Diophantine Approximation and the Geometry of Limit Sets in Gromov Hyperbolic Metric Spaces

ISBN-13: 9781470428860 / Angielski / Miękka / 2018 / 137 str.

Lior Fishman; David Simmons; Daniel Tanré

Opis

Provides a complete theory of Diophantine approximation in the limit set of a group acting on a Gromov hyperbolic metric space. This summarizes and completes a long line of results by many authors, from Patterson's 1976 paper to more recent results of Hersonsky and Paulin. The authors consider concrete examples of situations which have not been considered before.

Spis treści

• Introduction;
• Gromov hyperbolic metric spaces;
• Basic facts about Diophantine approximation;
• Schmidt's game and McMullen's absolute game;
• Partition structures; Proof of Theorem 6.1 (Absolute winning of $\mathrm_\xi$);
• Proof of Theorem 7.1 (Generalization of the Jarni-Besicovitch Theorem);
• Proof of Theorem 8.1 (Generalization of Khinchin's Theorem);
• Proof of Theorem 9.3 ($\mathrm_d$ has full dimension in $\Lambda_{\mathrm r} (G)$);
• References.