1. Given a Markov measure µ, what is its pushforward g ? µ by the action of a finite-state automaton? When are µ and g ? µ singular? When is g ? µ Gibbsian? (The question was explored for Bernoulli measures by R. Kravchenko). This question is answered fully for the case when g is invertible in our upcoming paper with R. Grigorchuk and V. Vorobets using the machinery related to the project below.
2. Given a transitive action of a Mealy machine A on the standard binary tree, one can, for a Mealy machine B consider the distribution of (directed graph) distances by which the vertices of the tree on a given level are moved by B in the cycle generated by the action of A. This leads to the definition of an interesting function, the automatic logarithm, and in some cases, gives rise to a shift-invariant measure (suggested by Y. Vorobets) with interesting properties.
Further research interests involve applying automata theory to groups – in particular, the Thompson groups F, T and V , and their n-dimensional generalizations nV introduced by Brin in . The question of whether F is an automaton group is still open.
While the conjugacy problem for F has been solved (in a particularly nice way by Belk in with strand diagrams), it is still open for nV .
I wrote software for computation in nV (), and plan to port it to GAP for a wider audience http://hdl.handle.net/1969.1/166707
To learn more, visit: http://romankogan.net/math
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Contact Roman at roman /dot/ kogan at-sign ronininstitute dot org