## Abstract

Let F, E ⊆ ℝ^{2} be two self-similar sets. First, assuming F is generated by an IFS Ф with strong separation, we characterize the affrne maps g: ℝ^{2} → ℝ^{2} 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 |G_{Ф}| = ∞ we show that any such self-embedding must be a similarity, and so (by the results of Elekes, Keleti and Máathé [9]) some power of its orthogonal part lies in G_{Ф}. When |G_{Ф}| = ∞ 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 |G_{Ф}| = ∞ then |G_{Ψ}| = ∞ and if |G_{Ф}| = ∞ then |G_{Ψ} lt; ∞.

Original language | English |
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Pages (from-to) | 695-757 |

Number of pages | 63 |

Journal | Journal d'Analyse Mathematique |

Volume | 140 |

Issue number | 2 |

DOIs | |

State | Published - 1 Mar 2020 |

Externally published | Yes |

### Bibliographical note

Funding Information:Supported by ERC grant 306494

Publisher Copyright:

© 2020, The Hebrew University of Jerusalem.

## ASJC Scopus subject areas

- Analysis
- Mathematics (all)