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T h e theory which we will develop in this part o f the thesis is a weakly hyper bolic version o f A n osov’s Stability lem m a [Ano70, Kat81, Shu87].
T h e central idea o f Anosov’s Stability lemma, as stated by K atok [Kat81] or Shub [Shu87], is that if we are given an action o f a hom eom orphism , A, on an topological space, A , and we are given an injection, i o f X into the manifold M for which the set t (A -) is contained in a neighbourhood o f a uniformly hyperbolic set, and m oreover for which the injection makes the action o f A on A and the action o f / on i ( A ) “almost” com m ute, then there is an injection, j which is close to i which makes the action o f A on A commute with the action o f / on j ( A ) .
T his idea is best represented by the pair o f diagrams
which “alm ost” and actually com m ute respectively. T h e fact that the first dia gram “ almost” com m utes means that the point / o t ( x ) and the point i o A (x) need not be equal but they must be close with respect to some uniform distance. This is the essentially the idea behind a fam ily o f pseudo-orbits. T h e topological space A acts as an index space which replaces the m ore com m on indexing by the integers used in the definition o f a single pseudo-orbit. T h e action o f A on the space A a cts as the index function which replaces the more com m on “ next integer” function.
T h e central idea in A nosov’s Stability lemma is then that any fam ily o f pseudo orbits which lie in a neighbourhood o f a uniformly hyperbolic invariant set is
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shadowed by another invariant set whose dynam ics is defined by the indexing function h. N ote that the original uniformly hyperbolic invariant set forms the support for the pseudo-orbit to ensure that the pseudo-orbit can be shadowed. In order to distinguish between them we will generally call this pair o f invariant sets the supporting and shadowing invariant sets respectively.
W e need t o extend this idea in two ways. Firstly, we would like to apply this lemma to weakly hyperbolic invariant sets. Secondly, we would like to obtain an estimate o f th e hyperbolicity o f the resulting shadowing invariant set.
The lack o f a hyperbolicity estimate in A nosov’s Stability lem m a is merely an oversight. In A nosov’s original setting, that o f A nosov diffeomorphisms, and in the standard A xiom -A setting, we know by other arguments, that the shadowing invariant set is as hyperbolic as the supporting invariant set. In the case o f an Anosov diffeomorphism, all o f the points in the m anifold are uniformly hyperbolic with the sam e bounds. In the case o f an A xiom -A diffeomorphism, any invariant set is uniform ly hyperbolic w ith the same bounds.
W e can extend A nosov’s Stability lemma to encompass weakly hyperbolic invariants sets for two reasons. Firstly, it is essentially a local result. T h e pseudo orbit is “close” to the supporting invariant set and the shadowing invariant set is “close” to the pseudo-orbit which is as a consequence “close" to the supporting invariant set. Secondly, by using a “metric” similar to the one Pesin used to prove his Stable manifold theorem for weakly hyperbolic invariant sets, we can similarly change the weak hyperbolicity into a uniform hyperbolicity.
Unfortunately, the “m etric” which we use is really a countable family o f m e t rics, or alternatively is a single metric on the disjoint union, M , o f a countable family o f cop ies o f the original manifold, M . This means that with respect to this “m etric” our invariant sets and pseudo-orbits tend to “hop” between cop ies o f the original manifold in M . Fortunately, since A nosov’s Stability lem m a is essentially a local result, we can best view this “hopping” more properly as the action o f a fibre bundle map on T M which is related to the original map / .
This suggests that the m ost natural way o f making our extensions to A n osov ’s stability lem m a is by explicitly formulating the concept o f a (fam ily o f) pseudo- orbit(s). Since an invariant set is a special case o f a pseudo-orbit it is then m ost
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natural to formulate the concept o f a hyperbolic (or even pseudo-hyperbolic) pseudo-orbit. Note that our formulation o f a hyperbolic pseudo-orbit encompasses precisely the fibre bundle structure with which it is m ost natural t o state and prove our extension o f A nosov’s Stability lemma.
Having explicitly form ulated the concept o f a pseudo-hyperbolic pseudo-orbit, it is then most natural to break A nosov’s rather m onolithic stability lem m a into a num ber o f separate sublemmas each o f which formulates an intuitively interesting aspect o f the “theory o f pseudo-orbits” . In particular our theory states that
• given any weakly pseudo-hyperbolic pseudo-orbit o f M , we can find a pseudo-hyperbolic pseudo-orbit o f M and a metric o f M with respect to which the new pseudo-hyperbolic pseudo-orbit is uniform ly pseudo- hyperbolic,
• given any uniformly pseudo-hyperbolic pseudo-orbit o f M there exists a unique splitting with respect to which the pseudo-orbit is hyperbolic,
• any uniformly hyperbolic invariant set has a neighbourhood, in A t,for which any other pseudo-orbit o f M which is contained in this neighbourhood is a uniformly pseudo-hyperbolic pseudo-orbit,
• any uniformly hyperbolic pseudo-orbit is shadowed by a uniform ly hyper b olic invariant set.
T here are six chapters in this part o f the thesis. T h e first two chapters define and explore the elementary properties o f pseudo-orbits and pseudo-hyperbolic pseudo-orbits respectively. T h e final four chapters contain the proofs o f the above four main parts o f our theory o f pseudo-hyperbolic pseudo-orbits. T h e statement and p roof o f our extension o f A nosov’s Stability lemma, which we choose to call the W eak Shadowing Stable Manifold theorem, can be found in the on ly chapter o f Part IV.