Anosov families, renormalization and non-stationary subshifts

Abstract

We introduce the notion of an Anosov family, a generalization of an Anosov map of a manifold. 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 2-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 rectangles for a given additive family. This codes the family by the non-stationary 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 additive Bratteli diagram. The resulting combinatorial space is then determined by the same sequence of non-negative matrices, as a non-stationary edge shift. This generalizes and provides a new proof for theorems of Adler and Manning.