Register shifts versus transitive f-cycles for piecewise monotone maps

Abstract

This paper investigates the family of continuous piecewise monotone functions which map a closed interval of the real line into itself. For these maps Preston \cite{1} and Blokh \cite{2} described the asymptotic behavior of the orbit of a “typical” point. Our results show that if the map is expanding on its intervals of monotonicity the dominant role is played by transitive \(f\)-cycles. Contrary to this for a “typical” map in a natural closure of the space of these maps there are no transitive \(f\)-cycles. Instead the behavior is dominated by the register shifts. This result is illustrated by an example.