Properties of the Escape Set and Keep Set

For eachthe escape setand the keep setof the basic quadratic functionhave the following properties

1. and

2. is open andis close

3. and

4. andare both invariant under

5. andare both symmetric under rotation byabout 0.

6. is connected andhas no holes in it

Proof

a) By the backwards form of the triangle inequality(1)

is the positive solution of the quadratic equationWe claim that if then(2) for

Indeed ifthen

as required by (2). (1) and (2) now givefor

Ifthen we can apply this inequality successively toto deduce thatis increasing andas

b) Supposethenasso for someLet Sinceandis a polynomialis continuous atso there existssuch thatimplieshenceIt follows thatimpliesasso thatandis open.

c) The setis not the whole ofbecause it does not include the fixed points of

d)as

as

as

e) Since the set is connected andit is sufficient to show that each pointcan be joined to some point ofby a path inThe proof is by contradiction.

Suppose thatcannot be joined toDefine thensinceis open because ifcan be joined tothen so can points of any open disc inwith centre andis connected because pairs of points incan be joined inviaR is thus a subset ofSincecannot be joined intoandis open we deduce

Now use the maximum principle. Ifthenelse we increase the size ofsofor

Applying the maximum principle to each polynomial functiononwe obtainforandwhich contradicts thathenceis connected.