Simple dynamical systems
Keywords:special points, non-ordinary points, critical points, order conjugacy
In this paper, we study the class of simple systems on R induced by homeomorphisms having finitely many non-ordinary points. We characterize the family of homeomorphisms on R having finitely many non-ordinary points upto (order) conjugacy. For x,y âˆˆ R, we say x âˆ¼ y on a dynamical system (R,f) if x and y have same dynamical properties, which is an equivalence relation. Said precisely, x âˆ¼ y if there exists an increasing homeomorphism h : R â†’ R such that h â—¦ f = f â—¦ h and h(x) = y. An element x âˆˆ R is ordinary in (R,f) if its equivalence class [x] is a neighbourhood of it.
L. S. Block and W. A. Coppel, Dynamics in One Dimension, Volume 1513 of Lecture Notes in Mathematics, Springer-Verlag, Berline, 1992. https://doi.org/10.1007/BFb0084762
L. Block and E. Coven, Topological conjugacy and transitivity for a class of piecewise monotone maps of the interval, Trans. Amer. Math. Soc. 300 (1987), 297-306. https://doi.org/10.1090/S0002-9947-1987-0871677-X
M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press, 2002. https://doi.org/10.1017/CBO9780511755316
R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley Publishing Company Advanced Book Program, Redwood City, CA, second edition, 1989.
R. A. Holmgren, A First Course in Discrete Dynamical Systems, Springer-Verlag, New York, 1996. https://doi.org/10.1007/978-1-4419-8732-7
S. Sai, Symbolic dynamics for complete classification, Ph.D Thesis, University of Hyderabad, 2000.
B. Sankara Rao, I. Subramania Pillai and V. Kannan, The set of dynamically special points, Aequationes Mathematicae 82, no. 1-2 (2011), 81-90. https://doi.org/10.1007/s00010-010-0066-6
A. N. Sharkovskii, Coexistence of cycles of a continuous map of a line into itself, Ukr. Math. Z. 16 (1964), 61-71.
How to Cite
This journal is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.