论文标题
关于基于换能器的cantor树的肉类着色
On the bijective colouring of Cantor trees based on transducers
论文作者
论文摘要
鉴于无限$ n $ n $ and cantor树的顶点着色（$ m $颜色（$ n，m \ geq 2 $），出现了自然问题：愿这种颜色引起无限路径从根部开始的无限路径的颜色，即每种无限的$ m $ m $颜色的路径都不是这些识别的路径？换句话说，我们询问上面的顶点着色是否可以定义相应的第托空间之间的徒图。我们表明，仅当且仅当$ n \ geq m $ $时，并且就这种自动机定义的功能提供了有效构造的肉体着色时，答案是积极的。我们还表明，有限的Mealy Automaton只能在微不足道的情况下定义这种肉类着色，即$ m = n $。
Given a vertex colouring of the infinite $n$-ary Cantor tree with $m$ colours ($n,m\geq 2$), the natural problem arises: may this colouring induce a bijective colouring of the infinite paths starting at the root, i.e., that every infinite $m$-coloured string is used for some of these paths but different paths are not coloured identically? In other words, we ask if the above vertex colouring may define a bijective short map between the corresponding Cantor spaces. We show that the answer is positive if and only if $n\geq m$, and provide an effective construction of the bijective colouring in terms of Mealy automata and functions defined by such automata. We also show that a finite Mealy automaton may define such a bijective colouring only in the trivial case, i.e. $m=n$.
