Graphs of nonsingular threshold transformations

Ergodic Theory: Nonsingular Transformations

Glossary 1 1. Definition of the subject and its importance 2 2. Basic Results 2 3. Panorama of Examples 8 4. Mixing notions and multiple recurrence 10 5. Topological group Aut(X,μ) 13 6. Orbit theory 15 7. Smooth nonsingular transformations 21 8. Spectral theory for nonsingular systems 22 9. Entropy and other invariants 25 10. Nonsingular Joinings and Factors 27 11. Applications. Connections wi...

Rank One Power Weakly Mixing Nonsingular Transformations

We show that Chacon's nonsingular type III transformation T , 0 < 1, is power weakly mixing, i.e., for all sequences of nonzero integers fk1; : : : ; krg, T k1 : : : T kr is ergodic. We then show that in in nite measure, this condition is not implied by in nite ergodic index (having all nite Cartesian products ergodic), and that in nite ergodic index does not imply 2-recurrence.

Nonsingular Block Graphs: An Open Problem

A block graph is a graph in which every block is a complete graph (clique). Let G be a block graph and let A(G) be its (0,1)-adjacency matrix. Graph G is called nonsingular (singular) if A(G) is nonsingular (singular). Characterizing nonsingular block graphs, other than trees, is an interesting open problem. In this article, we give some classes of nonsingular and singular block graphs and rela...

Double Ergodicity of Nonsingular Transformations and In nite measure-preserving Staircase Transformations

A nonsingular transformation is said to be doubly ergodic if for all sets A and B of positive measure there exists an integer n > 0 such that (T n(A) \ A) > 0 and (T n(A) \ B) > 0. While double ergodicity is equivalent to weak mixing for nite measure-preserving transformations, we show that this is not the case for in nite measure preserving transformations. We show that all measure-preserving ...

On nonsingular loop transformations using SUIF's dependence abstraction