Weighted kneading theory of one-dimensional maps with a hole

The purpose of this paper is to present a weighted kneading theory for one-dimensional maps with a hole. We consider extensions of the kneading theory of Milnor and Thurston to expanding discontinuous maps with a hole and introduce weights in the formal power series. This method allows us to derive techniques to compute explicitly the topological entropy, the Hausdorff dimension, and the escape rate.

Isentropic Real Cubic Maps

Given a family of bimodal maps on the interval, we need to consider a second topological invariant, other than the usual topological entropy, in order to classify it. With this work, we want to understand how to use this second invariant to distinguish bimodal maps with the same topological entropy and, in particular, how this second invariant changes within a given type of topological entropy level set. In order to do that, we use the kneading theory framework and introduce a symbolic product * between kneading invariants of maps from the same topological entropy level set, for which we show that the second invariant is preserved. Finally, we also show that the change of the second invariant follows closely the symbolic order between bimodal kneading sequences.

A Unified Renormalization Scheme for Two-Piecewise Monotonous Maps of the Interval

In this paper we study the relation between symbolic dynamics and renormalization of maps from the interval to itself, with one discontinuity and two branches of continuity and monotonicity. We introduce a symbolic formalism based on kneading theory that permits us to deal, in a unified way, with the four cases that can occur, at the same time. Using this formalism we also study properties of similarity from the four spaces of kneading invariants.

DEVIL'S STAIRCASE OF GAP MAPS

This study demonstrates the devil's staircase structure of topological entropy functions for one-dimensional symmetric unimodal maps with a gap inside. The results are obtained by using kneading theory and are helpful in investigating the communication of chaos.