Continuity of Hausdorff Dimension at Hopf Bifurcation

Abstract

We investigate the continuity of Hausdorff dimension and box dimension (limit capacity) of non-hyperbolic repellers of diffeomorphisms derived from transitive Anosov diffeomophisms through a Hopf bifurcation studied by Horita and Viana (see Discret. Contin. Dyn. Syst., 13 (2005), 1125-1137). Here, we extend their work showing that both dimensions are continuous at paremeter bifurcation. In the proof, we consider maps with holes introduced by Horita and Viana in Journal of Statistical Physics 105(2001), 835-862 and further developed by Dysman in Journal of Statistical Physics 120(2005),479-509, relating the Hausdorff dimension with the volume of the hole.