Texas A&M
University, Department of
Mathematics, 216 Milner Hall, 30th of Sepember 2009, 3:00-3:50
Groups and Dynamics Seminar
Measures on limit spaces of self-similar groups
Rostyslav Kravchenko of Texas A&M University
Let G be a group and f be a contracting homomorphism from a subgroup H
of G of finite index to G. V.Nekrashevych associated with the pair (G,
f) the limit dynamical system (JG, s) and the limit G-space XG together
with the covering of XG by the translates of the tile T. We develop the
theory of self-similar measures m on these limit spaces. It is shown
that (JG, s, m) is conjugated to the one-sided Bernoulli shift. We
prove that the tile T has integer measure and we give an algorithmic
way to compute it. In addition we give an algorithm to find the measure
of the intersection of tiles T and (Tg) for g in G. We present
application to the evaluation of the Lebesgue measure of integral
self-affine tiles.