A Uniform Dichotomy for Generic Sl(2,r) Cocycles over a Minimal Base

In this paper wewill consider SL(2,R)-valued cocycles over aminimal homeomorphism f : K → K of a compact set K. Such a cocycle can be defined as a pair ( f ,A) where A : K → SL(2,R) is continuous. The cocycle acts on K ×R2 by (x, y) 7→ ( f (x),A(x) · y). The iterates of the cocycle are denoted ( f ,A) = ( f ,An). We say that ( f ,A) is uniformly hyperbolic if there exists ε > 0 and N > 0 such that ‖An(x)‖ ≥ e for every x ∈ K, n ≥ N. This is equivalent to the existence of a continuous invariant splitting R = E(x)⊕E(x) such that vectors in E(x) are exponentially contracted by forward iteration and vectors in E(x) are exponentially contracted by backwards iteration – see [Y, proposition 2]. We say that ( f ,A) has uniform subexponential growth if for every ε > 0 there exists N > 0 such that ‖An(x)‖ ≤ e for every x ∈ K, n ≥ N. This condition is equivalent to the vanishing of the Lyapunov exponent for all f -invariant probability measures (see propostion 1 below). We recall that the Lyapunov exponent of the cocycle ( f ,A) with respect to an f -invariant probability measure μ is defined as L( f ,A, μ) = lim 1 n ∫