# One-sided nonexpansiveness and periodic decomposition under Nivat’s Conjecture setting

Cleber Fernando COLLE
Rio de Janeiro State University (UERJ)

Date(s) : 12/10/2021   iCal
11 h 00 min - 12 h 00 min

A few years ago, Kari and Szabados showed that, for any low pattern complexity configuration η∈A^ℤᵈ , with A ⊂ Z a finite alphabet, there exist periodic configurations η₁, …, ηₖ∈ℤ^ℤᵈ such that η=η₁+…+ηₖ. In the two-dimensional case (d = 2), such a decomposition theorem is naturally related to nonexpansiveness.
For instance, it is known that if X_η denotes the closure of the ℤ²-orbit of η and l is a one-dimensional nonexpansive subspace on X_η , then l contains a vector period for ηᵢ, where 1≤i≤k. In his Ph.D. thesis, Szabados conjectured that, if η is not doubly periodic and k is the minimal possible number of periodic configurations such that η can be decomposed, then l is a one-dimensional nonexpansive subspace on X_η if, and only if, l contains a vector period for ηᵢ , where 1≤i≤k.
Recent advances related to Nivat’s conjecture made use of a refined version of nonexpansiveness, a so-called one-sided nonexpansive direction. By imposing a strong restrictive condition on the complexity function, Cyr and Kra showed that a given line through the origin in ℝ² (one-dimensional subspace) is nonexpansive on X_η if, and only if, the same line endowed of any given orientation is a one-sided  nonexpansive direction on X_η . This leads us to the following natural question: For a low convex pattern complexity configuration η∈A^Z² , do one-sided nonexpansive directions on X η arise in pairs?
In this seminar, we will show in detail that, for a low convex pattern complexity configuration η∈A^ℤ², one may assume, without loss of generality for the proof of Nivat’s conjecture, that: