# Problem 31

## Primary tabs

Anosov diffeos

1. Hypothesis on $H_1(M)$
2. Fixed points
3. $\Omega = M$

## Tags

### John Franks and Bob Williams

John Franks and Bob Williams found an Anosov flow  on a manifiold $M$ whose chain recorrent set is not all of $M$.

### Under some pinching

Under some pinching assumptions on the spectrum there are some answers. See Brin, M.Manning, A.
Anosov diffeomorphisms with pinched spectrum. Dynamical systems and turbulence, Warwick 1980 (Coventry, 1979/1980), pp. 48–53,
Lecture Notes in Math., 898, Springer, Berlin-New York, 1981.

### See PORTEOUS, Hugh L. Anosov

See PORTEOUS, Hugh L. Anosov diffeomorphisms of flat manifolds. Topology, 1972, vol. 11, no 3, p. 307-315 for examples of manifolds admitting Anosov diffeomorphisms with first Betti number zero.

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