Transitivity of Euclidean extensions of Anosov diffeomorphisms

Abstract

We consider the class of R n extensions of Anosov diffeomorphisms on infranilmanifolds, and find necessary and sufficient conditions for topological transitivity. In particular, if the fiber is R, the existence of a semi-orbit with the projection on R unbounded from above and from below is equivalent to topological transitivity. We also show that in the above class topological transitivity and stable topological transitivity are equivalent.

The Mañé–Conze–Guivarc’h lemma for intermittent maps of the circle

“…A similar result was previously shown by G. Yuan and B. R. Hunt under more restrictive dynamical assumptions [25]. Mañé-Conze-Guivarc'h-type lemmas have also been found useful in circumstances that are not a priori related to maximizing measures [20].…”

Transitivity of Euclidean-type extensions of hyperbolic systems

“…, u k } is the whole of R n . Moreover, U does not lie in a halfspace (since 0 ∈ Int co U) so it follows from Niţicȃ & Pollicott [8] or [6,Lemma 2.12], that it suffices to arrange that the closed group generated by U = {u 2 , . .…”

“…The positive Livšic theorem of Bousch [1,Section 4] and the compactness of the set of hyperplanes imply that there are finitely many periodic orbits P j , 1 ≤ j ≤ F , such that the set {πβ(P j ) : 1 ≤ j ≤ F } does not lie in a halfspace of Fix G, and thus 0 ∈ Int co{πβ(P j ) : 1 ≤ j ≤ F }. For any non-negative integers n j that are not all zero, one can use shadowing arguments (see, for example [8,Sections 5 and 6]) to obtain periodic orbits Q n such that πβ(Q n ) = n( F j=1 n j πβ(P j )) + O(1). Thus, one can obtain d + 1 periodic orbits P such that 0 ∈ Int co{πβ(P ) : 1 ≤ ≤ d + 1}, proving (i).…”

“…The main task in this paper is to prove Theorem 1.2 for the class I groups. The general result is then proved by incorporating ideas of [4,8].…”

“…In [5] we proposed a general conjecture about transitivity in the class of Hölder cocycles: namely that modulo obstructions appearing from the fact that the range of the cocycle is included in a semigroup, transitivity is open and dense. The conjecture is proved for various classes of Lie groups, mostly semidirect products of compact and Euclidean, in [2,4,5,6,8]. In addition [5] exhibits open sets of C r transitive cocycles with fiber Sp(n).…”