Abstract
Suppose E×E is a planar self-similar set, where E⊆R is a self-similar set satisfying the strong separation condition with 0<dimHE<1. By revealing a relationship between algebraic numbers cos2πℓ and sin2πℓ with ℓ∈Z+, we prove that for all generating iterated function systems (IFSs) of E×E, the associated orthogonal matrices have at most 8 possibilities. Additionally, we investigate all generating IFSs of two typical planar self-similar sets, one has a minimal presentation (every other generating IFS is an iteration of this one) while the other has not.
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