By John M. Franks

ISBN-10: 0821817000

ISBN-13: 9780821817001

Those lectures provide a transparent and unified exposition of an enormous quarter of present examine at the connections among dynamics and topology, treating the basic problem--what dynamics can take place in a prescribed homological setting--via algebraic chain complexes derived from the volatile manifold decomposition of dynamics.

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**Example text**

5), or directly, it is easy to see that we can choose G and the topological conjugacy h: EG 11j so that O(x) is constant on h(C(j)) where C(j) is the cyclinder set {c E EGIco = j}. Let Ak be the value of 0(x) on h(C(k)). 10) DEFINITION. The matrix A defined by Aik = AkGik will be called a structure matrix for the basic set Stj. Since G = IAI, a structure matrix for Stj contains a description off If2j up to topological conjugacy. But A contains the additional information of how Df acts on the orientation of E"(SZI).

So, s . . . ) E E. 2). This is because if y E hi and x E si C h1, then f(W, (x)) intersects Wi (y) in Gi1 points, one point in each of the components in Sit, and consequently one point in so. The fact that f is uniformly contracting on each WW(y) by a JOHN M. FRANKS 24 factor of A means that in the W direction so has diameter < Ad and so n f(s,) has diameter < X2d where d is the diameter of the DTs. Of course in W" direction the diameter of so nf(s,)is d. An induction on n shows that K = flk_ofk(sk) crosses h, in the W" direction and similar argument shows K, _ nki _"fk(sk) has diameter in the W' direction < X"d.

Let Ak be the value of 0(x) on h(C(k)). 10) DEFINITION. The matrix A defined by Aik = AkGik will be called a structure matrix for the basic set Stj. Since G = IAI, a structure matrix for Stj contains a description off If2j up to topological conjugacy. But A contains the additional information of how Df acts on the orientation of E"(SZI). We now give a result relating structure matrices to homological invariants off and an associated filtration. , the map induced by e on the quotient V/Vo where Vo = (u E Vlek(u) = 0 for some k > 0}.

### Homology and dynamical systems by John M. Franks

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