Ergodic theorems for free semigroup and group actions
by
Alexander Bufetov
Independent University of Moscow

New ergodic theorems are obtained for actions of finitely generated free semigroups and groups by measure preserving transformations of Lebesgue spaces. For such actions, a new averaging process is suggested.

Start with an arbitrary shift-invariant Markov measure on the space of one-sided infinite sequences of free generators of our semigroup (or group). To each element of the semigroup, written as a word of generators, assign the measure of the cylinder of sequences beginning with this word. Then average with these weights by spheres of increasing radius in our semigroup, and consider the Cesaro averages of the sphere averages. For these "time-averages" analogues of the mean and pointwise ergodic theorems are proved. Under special nondegeneracy conditions on the Markov measure, the limit function is semigroup-invariant. This work continues the earlier research of Kalutani, Oseledec, Grigorchuk and Nevo-Stein.

Date received: August 11, 1999