Invariant measures for actions of higher rank abelian groups (Q2778789)

The linear group \(\text{GL}(m,\mathbb{Z})\) acts naturally on the torus \(T^m=\mathbb{R}^m/ \mathbb{Z}^m\). The purpose of the paper is to describe Borel probability measures on \(T^m\) that are invariant under the action of a subgroup of \(\text{GL} (m,\mathbb{Z})\) that is isomorphic to \(\mathbb{Z}^k\) for some \(k\geq 2\), does not possess nontrivial rank one factors and contains an Anosov element. NEWLINENEWLINENEWLINESince the elements of \(\mathbb{Z}^k\) commute, their generalized eigenspaces are mutually invariant and can be used to define the Lyapunov decomposition for the action. This Lyapunov decomposition induces various invariant foliations on \(T^m\). If \(\mu\) is ergodic then it is shown that under some additional assumptions on the partition into ergodic components of the generic singular elements \(a_1,\dots,a_k\) of the \(\mathbb{Z}^k\)-action the following dichotomy holds: Either \(\mu\) has zero entropy for all elements of the action or \(\mu\) decomposes as \(\mu=\frac 1N(\mu_1+ \cdots+\mu_N)\) where the measures \(\mu_i\) are invariant under the action of a finite index subgroup \(\Gamma\) and such that \((\Gamma, \mu_i)\) are algebraically isomorphic. Moreover, each \(\mu_i\) is an extension of a zero entropy measure in an algebraic factor for \(\Gamma\) of smaller dimension with Haar conditional measures in the fibres.NEWLINENEWLINENEWLINEThis result was presented earlier in a paper of \textit{A. Katok} and \textit{R. J. Spatzier} [Ergodic Tehory Dyn. Syst. 16, 751-778 (1996; Zbl 0859.58021)]. However, the statement of the result in these papers is slightly incorrect, and the proof contains several gaps. The present paper is based on the thesis of the first author who gave a rigorous proof based on the original ideas.NEWLINENEWLINENEWLINEThe paper also contains an extensive introduction to the subject and the discussion of the easiest nontrivial example as well as a discussion of nonuniformly hyperbolic actions.NEWLINENEWLINEFor the entire collection see [Zbl 0973.00044].