Multifractal analysis of the symmetry of a strictly isospectral energy landscape on a square lattice

We use the Holder regularity analysis to study the symmetry breaking and recovery due to a parametric potential generated via the strictly isospectral factorization method. The initial potential is two-dimensional and periodic in the two Cartesian directions, with the symmetry group $P_4mm$. The resulting parametric isospectral potential display a P_m symmetry for values of the parameter moderately close to the singular value gamma_s. However, at large values of the parameter, visually around gamma=gamma_s+110, the original symmetry is recovered. For a much higher precision value of the parameter for this symmetry recovery, we show that the multifractal spectrum of the parametric potential can be conveniently used. In the latter case, we obtain gamma=gamma_s+201.085 for three decimal digits precision.