Detecting topological and Banach fractals among zero-dimensional spaces

作者：Banakh Taras; Nowak Magdalena; Strobin Filip

来源：Topology and Its Applications, 2015, 196: 22-30.

DOI：10.1016/j.topol.2015.09.003

• Topological fractal
• Banach fractal
• Attractor
• Iterated function system
• Contracting function system
• Fixed point
• Zero-dimensional compact space

A topological space X is called a topological fractal if X = boolean OR(f is an element of F) f (X) for a finite system F of continuous self-maps of X, which is topologically contracting in the sense that for every open cover U of X there is a number n is an element of N such that for any functions f(1),...,f(n) is an element of F, the set f(1)circle...circle f(n)(X) is contained in some set U is an element of U. If, in addition, all functions f is an element of F have Lipschitz constant < 1 with respect to some metric generating the topology of X, then the space X is called a Banach fractal. It is known that each topological fractal is compact and metrizable. We prove that a zero-dimensional compact metrizable space X is a topological fractal if and only if X is a Banach fractal if and only if X is either uncountable or X is countable and its scattered height <(h)over bar>(X) is a successor ordinal. For countable compact spaces this classification was recently proved by M. Nowak.

• 出版日期2015-12