Periodic points of solenoidal automorphisms in terms of inverse limits




solenoid, periodic points, inverse limits, Pontryagin dual


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.


Download data is not yet available.

Author Biography

Sharan Gopal, Birla Institute of Technology and Science - Pilani, Hyderabad campus

Assistant Professor,

Department of Mathematics


J. M. Aarts and R. J. Fokkink, The Classification of solenoids, Proc. Amer. Math. Soc. 111 (1991), 1161-1163.

L. M. Abramov, The entropy of an automorphism of a solenoidal group, Theory of Probability and its Applications 4 (1959), 231-236.

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.

D. M. Arnold, Finite Rank Torsion Free Abelian Groups and Rings, Vol. 931, Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1982.

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.

L. Block, Periods of periodic points of maps of the circle which have a fixed point, Proc. Amer. Math. Soc. 82 (1981), 481-486.

R. Bowen and J. Franks, The periodic points of maps of the disk and the interval, Topology 15 (1976), 337-342.

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.

A. Clark, Linear flows on κ-solenoids, Topology and its Applications 94 (1999), 27-49.

A. Clark, The rotation class of a flow, Topology and its Applications 152 (2005), 201-208.

J. W. England and R. L. Smith, The zeta function of automorphisms of solenoid groups, J. Math. Anal. Appl. 39 (1972), 112-121.

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.

M. C. McCord, Inverse limit sequences with covering maps, Trans. Amer. Math. Soc. 114 (1965), 197-209.

R. Miles, Periodic points of endomorphisms on solenoids and related groups, Bull. Lond. Math. Soc. 40 (2008), 696-704.

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.

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.

A. M. Wilson, On endomorphisms of a solenoid, Proc. Amer. Math. Soc. 55 (1976), 69-74.




How to Cite

S. Gopal and F. Imam, “Periodic points of solenoidal automorphisms in terms of inverse limits”, Appl. Gen. Topol., vol. 22, no. 2, pp. 321–330, Oct. 2021.