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The Classification of the Conjugacy Classes of the Full Group of Homeomorphisms of an Open Interval and the General Solution of Certain Functional Equations
Author(s) -
McCarthy Patrick J.,
Stephenson W.
Publication year - 1985
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-51.1.95
Subject(s) - conjugacy class , mathematics , iterated function , bijection , interval (graph theory) , group (periodic table) , class (philosophy) , combinatorics , integer (computer science) , pure mathematics , discrete mathematics , mathematical analysis , chemistry , organic chemistry , artificial intelligence , computer science , programming language
We present a classification of the conjugacy classes of the full group G of self‐homeomorphisms of an open interval I of the real line, and apply this classification to the solution of certain functional equations. Namely, for β: I → I a given continuous bijection and n a positive integer we find, explicitly, the general continuous solution f : I → I of the functional equation f n = β (F), where the power denotes iterated function composition. The given β is necessarily increasing or decreasing, and the solutions of (F) are naturally grouped into conjugacy classes of G . We find that (i) for n odd, (F) admits exactly one conjugacy class of solutions; (ii) for n even and β increasing, there is only one conjugacy class of increasing solutions of (F), whilst the number of conjugacy classes d of decreasing solutions of (F) depends on the symmetry properties of β and can be d = 0, 1, 2, 3,…, ∞ (iii) for n even and β decreasing, (F) admits no continuous solutions at all and, in this case, we show that any solution of (F) must have infinitely many discontinuities.