 By Thesleff H

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Extra info for AN INTRODUCTION TO THE PYTHAGOREAN WRITINGS OF THE HELLENISTIC PERIOD

Example text

Of G in a single Z(P G) is closed. consists of finite orbits. Taking the (cf. e. G Z(P G) is PG minimal, closed also in this case. d) Finally, the proposition holds trivially when that is Z(P G) = F. 0 We observe that for Ge Homeo+(I) lS F = I the only minimal sets of are stationary points. From the preceding proof we deduce two corollaries. - for any Gx P~oo6: g E G x then g l! - Z(P G) is Clearly cG G x Z(P G) keeps the orbit for G(x) then thus showing ~ for ~ or i 2) then x E F. x E Z(P G). Conversely, if 0 is another closed orbit o l!

Iii) The following conditions are equivalent: (I) L E F is proper. (2) The manifold topology of the topology of L is the same as that induced on L by M. iv) Show that a closed leaf is proper. The converse is not true; give an example. 2). a) L E F is closed in M if and only if L n Q is closed in Q. sely. vi) A minimal set Mc M is always connected for the induced to- pology. It is arcwise connected if and only if 3. d bl1l1dl~; M is not exceptional. ~. We want to illustrate the notions of the preceding paragraph when the foliation is a foliated bundle.

FIIB I ) diffeomorphisms. Then the foliations (M,F) = (Mo,F o ) U O·1 1,F I ) and IP are isomorphie by ~ Cr diffeomorphism ~ keeping M o and : (M,F) -> (M',F') invariant. - a) and i) Gluing is also possible in the following cases: Bo same foliated manifold or transverse to Bo diffeomorphism between on Bo and b) BI are diffeomorphic boundary components of the (M,F) and B BI' o and and F is either tangent to Bo and BI Clearly in the latter case the gluing must preserve the induced foliations BI' B is a boundary component of (M,F) omorphism we take a fixed point free involution of induced foliation when F is transverse to and as gluing diffeB, preserving the B.