91 research outputs found
Periodic and eventually periodic points of affine infra-nilmanifold endomorphisms
In this paper, we study the periodic and eventually periodic points of affine
infra-nilmanifold endomorphisms. On the one hand, we give a sufficient
condition for a point of the infra-nilmanifold to be (eventually) periodic. In
this way we show that if an affine infra-nilmanifold endomorphism has a
periodic point, then its set of periodic points forms a dense subset of the
manifold. On the other hand, we deduce a necessary condition for eventually
periodic points from which a full description of the set of eventually periodic
points follows for an arbitrary affine infra-nilmanifold endomorphism.Comment: 18 page
The mathematical research of William Parry FRS
In this article we survey the mathematical research of the late William (Bill) Parry, FRS
Measure and cocycle rigidity for certain non-uniformly hyperbolic actions of higher rank abelian groups
We prove absolute continuity of "high entropy" hyperbolic invariant measures
for smooth actions of higher rank abelian groups assuming that there are no
proportional Lyapunov exponents. For actions on tori and infranilmanifolds
existence of an absolutely continuous invariant measure of this kind is
obtained for actions whose elements are homotopic to those of an action by
hyperbolic automorphisms with no multiple or proportional Lyapunov exponents.
In the latter case a form of rigidity is proved for certain natural classes of
cocycles over the action.Comment: 28 page
Differential Rigidity of Anosov Actions of Higher Rank Abelian Groups and Algebraic Lattice Actions
We show that most homogeneous Anosov actions of higher rank Abelian groups
are locally smoothly rigid (up to an automorphism). This result is the main
part in the proof of local smooth rigidity for two very different types of
algebraic actions of irreducible lattices in higher rank semisimple Lie groups:
(i) the Anosov actions by automorphisms of tori and nil-manifolds, and (ii) the
actions of cocompact lattices on Furstenberg boundaries, in particular,
projective spaces. The main new technical ingredient in the proofs is the use
of a proper "non-stationary" generalization of the classical theory of normal
forms for local contractions.Comment: 28 pages, LaTe
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