Fractal Dimension of Julia Set for Non-analytic Maps

TL;DR

This paper investigates the Hausdorff dimensions of Julia sets for specific non-analytic maps, showing that Ruelle's formula does not extend to non-analytic cases, with calculations performed perturbatively for small epsilon.

Contribution

It demonstrates the limitations of Ruelle's formula for non-analytic maps and provides perturbative calculations of Julia set dimensions for these maps.

Findings

Ruelle's formula does not generalize to non-analytic maps

Perturbative calculations of Julia set dimensions for small epsilon

Hausdorff dimensions differ from analytic map predictions

Abstract

The Hausdorff dimensions of the Julia sets for non-analytic maps: f(z) = z^2 + epsilon z^* and f(z) = {z^*}^2 + epsilon are calculated perturbatively for small epsilon. It is shown that Ruelle's formula for Hausdorff dimensions of analytic maps can not be generalized to non-analytic maps.