Entropy-Minimizing Diffeomorphisms on a $G_2$-Manifold

TL;DR

The paper exhibits infinitely many entropy-positive diffeomorphisms on a Joyce G_2-manifold M that minimize topological entropy in their homotopy class, and proves these maps act freely on a connected component of the Teichmüller space of torsion-free G_2-structures. Building on Farb–Looijenga’s strategy for K3 surfaces, the authors construct explicit f_T from SL(3,Z) times Z/2, analyze entropy via the spectral data of T, and verify the Yomdin bound is sharp in this setting. They further develop a global deformation theory for G_2-structures, showing a continuous path in the torsion-free moduli space from φ to f^*φ using Joyce’s gluing approach and a families gluing theorem of Crowley–Goette–Hertl. Finally, they discuss a local and global framework for dynamics on G_2-manifolds, including a Verbitsky-type operator, harmonic form decompositions, and a key result restricting dynamics by showing that any Q-preserving diffeomorphism is an isometry, thus finite in irreducible closed G_2 cases, with open questions about broader realizability.

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 Teichmüller space of $G_2$-metrics on $M$. 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.

Theorems & Definitions (18)

• Theorem 1.1
• proof : Proof of (2).
• Remark 1.2
• Lemma 3.1
• proof
• Lemma 3.2: FL
• Theorem 3.3
• Remark 3.4
• proof
• proof : Proof of (1) from Theorem 1.1