Virtual centers of hyperbolic components in the tangent family: Hausdorff dimension outlook

Abstract

The tangent family $f_\lambda(z)=\lambda\tan z$ ($\lambda\in\mathbb{C}\setminus\{0\}, z\in\mathbb{C}$) is considered. It follows from Kotus and Urbański (Math. Ann.324 (2002), 619–656) that the function ascribing to each parameter $\lambda$ the Hausdorff dimension of the Julia set of $f_\lambda$ is continuous at all hyperbolic parameters $\lambda$. Now, we prove that the hyperbolic dimension of the Julia set at each parameter $\lambda_0$ that is a virtual center of a hyperbolic component (in the sense of Keen and Kotus (Conf. Geom. Dyn.1 (1997), 28–57)) is equal to the limit of hyperbolic dimensions (which are also equal to ordinary Hausdorff dimensions) of the Julia sets at hyperbolic parameters $\lambda$ canonically approaching $\lambda_0$ within this component. It is also shown that the Hausdorff dimension of the Julia set of $f_{\lambda_0}$ is strictly larger than this limit.