Periodic points of solenoidal automorphisms in terms of inverse limits
DOI:
https://doi.org/10.4995/agt.2021.14589Keywords:
solenoid, periodic points, inverse limits, Pontryagin dualAbstract
In this paper, we describe the periodic points of automorphisms of a one dimensional solenoid, considering it as the inverse limit, limâ†k (S 1 , γk) of a sequence (γk) of maps on the circle S 1 . The periodic points are discussed for a class of automorphisms on some higher dimensional solenoids also.
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J. M. Aarts and R. J. Fokkink, The Classification of solenoids, Proc. Amer. Math. Soc. 111 (1991), 1161-1163. https://doi.org/10.1090/S0002-9939-1991-1042260-7
L. M. Abramov, The entropy of an automorphism of a solenoidal group, Theory of Probability and its Applications 4 (1959), 231-236. https://doi.org/10.1137/1104025
K. Ali Akbar, V. Kannan, S. Gopal and P. Chiranjeevi, The set of periods of periodic points of a linear operator, Linear Algebra and its Applications 431 (2009), 241-246. https://doi.org/10.1016/j.laa.2009.02.027
D. M. Arnold, Finite Rank Torsion Free Abelian Groups and Rings, Vol. 931, Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1982. https://doi.org/10.1007/BFb0094245
R. H. Bing, A simple closed curve is the only homogeneous bounded plane continuum that contains an arc, Canad. J. Math. 12 (1960), 209-230. https://doi.org/10.4153/CJM-1960-018-x
L. Block, Periods of periodic points of maps of the circle which have a fixed point, Proc. Amer. Math. Soc. 82 (1981), 481-486. https://doi.org/10.1090/S0002-9939-1981-0612745-7
R. Bowen and J. Franks, The periodic points of maps of the disk and the interval, Topology 15 (1976), 337-342. https://doi.org/10.1016/0040-9383(76)90026-4
P. Chiranjeevi, V. Kannan and S. Gopal, Periodic points and periods for operators on Hilbert space, Discrete and Continuous Dynamical Systems 33 (2013), 4233-4237. https://doi.org/10.3934/dcds.2013.33.4233
A. Clark, Linear flows on κ-solenoids, Topology and its Applications 94 (1999), 27-49. https://doi.org/10.1016/S0166-8641(98)00023-6
A. Clark, The rotation class of a flow, Topology and its Applications 152 (2005), 201-208. https://doi.org/10.1016/j.topol.2004.10.019
J. W. England and R. L. Smith, The zeta function of automorphisms of solenoid groups, J. Math. Anal. Appl. 39 (1972), 112-121. https://doi.org/10.1016/0022-247X(72)90228-4
S. Gopal and C. R. E. Raja, Periodic points of solenoidal automorphisms, Topology Proceedings 50 (2017), 49-57.
J. Keesling, The group of homeomorphisms of a solenoid, Trans. Amer. Math. Soc. 172 (1972), 119-131. https://doi.org/10.1090/S0002-9947-1972-0315735-6
M. C. McCord, Inverse limit sequences with covering maps, Trans. Amer. Math. Soc. 114 (1965), 197-209. https://doi.org/10.1090/S0002-9947-1965-0173237-0
R. Miles, Periodic points of endomorphisms on solenoids and related groups, Bull. Lond. Math. Soc. 40 (2008), 696-704. https://doi.org/10.1112/blms/bdn052
S. A. Morris, Pontryagin Duality and the Structure of Locally Compact Abelian Groups, London Mathematical Society Lecture Note Series, no. 29, Cambridge Univ. Press, 1977. https://doi.org/10.1017/CBO9780511600722
A. N. Sharkovsky, Coexistence of cycles of a continuous map of the line into itself, Ukrain. Mat. Zh. 16 (1964), 61-71. (Russian)
English translation: International Journal of Bifurcation and Chaos in Appl. Sci. Engg. 5 (1995), 1263-1273.
T. K. Subrahmonian Moothathu, Set of periods of additive cellular automata, Theoretical Computer Science 352 (2006), 226-231. https://doi.org/10.1016/j.tcs.2005.10.050
A. M. Wilson, On endomorphisms of a solenoid, Proc. Amer. Math. Soc. 55 (1976), 69-74. https://doi.org/10.1090/S0002-9939-1976-0390181-7
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