Affine embeddings of Cantor sets on the line

  • Amir Algom

    The Hebrew University of Jerusalem, Israel

Abstract

Let , and let be a self similar set such that . We prove that there exists such that if admits an affine embedding into a homogeneous self similar set and then (under some mild conditions on and ) the contraction ratios of and are logarithmically commensurable. This provides more evidence for Conjecture 1.2 of Feng, Huang, and Rao [7], that states that these contraction ratios are logarithmically commensurable whenever admits an affine embedding into (under some mild conditions). Our method is a combination of an argument based on the approach of Feng, Huang, and Rao in [7] with a new result by Hochman [10], which is related to the increase of entropy of measures under convolutions.

Cite this article

Amir Algom, Affine embeddings of Cantor sets on the line. J. Fractal Geom. 5 (2018), no. 4, pp. 339–350

DOI 10.4171/JFG/63