Estimates of Hausdorff dimension (and its derivatives)

Question

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.

Answer 1

First, I believe that you mean to assume that $T$ is affine on the intervals $I_1$ and $I_2$ for, otherwise, Moran's formula need not hold. Assuming so, your question essentially boils down to how we can deal with the function $s(x,y)$ numerically. Given $x$ and $y$, the equation $$x^s + y^s = 1$$ can easily be solved for $s$ using Newton's method and there are plenty of software tools that can be used to do so. Here is a simple way to define $s(x,y)$ and plot it using Mathematica.

sol[x_, y_] := s /. First[
  FindRoot[x^s + y^s == 1, {s, 1}]];
Plot3D[sol[x, y], {x, 0, 1}, {y, 0, 1},
  RegionFunction -> Function[{x, y}, x + y < 1]]

If we're a little more clever and force the graph to have the correct values on the boundary, we get something like the following:

Once we have a reliable way to compute $s(x,y)$, there are several options to compute its second derivatives. There are difference quotients that approximate the second derivatives for example. Alternatively, we might approximate $s(x,y)$ with a piecewise smooth interpolation and then compute the second derivatives based on this interpolation. Taking this second approach, I was able to construct the following graphs of $s_{xx}$ and $s_{xy}$. Of course, $s_{yx}=s_{xy}$ and $s_{yy}(x,y)=s_{xx}(y,x)$ by symmetry.

There's plenty more that could be done but it's hard to say exactly what more should be done without more specifics.