In this chapter we are interested in showing that corresponding to any appropriate pseudo-hyperbolic pseudo-orbit, 2L o f M and / , there exists an “adapted” C°° Riemannian m etric on the classifying m anifold, M , with respect to which the eventual contraction o f the pseudo-orbit SI with respect to the original metric becom es im m ediate contraction with respect to the adapted metric.
10.1
A d a p t e d m e trics
Given an invariant set A o f a Riemannian manifold M which is C-uniform ly A- hyperbolic, Mather [Mat6S]* has shown that for any A < \ < 1 < Ò there exists at least one Riemannian m etric, which is C °° on the whole o f M , for which the invariant set A is (^-uniformly À-hyperbolic with respect to the adapted metric. Since the splitting T \M = E\ ®a can be made arbitrarily close to being perpendicular with respect to the adapted metric, conditions 9.1a, 9.1b and 9.1c are satisfied with C = Ò arbitrarily close to 1. This means that the eventual contraction (expansion) with respect to the original metric can be changed to im m ediate contraction (expansion) with respect to the adapted metric.
Just as for uniformly hyperbolic sets, Pesin [Pes76, FHY83, PS89] has shown that there is at least one metric associated to any a- A-hyperbolic invariant set l
l See also [Kat81].
1 4 7
with respect to which the eventual contraction (expansion) with respect to the original metric is changed to im m ediate contraction (expansion) with respect to Pesin’s adapted “metric” . Unfortunately, in this case, the m etric is only defined in the tangent bundle o f the invariant set itself, and moreover the metric is not C °° nor even C °. Pesin was, however, able to use this m etric to prove a Stable M anifold theorem for /c-A-hyperbolic sets.
Com bining the techniques o f both M ather and Pesin, we will show, in the next theorem , that given any C °° Riemannian metric o f M and for any
0 < ¿, 0< A / e < A < l < / c , 1 < ho, and 1 < h0
there exists a * 4-slowly decreasing sequence ( Sn) for which for any (¿„)-*-A-pseudo hyperbolic pseudo-orbit, SI o f Af, which has /i0-hyperbolic blocks, there exists a C °° Riemannian metric o f the classifying manifold A f with respect to which SI is a 5-1-A-pseudo hyperbolic pseudo-orbit with /»o-hyperbolic blocks. W e call this metric o f A f the adapted m etric o f A f with respect to the pseudo-hyperbolic pseudo-orbit SL
A n important property o f the adapted metric o f A# is that while the topological equivalence between the Afn with the adapted m etric and Afn with the original m etric gets progressively poorer, it does so only «2-slowly. W e will make repeated use o f this slow change in the equivalence relationship to translate results proven relative to the adapted m etric back to results for the original metric.
T h e o r e m 1 0.1 (A d a p t e d m e t r i c s ) Consider
0 < ¿, 0< A / c < A < l < / e , 1 < ho, and 1 < ho,
let p = ~ , and let Sn < — pl and note that (£ „) is a k4-slowly decreasing sequence. Finally, assume that SI is an aligned factored ( Sn)-K-X-pseudo hyper bolic pseudo-orbit o f A t with ho-hyperbolic blocks with respect to the original C°° Riemannian metric o f A f.
Then there exists a C °° Riemannian metric o f A t with respect to which both the pseudo-orbit SI and the pseudo-orbit’s maximally shifted closure, 3 . are 6- 1 -A -pseudo hyperbolic pseudo-orbits o f A f and f with h0-hyperbolic blocks. We
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call this m etric the adapted metric o f M fo r the pseudo-orbit 91. W e denote this m etric and its associated noi~m by ((•,•)) and [•] respectively.
There exists a k2 -slowly increasing sequence, B n = h\ ^ , such that, M with the adapted m etric is - j - - ( B n)-related to M with the original metric. Moreover, there is a neighbourhood U o f i ( X ) in M f o r which i f p & U then [t>] = |v| f o r all v € T„M .
I f 91 is an invariant set, there exist constants K + and / f _ which only depend on X, h0, 6, supr€X {||Fx||}, and supr€X {||F/I ||}, such that we have
M i < <u.a, |[/7‘ | < K „
f o r all x € X .
M oreover, i f x £ P n C X , then ] I L . , „ - . , > s 1 >o r all m > n.
P r o o f : This is the m ost tedious and difficult p roof in this part o f the thesis. N ote that the statement o f this theorem specifically allows k = 1. In this case the /c2-slowly increasing sequence, (B „ ) , is a constant sequence.
W hile we have chosen to state the theorem in terms o f a single Riemannian metric o f M , it is more natural to prove the theorem in terms o f a countable collection o f Riemannian m etrics, one on each copy, M n, o f M in M . In order to distinguish the various m etrics contained in this countable collection o f metrics, we will denote the m etric and norm corresponding to A /„ by ((•,•))„, and (■)„ respectively. Furthermore, we will use the notation, ((•, •))(„.*)> an<i to stress in which tangent space the inner product or norm is taken. W ith this notation, in order to prove this theorem, we must prove
1. For every 0 < n there exists a C00 Riemannian m etric, ((•, -))n, on M n. This Riemannian m etric induces a norm, (•)„, on T\fnM , and a distance metric, ^((•»•))n’ on $2(T M n). W ith respect to these ob jects we have that for all x € i ( X n) C M n