de Mathématiques de Luminy
Arnoux Pierre, Fisher Albert
Anosov families, renormalization and nonstationary subshifts.
We introduce the notion of an Anosov family, a generalization of an
Anosov map of amanifold. This is a sequence of diffeomorphisms along
compact Riemannian manifolds such that the tangent bundles split into
expanding and contracting subspaces.
We develop the general theory, studying sequences of maps up to a notion
of isomorphism and with respect to an equivalence relation generated
by two natural operations, gathering and dispersal.
Then we concentrate on linear Anosov families on the two-torus. We study
in detail a basic class of examples, the multiplicative families, and
a canonical dispersal of these, the additive families.These form a natural
completion to the collection of all linear Anosov maps.
A renormalization procedure constructs a sequence of Markov partitions
consisting of two rect-angles for a given additive family. This codes
the family by the nonstationary subshift of finite type determined by
exactly the same sequence of matrices.
Any linear positive Anosov family on the torus has a dispersal which
is an additive family. The additive coding then yields a combinatorial
model for the linear family, by telescoping the additiveBratteli diagram.
The resulting combinatorial space is then determined by the same sequence
of nonnegative matrices, as a nonstationary edge shift. This generalizes
and provides a new proof fortheorems of Adler and Manning.
Last update : february 14, 2003, EL.