We present the different distances on tilings of Rd that exist in the literature, we prove that (most of) these definitions are correct (i.e. they indeed define metrics on tilings of Rd ). We prove that for subshifts with finite local complexity (FLC) these metrics are topologically equivalent and even metrically equivalent, and also we present classical results of compacity and completeness. Note that, excluding the equivalence of these metrics, all of the results presented here are known (see for example the survey [Rob04]) however we were unable to find a reference with complete proofs for some of these results so we decided to write this notice to clarify some definitions and give full proofs.
翻译:我们介绍了文献中存在的Rd砖瓦的不同距离,我们证明(大多数)这些定义是正确的(即它们确实定义了Rd砖瓦的衡量标准 )。我们证明,对于局部复杂程度有限的次轮班(FLC),这些衡量标准在地表上是等同的,甚至是等同的,我们还介绍了可比性和完整性的经典结果。请注意,除这些衡量标准等同性外,这里提出的所有结果都已经为人所知(例如,调查[Rob04]),然而,我们无法找到有这些结果的完整证据的参考,因此我们决定写这份通知,以澄清某些定义并提供充分的证据。