Characterizing expansivity through $C^*$-algebras
S. Bautista, W. Jung, C. A. Morales
TL;DR
The paper develops a C*-algebraic framework to study expansivity by introducing expansive observables $Exp(f)\subset C(X)$ for a compact metric space $X$ and a homeomorphism $f$. It proves $Exp(f)$ is an $F_\sigma$ subalgebra containing constants, with topological conjugacy invariance, and characterizes expansivity-friendly observables for equicontinuous connected systems as locally constant off the nonwandering set, linking expansivity to a common observability property. Using this setup, the authors derive $C^*$-algebraic proofs that the set of periodic points is countable and that an expansive and equicontinuous continuum must be degenerate, among other results; they also show no circle or interval homeomorphism can have a dense set of expansive observables. The results bridge dynamical and operator-algebraic viewpoints and suggest invariants under conjugacy for expansive/dynamical properties.
Abstract
We study expansive homeomorphisms of a compact metric space $X$ through the lens of the commutative $C^*$-algebra $C(X)$ of continuous complex-valued functions, viewed as observables of the system. We introduce the notion of expansive observables: elements of $C(X)$ whose level sets distinguish distinct orbits. We prove that the expansive observables form an F$_σ$-subalgebra of $C(X)$, and we characterize them completely for connected equicontinuous homeomorphisms, showing that only constant observables are expansive in this setting. Furthermore, we establish that topologically conjugate homeomorphisms share the same algebra of expansive observables. Using this framework, we show that the set of periodic points intersects at most countably many level sets of any expansive observable. This provides $C^*$-algebraic proofs of well-known facts like for instance that the set of periodic points of an expansive homeomorphism is countable or that the sole continuum exhibiting homeomorphisms which are both expansive and equicontinuous are the degenerated ones. Finally, we prove that no homeomorphism of the circle or the unit interval admits a dense set of expansive observables, yielding a $C^*$-algebraic demonstration of the nonexistence of expansive homeomorphisms in these spaces.