On univoque points for self-similar sets

Simon Baker ¹   ; Karma Dajani ²   ; Kan Jiang ²

¹ School of Mathematics University of Manchester Oxford Road Manchester, M13 9PL, UK
² Department of Mathematics Utrecht University Budapestlaan 6 3508TA Utrecht, The Netherlands

Fundamenta Mathematicae, Tome 228 (2015) no. 3, pp. 265-282

Résumé

Let $K\subseteq \mathbb {R}$ be the unique attractor of an iterated function system. We consider the case where $K$ is an interval and study those elements of $K$ with a unique coding. We prove under mild conditions that the set of points with a unique coding can be identified with a subshift of finite type. As a consequence, we can show that the set of points with a unique coding is a graph-directed self-similar set in the sense of Mauldin and Williams (1988). The theory of Mauldin and Williams then provides a method by which we can explicitly calculate the Hausdorff dimension of this set. Our algorithm can be applied generically, and our result generalises the work of Daróczy, Kátai, Kallós, Komornik and de Vries.

Détail

DOI : 10.4064/fm228-3-4

Keywords: subseteq mathbb unique attractor iterated function system consider where interval study those elements unique coding prove under mild conditions set points unique coding identified subshift finite type consequence set points unique coding graph directed self similar set sense mauldin williams nbsp theory mauldin williams provides method which explicitly calculate hausdorff dimension set algorithm applied generically result generalises work dar czy tai kall komornik vries