Amenable purely infinite actions on the non-compact Cantor set

Rohlin Properties for Z Actions on the Cantor Set

We study the space H(d) of continuous Z-actions on the Cantor set, particularly questions on the existence and nature of actions whose isomorphism class is dense (Rohlin’s property). Kechris and Rosendal showed that for d = 1 there is an action on the Cantor set whose isomorphism class is residual; we prove in contrast that for d ≥ 2 every isomorphism class in H(d) is meager. On the other hand,...

Orbit Inequivalent Actions of Non-amenable Groups

This is a preliminary note containing the proof that every countable non-amenable group admits continuum many orbit inequivalent free, measure preserving, ergodic actions on a standard Borel space with probability measure. 1. Summary and Notation Let Γ1 y α1 (X1, μ1),Γ2 y α2 (X2, μ2) be Borel measure preserving actions. α1 and α2 are orbit equivalent if there exist conull subsets A1 ⊂ X1, A2 ⊂ ...

Bootstrap Percolation on Infinite Trees and Non-Amenable Groups

Bootstrap percolation on an arbitrary graph has a random initial configuration, where each vertex is occupied with probability p, independently of each other, and a deterministic spreading rule with a fixed parameter k: if a vacant site has at least k occupied neighbors at a certain time step, then it becomes occupied in the next step. This process is well-studied on Z; here we investigate it o...

P-Amenable Locally Compact Hypergroups

The Cantor Set

Analysis is the science of measure and optimization. As a collection of mathematical fields, it contains real and complex analysis, functional analysis, harmonic analysis and calculus of variations. Analysis has relations to calculus, geometry, topology, probability theory and dynamics. We will focus mostly on ”the geometry of fractals” today. Examples are Julia sets which belong to the subfiel...