Polygon cyclic fixpoint conjecture

[tommy1729]

Consider a real-analytic function f.

Consider An nth cyclic fixpoint A.

N >= 4.

Connect those n fixpoints : A , f(A) , ... With a straith line.
That makes a polygon.

Consider the cyclic points that make convex polygons.
Call them convex cyclic points.
Call the polygons : cyclic polygons.

Conjecture : Every cyclic polygon within a cyclic polygon of order n , is cyclic of order m :

M =< N.

Regards

Tommy1729

[tommy1729]

Consider a real-analytic function f.

Consider An nth cyclic fixpoint A.

N >= 4.

Connect those n fixpoints : A , f(A) , ... With a straith line.
That makes a polygon.

Consider the cyclic points that make convex polygons.
Call them convex cyclic points.
Call the polygons : cyclic polygons.

Conjecture : Every cyclic polygon within a cyclic polygon of order n , is cyclic of order m :

M =< N.

Regards

Tommy1729

I assume there are conditions that we need to add.
True for all f would be surprising.

I guess it is more of An intresting property than conjecture.
And a quest for examples and counterex.

Is it true for f = exp ??


Also , is every 2cycle close to a fixpoint for exp ?

I guess so.


Regards

Tommy1729