## Properties of the Escape Set and Keep Set

For each the escape set and the keep set of the basic quadratic function have the following properties

1. and 2. is open and is close

3. and 4. and are both invariant under 5. and are both symmetric under rotation by about 0.

6. is connected and has no holes in it

Proof

a) By the backwards form of the triangle inequality (1) is the positive solution of the quadratic equation We claim that if then (2) for Indeed if then as required by (2). (1) and (2) now give for If then we can apply this inequality successively to to deduce that is increasing and as b) Suppose then as so for some Let Since and is a polynomial is continuous at so there exists such that implies hence It follows that implies as so that and is open.

c) The set is not the whole of because it does not include the fixed points of d) as  as  as  e) Since the set is connected and it is sufficient to show that each point can be joined to some point of by a path in The proof is by contradiction.

Suppose that cannot be joined to Define then since  is open because if can be joined to then so can points of any open disc in with centre and is connected because pairs of points in can be joined in via R is thus a subset of Since cannot be joined in to and is open we deduce Now use the maximum principle. If then else we increase the size of so for Applying the maximum principle to each polynomial function on we obtain for and which contradicts that hence is connected. 