Affine embeddings of Cantor sets on the line

Let F, E ⊆ ℝ2 be two self-similar sets. First, assuming F is generated by an IFS Ф with strong separation, we characterize the affrne maps g: ℝ2 → ℝ2 such that g(F) ⊆ F. Our analysis depends on the cardinality of the group GФ generated by the orthogonal parts of the similarities in Ф. When |GФ| = ∞ we show that any such self-embedding must be a similarity, and so (by the results of Elekes, Keleti and Maathe [9]) some power of its orthogonal part lies in GФ. When |GФ| = ∞ and Ф has a uniform contraction λ, we show that the linear part of any such embedding is diagonalizable, and the norm of each of its eigenvalues is a rational power of λ. We also study the existence and properties of affrne maps g such that g(F) ⊆ E, where E is generated by an IFS Ψ In this direction, we provide more evidence for a Conjecture of Feng, Huang and Rao [17], that such an embedding exists only if the contraction ratios of the maps in O are algebraically dependent on the contraction ratios of the maps in Ψ Furthermore, we show that, under some conditions, if |GФ| = ∞ then |GΨ| = ∞ and if |GФ| = ∞ then |GΨ lt; ∞.