### Geometry and Dynamics Seminar

# Marten Fels

### (Hamburg)

## "On the Hölder continuity of the core entropy (Second Part)."

** Date: ** |
Tue, January 31, 2017 |

** Time: ** |
14:15 |

** Place: ** |
Research I Seminar Room |

**Abstract:** The core entropy of a postcritically finite quadratic polynomial is the topological entropy of the corresponding Hubbard tree. The core entropy of a rational external angle is the core entropy of the corresponding postcritically finite quadratic polynomial. The resulting function on the set of rational angles can be extended in a natural manner to a function on the whole unit circle.

Dudko and Schleicher resp. Tiozzo have shown that this function is continuous everywhere and Hölder continuous at those rational angles at which the core entropy is positive. Using ideas of Thurston and the concept of energy flow in a directed graph we are able to show that this function is in fact Hölder continuous (exactly) at all angles with positive core entropy, and also to determine the precise Hölder exponent at any such angle.