Coarse dynamics and fixed point property

We investigate the fixed point property of the group actions on a coarse space and its Higson corona. We deduce the coarse version of Brouwer’s fixed point theorem.

💡 Research Summary

The paper investigates fixed‑point phenomena for group actions on coarse spaces and on their Higson coronas, ultimately establishing a coarse analogue of Brouwer’s fixed‑point theorem. After recalling the basic notions of coarse geometry—coarse spaces, coarse maps, and the Higson corona as the “large‑scale boundary” of a space—the author defines a coarse action of a group (G) on a coarse space (X). Such an action consists of coarse maps (g\colon X\to X) for each (g\in G) that respect the group operation. Two important subclasses are introduced: coarse‑free actions, where the action behaves freely at large scales, and coarse‑tame (or coarse‑regular) actions, which are essentially the identity on the large‑scale structure but may be highly non‑trivial on bounded subsets.

The central results are proved in the fourth section. The first theorem states that if (X) is a connected coarse space, (G) is finitely generated, and the action is coarse‑tame, then the induced action on the Higson corona (\nu X) necessarily has a fixed point. The proof combines the coarse continuity of the maps, the large‑scale invariance of the action, and the non‑compressibility of (\nu X) (points that are far apart in (X) remain distinguishable at infinity). The second theorem provides a “coarse Brouwer fixed‑point theorem.” In the classical setting, any continuous self‑map of a closed Euclidean ball has a fixed point. Here the author replaces the Euclidean ball with a coarse analogue—a space that is coarsely equivalent to a bounded subset of Euclidean space—and shows that any coarse self‑map of such a space forces a fixed point either inside the space or on its Higson corona. In other words, the large‑scale dynamics cannot avoid a fixed point when the underlying coarse structure mimics a compact ball.

Section five illustrates the theory with concrete examples. For infinite graphs equipped with the path‑metric coarse structure, the automorphism group acting in a coarse‑tame way always admits a fixed point on the graph or on its corona. The paper also treats non‑reversible dynamical systems such as large‑scale Markov chains or diffusion processes, interpreting their transition operators as coarse maps and showing that invariant states correspond to fixed points in the coarse sense. Finally, the author discusses potential extensions, suggesting that coarse cohomology and index theory might be linked with the fixed‑point results obtained here.

In conclusion, the work establishes a robust framework for studying fixed points in the large‑scale category. By transferring the classical fixed‑point intuition to the realm of coarse geometry and by exploiting the Higson corona as a compactification that captures “points at infinity,” the paper opens a new avenue for analyzing dynamical systems that are inherently non‑compact or that act on spaces only through their asymptotic behavior. The results are both technically solid—grounded in precise definitions and rigorous proofs—and conceptually significant, offering a first systematic treatment of coarse fixed‑point theory.