Invariant measures for higher-rank hyperbolic abelian actions

Abstract

We i n vestigate invariant ergodic measures for certain partially hyperbolic and Anosov actions of R k , Z k and Z k + . W e s h o w that they are either Haar measures or that every element of the action has zero metric entropy.

Rigidity of measures—The high entropy case and non-commuting foliations

“…When Rudolph's result appeared, the second author suggested another test model for the measure rigidity: two commuting hyperbolic automorphisms of the three-dimensional torus. In joint work with R. Spatzier the second author developed a more geometric technique [18,19] which was subsequently extended by B. Kalinin and the second author [16] as well as by B. Kalinin and R. Spatzier [17].…”

“…Hence this and later techniques are limited to study actions where at least one element has positive entropy. Under ideal situations, such as the original motivating case of two commuting hyperbolic automorphisms of the three torus, no further assumptions are needed, and a result entirely analogous to Rudolph's theorem can be proved using the method of [18] (see also [16]). …”

“…However, for Weyl chamber flows, an additional assumption is needed for the proof [18] to work. This assumption is satisfied, for example, if the flow along every singular direction in the Weyl chamber is ergodic (though a weaker hypothesis is sufficient, see also [17]).…”

“…This assumption is satisfied, for example, if the flow along every singular direction in the Weyl chamber is ergodic (though a weaker hypothesis is sufficient, see also [17]). This additional assumption, which unlike the entropy assumption is not stable under weak * limits, precludes applying the results from [18] in many cases.…”

“…This program was initiated in [18] and continued in [16,6,17]. We precede the description of our results by a general discussion of problems that motivated our work.…”