Hausdorff dimension for horseshoes in $\mathbb{R}^3$

Abstract

Using pressure formulas we compute the Hausdorff dimension of the basic set of ‘almost every’ ${\cal C}^{1+\alpha}$ horseshoe map in $\mathbb{R}}^3$ of the form $F(x,y,z)= (\gamma(x,z), \tau(y,z), \psi(z))$, where $|\psi'|>1$ and $0< |\gamma'_x|, |\tau'_y| < \frac{1}{2}$ on the basic set. Similar results are obtained for attractors of nonlinear ‘baker's maps’ in $\mathbb{R}}^3$.