On iteration of bijective functions with discontinuities
Abstract
We present three different types of bijective functions f:I→I on a compact interval I with finitely many discontinuities where certain iterates of these functions will be continuous. All these examples are strongly related to permutations, in particular to derangements in the first case, and permutations with a certain number of successions (or small ascents) in the second case. All functions of type III form a direct product of a symmetric group with a wreath product. It will be shown that any iterative root F:J→J of the identity of order k on a compact interval J with finitely many discontinuities is conjugate to a function f of type III, i.e., F = ϕ-1◦f◦ϕ where ϕ is a continuous, bijective, and increasing mapping between J and [0,n] for some integer n.
Keywords
iteration of functions; bijective functions on a compact interval; derangements; permutations with successions; wreath products; iterative roots of the identity; enumeration
References
2. L. Comtet, Advanced Combinatorics, D. Reidel Publishing Co., Dordrecht, 1974.
3. G. James and A. Kerber, The Representation Theory of the Symmetric Group, Encyclopedia of Mathematics and its Applications, 16, Addison-Wesley Publishing Co., Reading, 1981.
4. A. Kerber, Applied Finite Group Actions, Algorithms and Combinatorics, 19, Springer-Verlag, Berlin, 1999.
5. M. Kuczma, B. Choczewski and R. Ger, Iterative Functional Equations, Encyclopedia of Mathematics and its Applications, 32, Cambridge University Press, Cambridge, 1990.
6. Report of Meeting, Aequationes Math. 91 (2017), no. 6, 1157–1204.
7. SYMMETRICA, 1987. A program system devoted to representation theory, invariant theory and combinatorics of finite symmetric groups and related classes of groups. Copyright by “Lehrstuhl II für Mathematik, Universität Bayreuth, 95440 Bayreuth”. Avaliable at http://www.algorithm.uni-bayreuth.de/en/research/SYMMETRICA/.
Institute of Mathematics and Scientific Computing, University of Graz, Austria Austria
https://orcid.org/0000-0001-7449-8532
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