Abstract. Let φ be a hyperbolic diffeomorphism on a basic set Λ and let G be a connected Lie group. Let f : Λ → G be Hölder. Assuming that f satisfies a natural partial hyperbolicity assumption, we show that if u : Λ → G is a measurable solution to f = uφ · u −1 a.e., then u must in fact be Hölder. Under an additional centre bunching condition on f , we show that if f assigns 'weight' equal to the identity to each periodic orbit of φ, then f = uφ · u −1 for some Hölder u. These results extend well-known theorems due to Livšic when G is compact or abelian.

We consider cocycles with negative Lyapunov exponents defined over a hyperbolic dynamical system. It is well known that such systems possess invariant graphs and that under spectral assumptions these graphs have some degree of Hölder regularity. When the invariant graph has a slightly higher Hölder exponent than the a priori lower bound on an open set (even on just a set of positive measure for certain systems), we show that the graph must be Lipschitz or (in the Anosov case) as smooth as the cocycle.

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Abstract. We generalize a series of topological Wiener-Wintner ergodic theorems due to Walters to the context of group extensions of measure-preserving transformations where the group is a non-abelian compact Lie group. Applications to random ergodic theorems for a shift map are given.

Weierstrass' example of an everywhere continuous but nowhere differentiable function is given by wThere is a well-known and widely accepted, but as yet unproven, formula for the Hausdorff dimension of the graph of w. Hunt [H] proved that this formula holds almost surely on the addition of a random phase shift. The graphs of Weierstrass-type functions appear as repellers for a certain class of dynamical system; in this paper we prove formulae analogous to those for random phase shifts of w(x) but in a dynamic context. Let T : S 1 → S 1 be a uniformly expanding map of the circle. Let λ : S 1 → (0, 1), p : S 1 → R and define the function w). The graph of w is a repelling invariant set for the skew-product transformation T (x, y) = (T (x), λ(x) −1 (y − p(x))) on S 1 × R and is continuous but typically nowhere differentiable. With the addition of a random phase shift in p, and under suitable hypotheses including a partial hyperbolicity assumption on the skew-product, we prove an almost sure formula for the Hausdorff dimension of the graph of w using a generalization of techniques from [H] coupled with thermodynamic formalism.