This paper gives a simple proof of the well-known fact that a codimension-one Anosov diffeomorphism of a closed manifold is topologically conjugate to a hyperbolic toral automorphism.
ABSTRACT. In this note we will show that a positively expansive map of an arbitrary closed topological manifold is topologically conjugate to an expanding infra-nil-endomorphism.Let r be a group generated by {71,..., ik}-Then each 7 e T is represented as a word 7f117f2 ■ • -7^' and the number |pi| + IP2I + • • • + |p¡| is called the length of the word. The norm \\i\\ is defined as the minimal length of the word representing 7. As properties of the norm we know that ||7|| = ||7_1|| and ||77'|| < ||7|| + \\l'\\-If ¿1,..., 6k is another system of generators in T, then the corresponding norm || ||' is not necessarily equal to || ||. But there exists c > 0 such that for each 7 G T Let r be a finitely generated group and fix its generators. We denote by B(t), í > 0, the ball of radius t centered at the identity element e, and by #B(r) the number of elements in B(t).We say that T has polynomial growth if there are two positive numbers d and c such that for all balls B(t), t > 1, #5(i) < ctd. If T has a nilpotent subgroup of finite index then T has polynomial growth (Wolf [10]).The following is one interesting result for our investigation.THEOREM A (M. GroMOV [5]). // a finitely generated group T has polynomial growth, then T contains a nilpotent subgroup of finite index.Let L be a simply connected nilpotent Lie group with a left invariant Riemannian metric and denote by Aff (L) the group of transformations of L generated by the left translations and by all automorphisms from L onto itself. Let T C Aff(L) be a group which acts freely and discretely on L. If the quotient X = L/T is compact, then it is called an infra-nil-manifold. Each expanding automorphism L -» L which respects V induces an expanding map X -* X. Such maps are called an expanding infra-nil-endomorphism.From the following results together with Theorem A, M. Gromov [5] proved that an expanding differentiable map of an arbitrary closed C°° manifold is topologically conjugate to an infra-nil-endomorphism.
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