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UID:6306@i2m.univ-amu.fr
DTSTART;TZID=Europe/Paris:20211012T110000
DTEND;TZID=Europe/Paris:20211012T120000
DTSTAMP:20241120T201359Z
URL:https://www.i2m.univ-amu.fr/evenements/one-sided-nonexpansiveness-and-
 periodic-decomposition-under-nivats-conjecture-setting/
SUMMARY:Cleber Fernando COLLE (Rio de Janeiro State University (UERJ)): One
 -sided nonexpansiveness and periodic decomposition under Nivat's Conjectur
 e setting
DESCRIPTION:Cleber Fernando COLLE: A few years ago\, Kari and Szabados show
 ed that\, for any low pattern complexity configuration η∈A^ℤᵈ \, wi
 th A ⊂ Z a finite alphabet\, there exist periodic configurations η₁\,
  …\, ηₖ∈ℤ^ℤᵈ such that η=η₁+…+ηₖ. In the two-dimensi
 onal case (d = 2)\, such a decomposition theorem is naturally related to n
 onexpansiveness.\nFor instance\, it is known that if X_η denotes the clos
 ure of the ℤ²-orbit of η and l is a one-dimensional nonexpansive subsp
 ace 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 pe
 riodic and k is the minimal possible number of periodic configurations suc
 h that η can be decomposed\, then l is a one-dimensional nonexpansive sub
 space on X_η if\, and only if\, l contains a vector period for ηᵢ \, w
 here 1≤i≤k.\nRecent advances related to Nivat’s conjecture made use 
 of a refined version of nonexpansiveness\, a so-called one-sided nonexpans
 ive direction. By imposing a strong restrictive condition on the complexit
 y 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  nonexpan
 sive direction on X_η . This leads us to the following natural question: 
 For a low convex pattern complexity configuration η∈A^Z² \, do one-sid
 ed nonexpansive directions on X η arise in pairs?\nIn this seminar\, we w
 ill show in detail that\, for a low convex pattern complexity configuratio
 n η∈A^ℤ²\, one may assume\, without loss of generality for the proof
  of Nivat’s conjecture\, that:\n(i) Szabados’s conjecture holds\;\n(ii
 ) one-sided nonexpansive directions on X_η arise in pairs.\n[su_spacer si
 ze="10"]En visio :\nhttps://webconf.lal.cloud.math.cnrs.fr/b/pie-hmu-en9\n
 \n\n&nbsp\;
CATEGORIES:Séminaire,Ernest
LOCATION:I2M Luminy - Ancienne BU\, Salle Séminaire2 (RdC)\, 163 Avenue de
  Luminy\, 13009 Marseille\, France\, 
X-APPLE-STRUCTURED-LOCATION;VALUE=URI;X-ADDRESS=163 Avenue de Luminy\, 1300
 9 Marseille\, France\, ;X-APPLE-RADIUS=100;X-TITLE=I2M Luminy - Ancienne B
 U\, Salle Séminaire2 (RdC):geo:0,0
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