Taken from J. Franks, Anosov diffeomorphisms, in the book ‘Global Analysis’, (1968) 61–93.

Question: given , for what does there exist a nontrivial such that ? Franks proved that for some diffeo, it reduces to a homotopy problem. So the definition:

A diffeo is a -diffeo, if given any homeo on a CW-complex with a continuous map such that from , there exists a unique base-point preserving map homotopic to such that .

(covering version)

Examples of Anosov: hyperbolic toral automorphisms, hyperbolic nil-manifold automorphisms, hyperbolic infra-nilmanifold automorphisms (and their endomorphisms)

Theorem 2.2. Every hyperbolic infra-nilmanifold automorphism is a -diffeo.

Problem: are all Anosov diffeo ?

Two diffeos and are -conjugate if there exists an isomorphism , such that from .

So two diffeos are topological conjugate iff they are -conjugate.

Theorem 3.6. Suppose is torsion-free and is on .

a. if is virtually nilpotent, then is topologically conjugate to a hyperbolic infra-nilmanifold automorphism.

b. if is nilpotent, then is topologically conjugate to a hyperbolic nilmanifold automorphism.

a. if is abelian, then is topologically conjugate to a hyperbolic toral automorphism.

Theorem 6.3. Every transitive codim=1 Anosov is a hyperbolic toral automorphism. Two such diffeo are topological conjugate iff they are -conjugate.

Theorem 8.2. If is virtually nilpotent and is expanding, then is topologically conjugate to a hyperbolic infra-nilmanifold endomorphism.

A key step is that if is expanding, then has polynomial growth.

Prop 1.1 Let be Anosov and be the universal covering of . Then the stable and unstable foliations can be lifted to , both being orientable.

Coro 1.3. Each deck transformation is an isometry and preserves the lifted bundles and lifted foliations. So they are independent of the choices of the lift.

An Anosov is splitting if the local product structure can be pasted to a global splitting on .

Prop 2.1. Let be hyperbolic. Then the the quotient map on is .

Proof. Let be a homeo on a CW-complex and with on .

Let be the hyperbolic splitting of , be the simply connected covering of , be the lift of with respect to a marked point , and , which is a Banach space admitting a splitting , invariant under the graph transform . Moreover hyperbolicity of implies that is invertible on .

Now let be a lift of , and . Then check that :

.

Now let and : . So is the lift of some base map . Moreover,

, or equivalently, . All being lifted maps, we get . Moreover such is unique and homotopic to .

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