A conservative Turing complete $S^4$ flow

Pablo Suárez-Serrato

TL;DR

A Turing complete, volume preserving, smooth flow on the $4$-sphere is presented.

We present a Turing complete, volume preserving, smooth flow on the $4$-sphere.

A Turing complete, volume preserving, smooth flow on the

-sphere is presented.

We present a Turing complete, volume preserving, smooth flow on the

-sphere.

Paper Structure (8 sections, 2 theorems, 13 equations, 2 figures)

This paper contains 8 sections, 2 theorems, 13 equations, 2 figures.

Theorem 1

There exists a Turing complete volume preserving smooth flow on $S^4$.

Figure 1: Here we illustrate the first couple of steps in a possible parametrization of Moore's diffeomorphism to realize it in an area preserving way. First, with a clockwise $\pi$ rotation $\rho$, followed by exchanging the sectors $b$ and $d$ while preserving the area of small neighborhoods of them with the map $\sigma$. These two transformations, $\rho$ and $\sigma$, all preserve area, so their composition $\sigma\circ \rho$ is also an area preserving diffeomorphism.
Figure 2: Here we show the last step in an area preserving version of Moore's diffeomorphism. The teal mass on the left side sector $a$ circulates to fill the area $\bar{a}$ shown in teal on the right. Likewise, the blue mass on the left side sector $d$ circulates to end occupying the blue area $\bar{d}$ on the lower right