Expansive systems on lattices (Q2222108)

Fix a bounded, distributive lattice \((\mathscr{L},\le)\) with \(0\not=1\). A cover of \(\mathscr L\) is a finite subset \(\mathscr U\subset\mathscr L\) such that \(\bigsqcup\mathscr U=\mathscr L\). An automorphism \(\lambda:\mathscr{L}\to\mathscr{L}\) is expansive provided there is a cover \(\mathscr{U}\) of \(\mathscr{L}\) such that for every cover \(\mathscr{V}\) there is \(N\in\mathbb N\) such that \(\bigwedge_{|n|\le N}\lambda^n\mathscr U\prec \mathscr V\): such a cover \(\mathscr U\) is an expansivity cover for \(\lambda\). A standard example is where \((X,\tau)\) is a compact Hausdorff space and \(\mathscr L_X=\tau\) ordered by inclusion in which case a homeomorphism \(h:X\to X\) induces an automorphism \(\lambda_h:\mathscr L_X\to\mathscr L_X\): then \(\lambda_h\) is expansive if and only if \(X\) is metrisable and \(h\) expansive in the usual sense. For a cover \(\mathscr U\) set \(\mathscr U^2=\{u_1\sqcup u_2\ /\ u_1,u_2\in\mathscr U \mbox{ and } u_1\sqcap u_2\not=0\}\). Generalising results for compact metric spaces, it is shown that if \(\mathscr L\) admits an expansive automorphism with an expansivity cover of the form \(\mathscr U^2\) then \(\dim\mathscr L\) is finite and that if \(\mathscr L\) admits a positively expansive automorphism then there is \(N\in\mathbb N\) such that every cover has a subcover with at most \(N\) elements. A notion of entropy for lattice morphisms is introduced and calculations for non-Hausdorff shifts are presented.