Modified Hanoi Towers Groups and Limit Spaces

TL;DR

This paper introduces generalized Hanoi automorphisms and self-similar groups, analyzes their limit spaces, and establishes conditions for contraction, expanding understanding of Hanoi groups' structure and dynamics.

Contribution

It generalizes Hanoi Towers groups to k-peg automorphisms, characterizes their limit spaces, and explores contraction conditions under symmetry assumptions.

Findings

Limit spaces of certain Hanoi groups are uniquely maximal under symmetry.

Conditions for contractivity of generalized Hanoi automorphisms are established.

Partial results on contraction with weaker symmetry are provided.

Abstract

We introduce the $k$-peg Hanoi automorphisms and Hanoi self-similar groups, a generalization of the Hanoi Towers groups, and give conditions for them to be contractive. We analyze the limit spaces of a particular family of contracting Hanoi groups, $H_c^{(k)}$, and show that these are the unique maximal contracting Hanoi groups under a suitable symmetry condition. Finally, we provide partial results on the contraction of Hanoi groups with weaker symmetry.