For example, the cookie cutter maps, say $T:I_1 \cup I_2 \subset [0,1] \to [0,1] $ is a $C^2$ map such that $|T'|>1$ and provided $I_1$ and $I_2$ are disjoint closed intervals and $T(I_i)=[0,1]$. Suppose $|I_1|=x$ and $|I_2|=y$, we know that the Hausdorff dimension $s$ of the repeller of $T$ satisfies the Morán formula: $$ x^s+y^s=1. $$

Moreover, we know that $s=s(x,y)$ is an analytic function. I'm interested in behavior of $s(x,y)$ and its second derivative of $s(x,y)$, or more precisely its Hessian form.

Does anybody know any reference about this? The estimate of the solution, $s(x,y)$, to $ x^s+y^s=1$ and the estimate of its derivatives?