.
|
We have the following qualitative result about the topological entropy
of the maps .
Proof.
For p=1 or 
, the curve 
is a square and
is integrable with integral I(s,y)=y+y', where 
.
The topological entropy is then zero.
For p=2, the
curve is a circle and is integrable having the integral
I(s,y)=y. On each invariant curve I=const, the map 
is a rotation. The topological entropy is also zero.
For p;SPMgt;2, there exist points on the curve where the curvature
is zero. A theorem of Mather [15] assures that there exists then
no homotopiclly nontrivial
invariant curve. Angenent's result [1] implies then that the
topological entropy is non-vanishing.
For p;SPMlt;2, there exist points on the table where the curvature is unbounded.
A theorem of Hubacher [10] assures that there exist no invariant
curves near the boundary of X. Again, Angenent's result leads to
positive topological entropy.
End of the proof.