I will introduce three examples of Smale spaces (Smale spaces are a class of hyperbolic dynamical systems). Group actions on these three examples will be introduced to motivate the study of group actions on a general Smale space. Finally, I will discuss such actions both in regard to a Smale space itself and the ${\displaystyle C^{*}}$-algebras associated to it. This talk is based on joint work with Karen Strung.

We will discuss the following paradox:

Let M be a strong idempotent linear Maltsev condition. Under mild restrictions on M the following statement is true: If A is a random finite algebra whose operations satisfy the identities in M, then asymptotically almost surely, A satisfies much stronger conditions than M.

An example of a strong idempotent linear Maltsev condition is the condition ``There is a ternary operation m which satisfies the identities m(x,x,y)=y=m(y,x,x)''. Every group satisfies this condition with the (derived) operation m(x,y,z)=xy^{-1}z.

Lusztig's a-function is an integer-valued function on the elements of a Coxeter group, and it is defined in terms of the structure constants of the Kazhdan--Lusztig basis of the associated Hecke algebra. We call a Coxeter group "a(k)-finite" if it has finitely many elements with a-value equal to k. A Coxeter group element has a-value 1 if and only if it is "rigid", meaning that it has a unique reduced expression. The classification of a(1)-finite Coxeter groups is known, and this talk will describe the classification of a(2)-finite Coxeter groups.

[This is a continuation of last week's talk, but it is not necessary to have attended the previous talk.]