Sub-actions for weakly hyperbolic one-dimensional systems

摘要

Let f be an expanding map of degree 2 in the unitary interval [0, 1], with an indifferent fixed point at x = 0 (I. E. F'_+(0) = 1), increasing, surjective and C~1 in each injective branch [0, c] and (c, 1], with inf{f(x)|x ∈[β, 1){c}}>1 for all β > 0. Let A: [0, 1] →R be a function α-Hoelder in each injective branch, monotone in a small neighbourhood of the origin, with A(0) < m and A(1) < m, where m is the maximum value of ∫ A dμ, over all invariant probability measures of f. Our main result is to prove that there exists a α-Hoelder function S: [0,1]→R, which satisfies the subcohomology equation S o f ≥ S + A - m. Using S we prove that, if A admits a unique measure which maximizes ∫A dμ, then this measure is uniquely ergodic. If A = log f, we are analysing the measure of maximal Lyapunov exponent. We also use the function f to define a bidimensional bijective function B in the unitary square [0, 1) x [0, 1), which is a modification of the Baker's Map, having an indifferent fixed point at the origin. If A: [0, 1) x [0, 1)→R is α-Hoelder we prove under conditions similar to the previous case that there exists a function S which satifies S o B≥S + A-m and is α/C-Hoelder, where C > 1, m = sup {∫A dμ;μ invariant for B}. The same results about maximizing measures are obtained.