Piecewise Monotonicity Research Articles

Least Squares Smoothing of Univariate Data to achieve Piecewise Monotonicity

Least Squares Smoothing of Univariate Data to achieve Piecewise Monotonicity

Entropy of continuous flows on compact 2-manifolds

THEOREM. Let Q, be a continuous flow on a compact 2-dimensional manifold M. Then h(cp,) = 0. The idea of the proof is as follows: Via an ergodic measure we can realize the flow as a suspension of part of a circle. The problem is then reduced to a l-dimensional one and the base transformation has entropy zero for the same reason that homeomorphisms of the circle have entropy zero. I wish to thank R. Bowen for suggesting this problem. LEMMA 1. Let D C S’ be a subset of the circle. Let f: D+ D be a piecewise monotonic bijection and m an f-invatiant Bore1 probability measure on D. Then the measure-theoretic entropy off, h,(f) = 0. Proof. Piecewise monotonicity of f means that S’ is the union of finitely many disjoint intervals {I,, Z,, . . ., Zk} such that f restricted to each (4 fl D) is order-preserving. Let Q = {A,, . . ., A,} be any finite partition of D with the properties that Ai = D n (some interval) and that each Aj C some Zi. For any two partitions /3 and y of this kind, card (~3 v y)scard (/3)+card (y). We show by induction that for each n, the number of nonempty elements in : f-‘a is at most (n + 1)s. Each element in “i’ fmia is of the form D n (an interval), so by our i=O i=O