Notes about Anosov diffeomorphisms

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

Question: given f:M\to M, for what g:N\to N does there exist a nontrivial h:N\to M such that h\circ g=f\circ h? Franks proved that for some diffeo, it reduces to a homotopy problem. So the definition:

A diffeo f:M\to M is a \pi_1-diffeo, if given any homeo g:K\to K on a CW-complex K with a continuous map h:K\to M such that h_\ast\circ g_\ast=f_\ast\circ h_ast from \pi_1(K)\to \pi_1(M), there exists a unique base-point preserving map \hat h:K\to M homotopic to h such that \hat h\circ g=f\circ\hat h.

(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 \pi_1-diffeo.

Problem: are all Anosov diffeo \pi_1?

Two diffeos f:M\to M and g:N\to N are \pi_1-conjugate if there exists an isomorphism \phi:\pi_1(N)\to \pi_1(M), such that \phi\circ g_\ast=f_\ast\circ \phi from \pi_1(N)\to \pi_1(M).

So two \pi_1 diffeos are topological conjugate iff they are \pi_1-conjugate.

Theorem 3.6. Suppose \pi_1(M) is torsion-free and f is \pi_1 on M.
a. if \pi_1(M) is virtually nilpotent, then f is topologically conjugate to a hyperbolic infra-nilmanifold automorphism.
b. if \pi_1(M) is nilpotent, then f is topologically conjugate to a hyperbolic nilmanifold automorphism.
a. if \pi_1(M) is abelian, then f 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 \pi_1-conjugate.

Theorem 8.2. If \pi_1(M) is virtually nilpotent and f is expanding, then f is topologically conjugate to a hyperbolic infra-nilmanifold endomorphism.
A key step is that if f:M\to M is expanding, then \pi_1(M) has polynomial growth.

Prop 1.1 Let f:M\to M be Anosov and \hat M be the universal covering of M. Then the stable and unstable foliations can be lifted to \hat M, 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 f:M\to M is splitting if the local product structure can be pasted to a global splitting on \hat M.

Prop 2.1. Let A\in \mathrm{SL}(n,\mathbb{Z}) be hyperbolic. Then the the quotient map f on \mathbb{T}^n is \pi_1.
Proof. Let g:K\to K be a homeo on a CW-complex and h:K\to\mathbb{T}^n with f_\ast\circ h_\ast=h_\ast\circ g_\ast on \pi_1(K)\to\pi_1(\mathbb{T}^n).

Let E^u\oplus E^s be the hyperbolic splitting of A, \hat K be the simply connected covering of K, \hat g be the lift of g with respect to a marked point \ast\in K, and \mathcal{B}=\{H\in C(K,\mathbb{R}^n,\ast)|H\circ\alpha=H,\forall \alpha\in\pi_1(K)\}, which is a Banach space admitting a splitting \mathcal{B}=\mathcal{B}^u\oplus \mathcal{B}^s, invariant under the graph transform \mathcal{F}(H)=A^{-1}\circ H\circ \hat g. Moreover hyperbolicity of A implies that \mathcal{F}-I is invertible on \mathcal{B}.

Now let \hat h:\hat K\to\mathbb{R}^n be a lift of h, and H=\mathcal{F}(\hat h)-\hat h. Then check that H\in \mathcal{B}:
H(\alpha+x)=A^{-1}\circ \hat h(\hat g(\alpha+x))-\hat h(\alpha+x)  =A^{-1}\circ \hat h(g_\ast(\alpha)+\hat g(x))-(h_\ast(\alpha)+\hat h(x))
=A^{-1}\circ A h_\ast(\alpha)+A^{-1}\hat h\hat g(x)-h_\ast(\alpha)-\hat h(x)=A^{-1}\hat h\hat g(x)-\hat h(x)=H(x).

Now let H'=(\mathcal{F}-I)^{-1}H\in \mathcal{B} and J=\hat h-H': J(\alpha+x)=\hat h(\alpha+x)-H(\alpha+x)=h_\ast(\alpha)+(\hat h(x)-H(x)). So J is the lift of some base map j:K\to\mathbb{T}^n. Moreover,
A^{-1}\circ J\circ \hat g-J=\mathcal{F}(\hat h-H')-(\hat h-H')=(\mathcal{F}-I)\hat h-(\mathcal{F}-I)H'=H-H=0, or equivalently, J\circ \hat g=A\circ J. All being lifted maps, we get j\circ g=f\circ j. Moreover such j is unique and homotopic to h.

$latex $

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