Continuity of Hausdorff Dimension at Hopf Bifurcation
Vanderlei Horita, Oyran Rayzzaro
TL;DR
The paper addresses the problem of continuity of Hausdorff and box dimensions for non-hyperbolic repellers arising at a Hopf bifurcation in diffeomorphisms derived from Anosov systems on $\mathbb{T}^3$. It extends Horita–Viana by coupling the dynamics to maps with holes and establishing a decay condition that links the volume of bad cylinders to the hole size, enabling lower bounds on dimension via induced expanding maps. The authors prove two main results: (A) under hypotheses $(\mathcal{A}_1)$ and $(\mathcal{A}_2)$, the Hausdorff dimension of the repeller is close to the ambient dimension, and (B) for the Hopf-unfolding family, the fractal dimensions $\operatorname{HD}(\Lambda_\mu)$ and $\operatorname{BD}(\Lambda_\mu)$ converge to the ambient dimension $3$ as $\mu \to \mu_*$ by projecting the dynamics to a 2D hole map and verifying $(\mathcal{A}_2)$. The method leverages the $C^{1+\delta}$ holonomy of the stable foliation and relates the 2D holes to the 3D repeller, yielding a precise continuity result with implications for open families of diffeomorphisms near Hopf 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.