论文标题
Peano Continua带有自我再生分形
Peano continua with self regenerating fractals
论文作者
论文摘要
我们处理了Masayoshi Hata的问题：每个Peano连续体都是拓扑分形吗？如果存在$ \ Mathcal {f} $，一个紧凑的空间$ x $是拓扑分形，这是$ x $上的有限自图家族，以至于$ x = \ bigcup_ {f \ in \ in \ nathcal {f}} f（x）f（x）所有地图$ f_1，\ dots，f_n \ in \ mathcal {f} $ set $ f_1 \ circ \ dots \ circ f_n（x）$包含在某些set $ u \ in \ mathcal {u} $中。在本文中，我们提出了一些想法，即扩展拓扑分形，我们表明，如果Peano Continuum含有拓扑分形，那么它将其含有所谓的自我再生分形和非空内部的分形。如果对于每一个非空的开放子集$ u $，$ a $对于某些$ a \ setminus u $的地图族常数，则Hausdorff拓扑空间$ a $是一种自我再生分形。自我再生分形的概念更好地反映了自相似性的直觉看法。我们提出了一些自我再生的古典分形。
We deal with the question of Masayoshi Hata: is every Peano continuum a topological fractal? A compact space $X$ is a topological fractal if there exists $\mathcal{F}$ a finite family of self-maps on $X$ such that $X=\bigcup_{f\in\mathcal{F}}f(X)$ and for every open cover $\mathcal{U}$ of $X$ there is $n\in\mathbb{N}$ such that for all maps $f_1,\dots,f_n\in\mathcal{F}$ the set $f_1\circ\dots\circ f_n(X)$ is contained in some set $U\in\mathcal{U}$. In the paper we present some idea how to extend a topological fractal and we show that a Peano continuum is a topological fractal if it contains so-called self regenerating fractal with nonempty interior. A Hausdorff topological space $A$ is a self regenerating fractal if for every non-empty open subset $U$, $A$ is a topological fractal for some family of maps constant on $A\setminus U$. The notion of self regenerating fractal much better reflects the intuitive perception of self-similarity. We present some classical fractals which are self regenerating.
