Abstract
Let I be an interval in the real line ℝ. Among the real polynomials that take I to I, we ask which ones do not commute with any increasing bijection of I other than identity. For this purely algebraic problem, the solution involves concepts in topological dynamics. Our main characterizations are in terms of full orbits of critical points and periodic points. Using these, we obtain simpler criterion, namely, that for no nontrivial subinterval K⊂I, the successive images {f n(K):n=0,1,2,…} form a pairwise disjoint collection. This problem is of interest in topological dynamics because it is about characterization of polynomials with unique self-topological-conjugacy.
Similar content being viewed by others
References
Block, L.S., Coppel, W.A.: Dynamics in One Dimension. Lecture Notes in Mathematics, vol. 1513. Springer, New York (1992)
Devaney, R.L.: An Introduction to Chaotic Dynamical Systems. Addison-Wesley, Reading (1989)
Holmgren, R.A.: A First Course in Discrete Dynamical Systems. Springer, New York (1996)
Subrahmonian Moothathu, T.K.: Sensitive maps on an interval. In: Proceedings of the First National Conference on Nonlinear Systems and Dynamics, IIT Kharagpur, pp. 255–256 (2003)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jimmie D. Lawson.
Rights and permissions
About this article
Cite this article
Kannan, V., Sankararao, B., Subramania Pillai, I. et al. Polynomials hardly commuting with increasing bijections. Semigroup Forum 76, 124–132 (2008). https://doi.org/10.1007/s00233-007-9031-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00233-007-9031-7