Journal of Applied Mathematics and Stochastic Analysis
Volume 14 (2001), Issue 3, Pages 257-264

Invariant measures for Chebyshev maps

Abraham Boyarsky and Paweł Góra

Concordia University, Department of Mathematics and Statistics, 7141 Sherbrooke Street, West Montreal H4B 1R6, Quebec, Canada

Received 1 April 2000; Revised 1 November 2000

Copyright © 2001 Abraham Boyarsky and Paweł Góra. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let Tλ(x)=cos(λarccosx), 1x1, where λ>1 is not an integer. For a certain set of λ's which are irrational, the density of the unique absolutely continuous measure invariant under Tλ is determined exactly. This is accomplished by showing that Tλ is differentially conjugate to a piecewise linear Markov map whose unique invariant density can be computed as the unique left eigenvector of a matrix.