Topological conjugacy of piecewise monotonic functions of nonmonotonicity height ≥1

Abstract

The conjugacy problem is an important topic in the theory of dynamical systems and functional equations. In this paper, we investigate a class of piecewise monotone and continuous maps with nonmonotonicity height ≥ 1. We give a sufficient and necessary condition under which any two of these maps are topologically conjugate, and construct a topological conjugacy with an extension method if such a conjugacy exists.

Topological conjugacy of PM functions with height equaling ∞

“…We remark that, the sequence (i n (t)) n∈N is different from (i k (x)) k∈N 0 in Ref. [11] since the set A f is not equal to S(f) in there. In fact, the sequence (i n (t)) n∈N is not necessarily monotone, now we present two examples to describe the monotone and non-monotone cases, respectively.…”

“…As a continuation of [18], Ref. [11] investigated the more generally cases H(f) 1, i.e., f([a, t 1 ]) ⊂ [a, t 1 ] and f(t) < t for all t ∈ (t 1 , b) (here, t 1 is the smallest fort). Under the assumptions that f is topologically conjugate to g by ϕ 0 on subinterval (a, t 1 ) and has the same sequences of endpoints and forts as that of g, they proved that f ∼ g by ϕ, an extension of ϕ 0 .…”

“…It is worth mentioning that we need not the assumption on the existence of the topological conjugacy on a subinterval, which is different from Refs. [11,18] and Ref. [12].…”

Conjugacy between piecewise monotonic functions and their iterative roots

“…In fact, [8] requires the characteristic interval to be on the left end of I, but we do not need such a requirement.…”

“…Thus, one problem was raised naturally: Are non-monotonic functions conjugate to their iterative roots? Notice that [8,13] investigated topological conjugacy between two non-monotonic functions. However, their conditions are hardly verified in our case because two functions can be given separately, while functions and their iterative roots have strong relations.…”