On the structure of isentropes of polynomial maps

Abstract

The structure of isentropes (i.e. sets of constant topological entropy) including the monotonicity of entropy has been studied for polynomial interval maps since the 1980s. We show that isentropes of multimodal polynomial families need not be locally connected and that entropy does in general not depend monotonically on a single critical value.

Keywords:

• interval maps
• topological entropy
• isentrope
• polynomial maps

Notes

1. where it is assumed that for every n, every separate branch of f ⁿ contains a bounded number of n-periodic points. That this assumption is necessary emerges e.g. from [5].