Measure preserving interval maps and topological conjugacy (Q6156199)

Continuous maps ${s_1}$ and ${s_2}:\,\,[0,1] \to [0,1]$ are topologically conjugate if there exists a homeomorphism $h:[0,1] \to [0,1]$ such that ${s_2} = h \circ {s_1} \circ {h^{ - 1}}$. The goal of this paper is to characterize a continuous map $s$ such that a measure preserving interval map $t$ exists to be topologically conjugate with $s$. In Section 2, the author formulates the problem and shows that under mild assumptions this problem is reduced to a Markov chain problem. Section 3 presents the main results of this paper, which are the conditions for such a $t$ to exist and be unique in variuous scenarios.