Theoretical explanation for the ~1.58 fractal dimension in the Christ–Lee model

Establish a theoretical explanation for why the measured Box-counting fractal dimension of the fractal structure arising in kink–antikink collisions in the Christ–Lee model approaches approximately 1.58 (the Hausdorff dimension of the Sierpinski triangle) as the parameter ε becomes sufficiently large.

Background

The authors compute Box-counting dimensions for the fractal structures observed in the relationships V_out versus V_in and V_out versus ε in the Christ–Lee model. Across a range of parameters, the measured dimensions lie roughly between 1.36 and 1.66 and increase with ln ε.

They observe that when ε is sufficiently large (approaching the φ^4 regime), the fractal dimension trends toward ~1.58, which coincides with the Hausdorff dimension of the Sierpinski triangle. Despite this empirical observation, they explicitly note that a theoretical explanation of this value is lacking.