Title: Entropy estimates for a family of expanding maps of the circle

Authors: R. de la Llave (1), M. Shub (2) and C. Sim\'o (3)

(1) Math Dept., University of Texas at Austin, USA.
(2) Math Dept, University of Toronto, 40St. George Street, Toronto,
    ON M5S 2E4, Canada.
(3) Dept. de Matem\`atica Aplicada i An\`alisi, Univ. de Barcelona,
    Gran Via, 585, 08007 Barcelona, Spain.
E-mail: llave@math.utexas.edu, {michael.shub@utoronto.ca, carles@maia.ub.es

Abstract

In this paper we consider the family of circle maps f_{k,alpha,eps}: S^1-->S^1
which when written mod 1 are of the form
f_{k,alpha,eps}: x --> k x + alpha + eps sin(2 pi x),
where the parameter alpha ranges in S^1 and k > 1. We prove that for small eps
the average over alpha of the entropy of f_{k,alpha,eps with respect to the
natural absolutely continuous measure is smaller than
int_0^1 log|Df_{k,0,eps}(x)|dx, while the max is larger. In the case of the
average the difference is on the order of eps^{2k+2}. This result is in contrast
to families of expanding Blaschke products depending on rotations where the
averages are equal and for which the inequality for averages goes in the other
direction when the expanding property does not hold. A striking fact for both
results is that the max of the entropies is greater than or equal to
int_0^1 log|Df_{k,0,eps}(x)|dx. These results should also be compared with
previous work, where similar questions are considered for a family of
diffeomorphisms of the two sphere.