Microorganismos presentes en los productos lácteos

W e are now interested in showing that the pair o f sections, g u and g ', intersect at a unique section, x , o f t : A rP 0 / / A TP —* H , and m oreover that this section is /-invariant. In order to d o this, we define, for the pair o f sections, <7U € Lipu (a) and o ' € Lip* (a) , to be the section o f sets o f intersection points o f o u and <7* in the fibres o f A rP 0 / / A TP .

L e m m a 7 .1 3 The section o f sets, o u# o ' , is a section o f single points. Moreover, this section, x = gu# g ' , is an f -invariant sectio n o f the fibre bundle x : A rP A rP - H .

P r o o f: Let su and s ' denote the fibre com p on en ts o f g u and g* respectively. Consider p € H . T h e sections g u and g' are defined over the whole o f ( ArP ) , and (A ft £ * )p respectively. For p < 1, Corollaries 7.5 and 7.9 im ply that over the proper subsets ( A rE u)p C ( ArE', )p, and ( A r£J")p C ( ArE ') p, the graphs o f gu and g* must lie in the box ( A r£ * 0 / / A rETt)p. This implies that there must be at least one transversal intersection o f g u and g ‘ inside the closure o f { A r P 0« A rP ) , .

Now chose any point, x, in this intersection, and place a pair o f transversal cones based at x with slopes o f A in the stable a n d unstable directions respectively. Since A < 1, these cones do not intersect except at the point x. Since y “ € Lipu (a) and g* € Lip' (a) and since, moreover they intersect at x, the sections g* and g u must lie in the stable and unstable cones based at x. This implies that the point x is the only point o f intersection o f gu and g * .

Since p € H was arbitrary, there exists exactly one point o f intersection o f gu and g ' in each fibre o f ArP 0w ArP which is m oreover contained in A rP 0 / / A rP - Together these intersection points form a section o f the bundle ArP 0w ArP . Let x denote this section.

91 In the b od y o f lemma 7.4 we showed that L ipj ( / , o <7) < A + A for all <7€ Lipu ( A ). This means that the fibre Lipschitz constant o f the fibre com ponent o f / o gu, is at most A. Lemma 7.10 shows that f o g * (A /j£ * ) C g* ( ArE*) and hence the fibre Lipschitz constant o f the fibre com ponent o f f o g * , is also at most A. Hence by placing stable and unstable cones based at f (g u# g ‘ ) = f o g u# f o g * with slopes A, we can again show that the sections, f o g * and / o gu intersect at exactly one point, namely f ( g u# g ‘ ).

Again By lemma 7.10 we know th a t / (gu# g * ) is contained in g* ( ArE 1). Since / ( g u ( A /?£ “ )) D gu ( A /jjE"), we see that the unique intersection g u# g * of the sections g u and g* is equal to the unique intersection f (g u# g * ) o f the sections f o g u and / o g*. That is / (gu# g t ) = yu#<7*. T h is means that the section,

x = gu# g* is /-invariant. ■

W hile the following lemma is not strictly required for the main theorems in this thesis, it does provide a useful characterization o f the section x as a Cauchy sequence o f the zero section o f E? ET1. T o simplify the notation, let 5“ = T" / (Oa„s- ) and g\ = r*y (Oa„k*). and finally, let s “ and s* denote the fibre com ponents o f <7“ and g\ respectively.

Lem m a 7 .1 4 Let 0 < then gnu#9m, ** a section o f the bundle x : A r£* ® // A rE? —* H. Moreover, f o r all 0 < m „ m „ j i , , # ? « , , is a Cauchy se­ quence with respect to m u, with respect t o m ,, and jo in tly with respect to

P r o o f: W e begin by showing that for fixed mu, >s a Cauchy sequence with respect to m t. Consider m „ m , > 0. Fix p € H . To simplify the following discussion we will work in the fibre ( A rE* ® // A rE u)p. As in figure 7.1, define

" ( p ) = (p ) * « • + « ! * *m, ( « . ) +

V

(

= V, +

= a*. (U.) + U„

«"(P) = 9m„#9‘m, * + W>~

Since 6L ip u(A ), we know that \to, — u,|p < A |u>„ — u„|p. Similarly, since g^' € L ip* (A ), we know that |u>„ — u»|p < A |u>, - u,|p. Let r* = - u„. Since

92

Figure 7.1: Diagram used to prove that for fixed m u, *8 a Cauchy sequence with respect t o m ,.

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g* is a Cauchy sequence we can make \zu\p arbitrarily small by choosing m „ m , large enough. C om bining the previous two inequalities we have

Hence in the b o x norm in the fibre ( ArF ® « ArEu)p the distance between 9mu#9m . and 9mu# 9 A , can be made arbitrarily small by choosing |zu|p small, and this can b e done by choosing m , and m , large enough.

In fact since g * is a Cauchy sequence in the space o f sections o f the bundle if,, the norm o f zu can be made arbitrarily small independently o f the specific, fixed, g^H ■ In particular this means that for all e > 0, there exists an M > 0 such that for all m u > 0, and all m ,,m , > M

Similar argum ents can be used to show that g^m is a Cauchy sequence with respect to m u fo r all m , > 0.

Now, to show that s£,„#<7m, *s a Cauchy sequence for both m u, and m , jointly, consider £ > 0. Choose M , > 0 for which for all m u > 0, and all m „ m , > M ,

Similarly ch oose M u > 0 for which for all m u,m „ > A /«, and all m , > 0, Hence, since A < 1, we have

Let M be the m axim um o f M u and M , and consider m u,m u, m ,,r h , > M . Then we have

9 4