Eigenfunction 53 and associated derived eigenfunctions on the Basilica with Equilibrium Measure

A horizontal eigenfunction whose first derived eigenfunction lives in an eigenspace of multiplicity 2.

A comment on Eigenvalues

First of all this is only exceptional if it is in fact and eigenspace of dimension 2, hence we consider the eigenvalues. For comparison we look at eigenvalues 7-8 a known 2D eigenspace. We observe that these differences are similar.

Eigenvalues of 105 and 106, 1.0e+04 *

Eigenvalues of 7 and 8, 1.0e+02 *

Picking a Basis

Working from the assumption that we in fact have a 2D eigenspace, our next goal is to choose a basis. In terms of our symmetries we want one eigenfunction each of types -+ and +-. By taking appropriate linear combinations of the eigenfunctions returned by Matlab we may create a basis of this type. (This seems to suggest that things aren't too far from the simple explanation-- if every eigenfunction in this space had type +- or ++ our simple explanation would fail to a greater extent).

Explanation of Exceptional Behavior

Similarity on Loops