Finite and inﬁnite measures for adic transformations
Alby Fisher (IME-USP)

An adic transformation, as deﬁned by Vershik, deﬁnes a dynamics on a one-sided nonstationary subshift of ﬁnite type which is transverse to the ususal shift dynamics, much as the horocycle ﬂow is transverse to the geodesic ﬂow: it acts on the stable manifolds of the shift space. The most classic example of adic transformation (the odometer) is, like the horocycle ﬂow of a compact Riemann surface, both minimal and uniquely ergodic. This happens more generally whenever the matrix is primitive (some power is strictly positive), in the stationary case. Things get more interesting in the nonstationary situation (a sequence of matrices) where, just as for interval exchanges, one can ﬁnd examples which are minimal but not uniquely ergodic. But the really fascinating things happen in the nonstationary, nonprimitive case, where one can also ﬁnd interesting inﬁnite measures. In this talk we sketch a classiﬁcation of such measures. (Precisely, we classify x the invariant Borel measures for adic transformations of ﬁnite rank which are ﬁnite on the path space of some sub-Bratteli diagram.) This extends and builds on work by Bezuglyi, Kwiatkowski, Medynets and Solomyak. An application is given to nested circle rotations, where our necessary and suﬃcient condition for the measure to be inﬁnite is expressed in terms of continued fractions. (Joint work with Marina Talet)