Direct topological factorization for topological flows

This paper considers the general question of when a topological action of a countable group can be factored into a direct product of a nontrivial actions. In the early 1980's D. Lind considered such questions for $\mathbb{Z}$-shifts of finite type. We study in particular direct factorizations of subshifts of finite type over $\mathbb{Z}^d$ and other groups, and $\mathbb{Z}$-subshifts which are not of finite type. The main results concern direct factors of the multidimensional full $n$-shift, the multidimensional $3$-colored chessboard and the Dyck shift over a prime alphabet. A direct factorization of an expansive $\mathbb{G}$-action must be finite, but a example is provided of a non-expansive $\mathbb{Z}$-action for which there is no finite direct prime factorization. The question about existence of direct prime factorization of expansive actions remains open, even for $\mathbb{G}=\mathbb{Z}$.Comment: 21 pages, some changes and remarks added in response to suggestions by the referee. To appear in ETD

A certain synchronizing property of subshifts and flow equivalence

We will study a certain synchronizing property of subshifts called $\lambda$-synchronization. The $\lambda$-synchronizing subshifts form a large class of irreducible subshifts containing irreducible sofic shifts. We prove that the $\lambda$-synchronization is invariant under flow equivalence of subshifts. The $\lambda$-synchronizing K-groups and the $\lambda$-synchronizing Bowen-Franks groups are studied and proved to be invariant under flow equivalence of $\lambda$-synchronizing subshifts. They are new flow equivalence invariants for $\lambda$-synchronizing subshifts.Comment: 28 page

Finitely dependent coloring

We prove that proper coloring distinguishes between block-factors and finitely dependent stationary processes. A stochastic process is finitely dependent if variables at sufficiently well-separated locations are independent; it is a block-factor if it can be expressed as an equivariant finite-range function of independent variables. The problem of finding non-block-factor finitely dependent processes dates back to 1965. The first published example appeared in 1993, and we provide arguably the first natural examples. More precisely, Schramm proved in 2008 that no stationary 1-dependent 3-coloring of the integers exists, and conjectured that no stationary k-dependent q-coloring exists for any k and q. We disprove this by constructing a 1-dependent 4-coloring and a 2-dependent 3-coloring, thus resolving the question for all k and q. Our construction is canonical and natural, yet very different from all previous schemes. In its pure form it yields precisely the two finitely dependent colorings mentioned above, and no others. The processes provide unexpected connections between extremal cases of the Lovasz local lemma and descent and peak sets of random permutations. Neither coloring can be expressed as a block-factor, nor as a function of a finite-state Markov chain; indeed, no stationary finitely dependent coloring can be so expressed. We deduce extensions involving d dimensions and shifts of finite type; in fact, any non-degenerate shift of finite type also distinguishes between block-factors and finitely dependent processes

On subshift presentations

We consider partitioned graphs, by which we mean finite strongly connected directed graphs with a partitioned edge set ${\mathcal E} ={\mathcal E}^- \cup{\mathcal E}^+$. With additionally given a relation $\mathcal R$ between the edges in ${\mathcal E}^-$ and the edges in $\mathcal E^+$, and denoting the vertex set of the graph by ${\frak P}$, we speak of an an ${\mathcal R}$-graph ${\mathcal G}_{\mathcal R}({\frak P},{\mathcal E}^-,{\mathcal E}^+)$. From ${\mathcal R}$-graphs ${\mathcal G}_{\mathcal R}({\frak P},{\mathcal E}^-,{\mathcal E}^+)$ we construct semigroups (with zero) ${\mathcal S}_{\mathcal R}({\frak P}, {\mathcal E}^-,{\mathcal E}^+)$ that we call ${\mathcal R}$-graph semigroups. We describe a method of presenting subshifts by means of suitably structured labelled directed graphs $({\mathcal V}, \Sigma,\lambda)$ with vertex set ${\mathcal V}$, edge set $\Sigma$, and a label map that asigns to the edges in $\Sigma$ labels in an ${\mathcal R}$-graph semigroup ${\mathcal S}_{\mathcal R}({\frak P}, {\mathcal E}^-, {\mathcal E}^-)$. We call the presented subshift an ${\mathcal S}_{\mathcal R}({\frak P}, {\mathcal E}^-, {\mathcal E}^-)$-presentation. We introduce a Property $(B)$ and a Property (c), tof subshifts, and we introduce a notion of strong instantaneity. Under an assumption on the structure of the ${\mathcal R}$-graphs ${\mathcal G}_{\mathcal R}({\frak P},{\mathcal E}^-, {\mathcal E}^-)$ we show for strongly instantaneous subshifts with Property $(A)$ and associated semigroup ${\mathcal S}_{\mathcal R}({\frak P},{\mathcal E}^-,{\mathcal E}^-)$, that Properties $(B)$ and (c) are necessary and sufficient for the existence of an ${\mathcal S}_{\mathcal R}({\frak P}, {\mathcal E}^-,{\mathcal E}^-)$-presentation, to which the subshift is topologically conjugate,Comment: 33 page

Combinatorics of r-Dyck paths, r-Parking functions, and the r-Tamari lattices

This paper's aim is to present recent combinatorial considerations on r-Dyck paths, r-Parking functions, and the r-Tamari lattices. Giving a better understanding of the combinatorics of these objects has become important in view of their (conjectural) role in the description of the graded character of the Sn-modules of bivariate and trivariate diagonal coinvariant spaces for the symmetric group.Comment: 36 pages, 12 figure