We introduce weaving of maps, which is a way of combining two dynamical systems. Roughly speaking, this means we have two dynamical systems $f$ and $g$ that are represented by the same shift-space via appropriate Markov partitions. We now consider $f$ as the left-shift and $g$ as the right-shift on bi-infinite sequences with identifications coming both from $f$ and $g$. Simple examples give the Baker-transformation and the tight horseshoe (as weaving of the angle-doubling respectively tent-map with itself). More complicated examples give Anosov maps on the $2$-torus (such as the cat map).

A folding map is obtained by a piece of paper (i.e., a $k$-gon) that is folded and mapped to itself. Such a map may be obtained from a real rational map where $z \in \mathbb{C}$ and its complex conjugate $\bar{z}$ are identified. Weaving this with a certain tent map yields a generalized (invertible) pseudo-Anosov map on the $3$-sphere.

This is joint work with Andre de Carvalho and Toby Hall.

Bio: Daniel Meyer is now at the University of Liverpool.