Entropy-Minimizing Diffeomorphisms on a $G_2$-Manifold

TL;DR

This paper constructs infinitely many entropy-minimizing diffeomorphisms on a Joyce $G_2$-manifold, showing they achieve homological entropy bounds and act freely on the moduli space, revealing complex topological and dynamical properties.

Contribution

It introduces a method to construct entropy-minimizing diffeomorphisms on Joyce manifolds that achieve Yomdin's entropy bounds and act freely on the $G_2$ structure moduli space.

Findings

Constructed infinitely many diffeomorphisms with minimal entropy.

Diffeomorphisms act freely on a component of the $G_2$ structure space.

Homotopy moduli space of $G_2$ structures has infinite fundamental group.

Abstract

In this paper, we construct infinitely many diffeomorphisms of a Joyce manifold $M$ which achieve Yomdin's homological lower bound for topological entropy, imitating a recent construction of Farb-Looijenga for K3 surfaces. Moreover, following a recent paper by Crowley-Goette-Hertl, we show these diffeomorphisms act freely on a connected component of the Teichm\"uller space of $G_2$ structures on $M$, and hence that the homotopy moduli space of $G_2$ structures on $M$ has infinite fundamental group. We also discuss a putative analogy between dynamics on a $G_2$ manifold and that of an algebraic surface, and prove a theorem about its limitations.