Errata to ‘Hausdorff dimension for horseshoes’

Extract

In our paper ‘Hausdorff dimension for horseshoes’ (H. McCluskey and A. Manning, Ergod. Th. & Dynam. Sys. (1983) 3, 251–260), § 3 entitled ‘Continuity across a bifurcation’ should be deleted since the proof of theorem 3 there is wrong.

The mistake is that does not in fact imply that δ_s → 1. This is because, for q = q₀, the function is is only non-positive rather than negative and so is not strictly decreasing function of t.(Thus the pressure curve in figure 3 may, for q = q₀, have become horizontal before t = l.) A similar mistake occurs in the second half of the same proof. Thus the claim of theorem 3 that the Hausdorff dimension of the basic set changes continuously in the bifurcation from Anosov to DA remains unproved. The rest of the paper is not affected.