English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Let $\Gamma$ be a countable Abelian group and $f \in \Z[\Gamma]$, where$\Z[\Gamma]$ denotes the integral group ring of $\Gamma$. Consider thePontryagin dual $X_f$ of the cyclic $\Z[\Gamma]$-module $\Z[\Gamma]/\Z[\Gamma]f$ and suppose that $f$ is weakly expansive (e.g., $f$ is invertible in$\ell^1(\Gamma)$, or, when $\Gamma$ is not virtually $\Z$ or $\Z^2$, $f$ iswell-balanced) and that $X_f$ is connected. We prove that if $\tau \colon X_f\to X_f$ is a $\Gamma$-equivariant continuous map, then $\tau$ is surjective ifand only if the restriction of $\tau$ to each $\Gamma$-homoclinicity class isinjective. We also show that this equivalence remains valid in the case when$\Gamma = \Z^d$ and $f \in \Z[\Gamma] = \Z[u_1,u_1^{-1}, \ldots, u_d,u_d^{-1}]$ is an irreducible atoral polynomial such that its zero-set $Z(f)$ iscontained in the image of the intersection of $[0,1]^d$ and a finite union ofhyperplanes in $\R^d$ under the quotient map $\R^d \to \T^d$ (e.g., when $d\geq 2$ such that $Z(f)$ is finite). These two results are analogues of theclassical Garden of Eden theorem of Moore and Myhill for cellular automata withfinite alphabet over $\Gamma$.

Homoclinically expansive actions and a Garden of Eden theorem for harmonic models