**Week 11 **

It follows from Theorem 2 of Week 10 and a standard
compact --> open argument that (keeping the same
context as that theorem) there exists a positive
*epsilon* such that *alpha*[(-*epsilon*,*epsilon*)
x *B*_{*a*/2}(*x*_0)] is contained in
*B*_*a*(*x*_0). So for |*t*| < *epsilon*,
and *x* in *B*_{*a*/2}(*x*_0),
the point *alpha*(*t*,*x*) is an admissible
initial condition for the problem considered in Theorem 1, i.e. for
|*s*| < *b*, *alpha*(*s*,
*alpha*(*t*,*x*)) is a point of *U*.

*Theorem 3.*(See Spivak, pp. 201,202).
For *x* in *B*_{*a*/2}(*x*_0),
|*t*| < *epsilon*,
and |*s*| and |*s*+*t*| < *b*,

*alpha*(*s*, *alpha*(*t*,*x*)) =
*alpha*(*s*+*t*,*x*).

It follows that for |*t*| < *epsilon* the map *phi*_*t*
defined by *phi*_*t* (*x*) = *alpha*(*t*,*x*)
is a homeomorphism
(in fact, a diffeomorphism when we can suppose that *f* is of class
C^{infty}) of *B*_{*a*/2}(*x*_0) onto its image, with inverse
*phi*_{-*t*}. Because when *s* and *t*
are sufficiently small
we have

*phi*_*s* o *phi*_*t* = *phi*_{*s*+*t*}

the ``group'' of these diffeomorphisms has the structure
of the additive group of real numbers in a neighborhood of 0.
The *phi*'s can be called a local one-parameter group of local
diffeomorphisms.

*Class exercise.* Show that the ``law of exponents''
(*e*^*s*)(*e*^*t*) = *e*^{*s*+*t*}
is a special case of this phenomenon.

All this analysis can be moved onto a manifold:

*Theorem 4.* (Spivak, Theorem 5.5)
Let *X* be a smooth vectorfield on a manifold *M*,
and *p* a point of *M*.
Then there exists a neighborhood *V* of *p*,
an *epsilon* > 0 and
and a local one-parameter group of local
diffeomorphisms *phi*_*t* : *V* --> *M*
(|*t*| < *epsilon*)
satisfying

(d/d

The two occurrences of ``local'' can be deleted in case
the vectorfield *X* has compact support (in particular,
when *M* is compact).

*Theorem 5.* (Spivak, Theorem 5.6)
Let *X* be a smooth vectorfield, with compact
support, on a manifold *M*.
Then there exists
one-parameter group of
diffeomorphisms *phi*_*t* : *M* --> *M*
satisfying

(d/d

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