The entropy of Chebyshev polynomials

Abstract

The purpose of this work is to compute the topological entropy of the vth Chebyshev polynomial Tv(x) considered as a map of [ -1,1] onto itself.The notation and basic definitions relevant to the concept of topological entropy are contained in [1] and are reviewed briefly below.For an open cover 51 of a compact space X,N(2l) denotes the minimum cardinality of all sub-covers of 31.//(2I) = logN(2l) is called the entropy of 21.A cover 23 is said to refine a cover 21 if every set of 23 is a subset of some set of 21; we use the notation 2I-<23.We define the join of two covers 21,23 to be the cover 21V 23 = {^4 n R ; ^4 e 21, Re23}.For a continuous map <p of X into itself we define h(<¡>, 21), the entropy of <p with respect to 21 to be lim /7(2I V <^_13i V •-V <T"+12I)/n; ÏI-.CO in [1] this limit is shown to exist.Finally h((p), the entropy of<p, is defined to be sup h ((p, 21) where the supremum is taken over all open covers 21 of X.In the sequel we use the following properties.(1) -< is transitive.