Eigenfunctions 212 and 213 on the Basilica with Equilibrium Measure

An eigenspace of dimension 2 spanned by a (primitive) horizontal eigenfunction and (primitive) vertical eigenfunction.

A 2-dimensional Eigenspace

We argue that our functions are part of a 2 dimensional eigenspace because of symmetry concerns (because we use numerical methods the eigenvalues are not exactly equal, see the note on eigenvalues at the bottom of the page). The symmetries of our Laplacian are horizontal and vertical reflection. We, therefore, expect our eigenfunctions to be either symmetric or antisymmetric with respect to these reflections. However 212 and 213, as spit out by Matlab, are neither. Hence we infer that our eigenfunctions live in a two dimensional eigenspace where such failures of symmetry are possible.

212 as given compared to 212 under horizontal reflection. If the difference were 0 we would have horizontal symmetry, if the sum were 0 we would have antisymmetry. However neither is the case.

213 as given compared to 213 under horizontal reflection. Again neither symmetry or antisymmetry holds.

Picking a Basis

Working from the assumption that we in fact have a 2D eigenspace, our next goal is to choose a basis. We again do this based on symmetry concerns. Our simple explanation of the spectrum would predict a primitive horizontal and a primitive vertical eigenfunction. In terms of our symmetries we want one eigenfunction each of types -+ and +-. By taking appropriate linear combinations of the eigenfunctions returned automatically by Matlab we may create a basis of this type. Further we program Matlab to do this future. (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).

$INSERT PICTURES OF CHOSEN BASIS FUNCTIONS$

Explanation of Exceptional Behavior

Similarity on Loops

A comment on Eigenvalues

We note that while we do not have exact agreement in computed eigenvalues for these functions we do not believe this contradicts our assertion that they form a two dimensional eigenspace. For comparison we look at eigenvalues 7-8 a known 2D eigenspace. We observe that these differences are similar. Moreover, for each of the the one dimensional eigenspaces the order of the eigenfunctions remains constant form level to level whereas 212 varies between appearing more horizontal or more vertical, with 213 having the opposite oscillation.

Eigenvalues of 212 and 213, 1.0e+04 *

	9	10	11	12
212	6.814000256189883	8.357378288117333	9.456748393301661	9.59914739688889
213	6.814000256190559	8.357378288144016	9.456748393337685	9.599147396926231

Eigenvalues of 7 and 8, 1.0e+02 *

	9	10	11	12
7	5.548154621594030	5.556615119914948		5.558898114340901
8	5.548154621597707	5.556615119928043		5.558898114476611

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