Topological conjugacy and transitivity for a class of piecewise monotone maps of the interval

Abstract:

We say that a continuous map $f$ of a compact interval to itself is linear Markov if it is piecewise linear, and the set of all ${f^k}(x)$, where $k \geqslant 0$ and $x$ is an endpoint of a linear piece, is finite. We provide an effective classification, up to topological conjugacy, for linear Markov maps and an effective procedure for determining whether such a map is transitive. We also consider expanding Markov maps, partly to motivate the proof of the more complicated linear Markov case.