Continuity of Hausdorff Dimension at Hopf Bifurcation

TL;DR

This paper proves that the Hausdorff and box dimensions of certain non-hyperbolic repellers remain continuous during a Hopf bifurcation, extending previous work by analyzing maps with holes and their dimensions.

Contribution

It extends prior research by demonstrating the continuity of dimensions at bifurcation points for non-hyperbolic repellers, using maps with holes and volume-dimension relations.

Findings

Hausdorff and box dimensions are continuous at bifurcation points.

The work relates Hausdorff dimension to the volume of holes in the maps.

It generalizes previous results on dimensions in dynamical bifurcations.

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.