Multi-invariant sets on tori

Abstract

Abstract. Given a compact metric group G, we are interested in those semigroups 2 of continuous endomorphisms of G, possessing the following property: The only infinite, closed, 2-invariant subset of G is G itself. Generalizing a one-dimensional result of Furstenberg, we give here a full characterization-for the case of finitedimensional tori-of those commutative semigroups with the aforementioned property.

Measurable rigidity and disjointness for {\mathbb Z}^k actions by toral automorphisms

“…We give a brief description of the algebraic structure of irreducible Z k actions on T m , see [1,2] for more details. Irreducible Z k actions on T m .…”

“…Irreducible Z k actions on T m . [1] Let α be a Z k action by automorphisms of T m . The action α is called irreducible if any non-trivial algebraic factor has finite fibres.…”

“…This implies that we can take a regular element b in the positive half-space P + sufficiently close to a so that 1 is again the part of W − b that corresponds to the Lyapunov exponents of α 1 . Since α 1 is a TNS action we see that all Lyapunov exponents of α 1 with kernel P are positively proportional.…”

Invariant Subsets and Invariant Measures for Irreducible Actions on Zero-Dimensional Groups

“…Topological rigidity has been studied since the seminal paper by Furstenberg [5], in which it was shown that there is no infinite closed proper subset A ⊂ T that is invariant under multiplication by two coprime (or multiplicatively independent) integers, other than Y = T. The proper generalization for irreducible algebraic Z dactions on T m was found in [1] and generalized in [2].…”

“…Clearly, ν (1) is the weak * limit for the sequence µ (1) k . The measures ν (1) and µ (1) k are invariant measures on the shift of finite type X 1 , and the desired second inequality follows at once, since metric entropy for a shift space is an upper semi-continuous function of the measure; see [14,Theorem 8.2].…”