Uniqueness of Parry Maps, and Invariants for Transitive Piecewise Monotonic Maps

Parry showed that every continuous transitive piecewise monotonic map τ of the interval is conjugate (by an order preserving homeomorphism) to a uniformly piecewise linear map T (i.e., one with slopes ±s). In the current article, it is shown that the map T is unique. This is proven by showing that any order-preserving conjugacy between two continuous transitive uniformly piecewise linear maps is the identity map. (This is false for non-transitive maps.) This generalizes analogous results for transitive Markov maps by Block and Coven. Analogous results hold for discontinuous transitive piecewise monotonic maps if the maps have positive entropy, but fail in the zero entropy case. A map τ : [0, 1] → [0, 1] is piecewise monotonic if there is a partition 0 = a0 < a1 < . . . < an = 1 such that τ is continuous and strictly monotonic on each interval (ai−1, ai) for 1 ≤ i ≤ n. An example of such a map is a uniformly piecewise linear map, i.e., a map τ for which there is a partition 0 = a0 < a1 < . . . < an = 1 such that τ is linear on (ai−1, ai) for 1 ≤ i ≤ n, with slope ±s for some s > 0. Throughout this article, we write I for the interval [0, 1], and τ : I → I will be a piecewise monotonic map of the unit interval, not necessarily continuous unless that is stated.. It is well known that the quadratic map given by τ(x) = 4x(1− x) is transitive, and is conjugate to the uniformly piecewise linear map T (x) = 1− |1− 2x|. Parry [6] showed that every (strongly) transitive piecewise monotonic map τ is conjugate to a uniformly piecewise linear map T , cf. Theorem 2. The map T is not unique, since two or more uniformly piecewise linear maps can be conjugate. For example, the map φ(x) = 1 − x is a homeomorphism that conjugates any uniformly piecewise linear map T onto another uniformly piecewise linear map φ ◦T ◦φ−1, which we call the reflection of T , cf. Definition 6. However, we will show that if τ is a continuous transitive piecewise monotonic map, this is the extent of the ambiguity: two conjugate continuous transitive uniformly piecewise linear maps are either the same or are reflections of each other, cf. Theorem 10. (This is false without the assumption of transitivity, cf. Example 13.) Thus for a continuous transitive piecewise monotonic map, there is an order preserving homeomorphism onto a unique uniformly piecewise linear map T . We will prove analogous results for discontinuous τ with the added hypothesis of positive entropy. For continuous τ , uniqueness of this “Parry map” T (up to reflection) was proven with the additional assumption that τ was Markov by Block and Coven [1]. Date: April 12, 2007. 2000 Mathematics Subject Classification. Primary 37E05.