Register shifts versus transitive \(F\)-cycles for piecewise monotone maps. (English) Zbl 0849.26002

Bobok, Jozef; Kuchta, Milan

Real Anal. Exch. 21(1995-96), No. 1, 134-146 (1996).

The family of continuous piecewise monotonic selfmappings of a real compact interval is considered. It is shown that if the mapping has a derivative on any subinterval of monotonicity bigger than some number greater than one, then the “typical” point is attracted by a periodic interval in which some orbit is dense (transitive \(f\)-cycle). Contrary to this, for a “typical” function from the space of piecewise monotonic functions with derivatives greater or equal to one a “typical” point is attracted by a register shift, i.e., has an asymptotic behavior similar to periodic one.

Reviewer: K.Janková (Bratislava)

MSC:

26A18	Iteration of real functions in one variable
54H20	Topological dynamics (MSC2010)
37B99	Topological dynamics

Keywords:

iteration; transitive \(f\)-cycle; continuous piecewise monotonic selfmappings; register shift