXiv: 1610.06278v3. |
| [13] |
L. Rempe, Dynamics of Exponential Maps, Ph.D thesis, Christian-Albrechts-Universität Kiel, 2003. |
| [14] |
L. Rempe,
On nonlanding dynamic rays of exponential maps, Ann. Acad. Sci. Fenn. Math., 32 (2007), 353-369.
|
| [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. |
| [16] |
D. Schleicher and J. Zimmer,
Escaping points of exponential maps, J. London. Math. Soc. (2), 67 (2003), 380-400.
doi: 10.1112/S0024610702003897. |
| [17] |
D. Schleicher and J. Zimmer,
Periodic points and dynamic rays of exponential maps, Ann. Acad. Sci. Fenn. Math., 28 (2003), 327-354.
|
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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