Loose almost blowing-up and related properties (Q5957368)

Let \(A\) be a finite set and \(x^n_m\) a finite sequence \(x_m,\dots, x_n\) of elements of \(A\). The per-symbol Hamming distance \(d_n(x^n_1, y^n_1)\) is the fraction of indices \(i\leq n\) for which \(x_i\neq y_i\), and the per-symbol substitution distance \(f_n(x^n_1, y^n_1)\) is \(1-t/n\), where \(t\) is the length of the longest subsequence of \(x^n_1\) that agrees with a subsequence of \(y^n_1\). For two measures \(\mu_n\), \(\nu_n\) on \(A^n\) let \(\overline d_n(\mu_n, \nu_n)\) be the minimum of the expectations \(E_\lambda(d_n(x^n_1, y^n_1))\) where the minimum is over all distributions \(\lambda\) on \(A^n\times A^n\) with marginals \(\mu_n\), \(\nu_n\). For stationary \(\mu\), and \(\nu\) with marginals \(\mu_n\), \(\nu_n\) let \(\overline d(\mu,\nu)= \lim\overline d_n(\mu_n, \nu_n)\). A stationary \(\mu\) is finitely determined if \(\overline d(\mu,\nu)\leq \varepsilon\) holds for any ergodic \(\nu\) for which the marginals and entropy are close enough to those of \(\mu\). In a joint paper with K. Morton, the author had characterized the property ``finitely determined'' and proved an approximate waiting time theorem. The principal theorem in the present paper presents analogues for ``finitely fixed'' processes, defined by replacing \(d_n\) by \(f_n\).