The Hausdorff dimension of sets invariant under conformal dynamical systems can often be realized as the zero of certain natural pressure equation (going back to Bowen). This pressure is usually continuous as a function of the defining dynamics, in the appropriate topology, and hence so is the Hausdorff dimension of the invariant set.

The situation is dramatically more complicated in the non-conformal situation where, nevertheless, a subadditive version of the pressure equation, involving singular values of a matrix cocycle, is crucial. A natural question is therefore whether this subadditive pressure is also a continuous function of the dynamics (or, what is the same, of the associated cocycle). We resolve this in the affirmative in many important situations, in particular answering a question of Falconer and Sloan.

This is joint work with De-Jun Feng (Chinese University of Hong Kong).