Geometry of hyperbolic Julia-Lavaurs sets (Q1602608)

This paper deals with the properties of the Lavaurs map. Indeed let \(J_\sigma\) be the Julia-Lavaurs set of a hyperbolic map \(g_\sigma\) and let \(h_\sigma\) be its Hausdorff dimension. The authors show that the upper ball-(box) counting dimension and the Hausdorff dimension of \(J_\sigma\) are equal. Moreover, they show, if \(g_\sigma\) is derived from the parabolic quadratic polynomial \(f(z)= z^2+ \frac 14\), then the Hausdorff dimension \(h_\sigma\) is a real-analytic function of \(\sigma\). To this end, the authors study analytic dependence of the Perron-Frobenius operator on the symbolic space with infinite alphabet.