Posts: 69

Threads: 29

Joined: Aug 2007

Without lose of generality, set the fixed point of

at zero. It is well known that

. Consider that

, then

. But this gives a functional equation for a geometrical progression solved at

by

.

Posts: 1,389

Threads: 90

Joined: Aug 2007

Daniel Wrote:It is well known that .

This is not well-known but one requirement for uniqueness.

Posts: 69

Threads: 29

Joined: Aug 2007

The idea here is that the Taylor series for an arbitrary dynamical system

can be derived. The second derivitive's dependancy on a geometrical progression results in the different types of fixed points. The higher derivitives also are built up from nested geometrical progressions. The regular formula for geometrical progressions breaks down when the roots of unity are involved which gives an explaination for why therre are different types of fixed points.

Posts: 1,389

Threads: 90

Joined: Aug 2007

Daniel Wrote:The regular formula for geometrical progressions

What do you mean by this?

Posts: 69

Threads: 29

Joined: Aug 2007

bo198214 Wrote:Daniel Wrote:The regular formula for geometrical progressions

What do you mean by this?

That

unless

for some integer

.

For

we have

.

Here

is the Lyponouv characteristic - the first derivitive at a fixed point. Note that the regular formula for geometrical progressions blows up for

which happens at

.