Metrizable extensions of dynamical systems from function algebras
Suppose that \(\varphi\) is a homeomorphism of a compact metric space \(X\). Then, one defines a dynamical system \((X,\varphi,{\mathbb Z})\). Another dynamical system \((X',\varphi',{\mathbb Z})\) is called an extension of \((X,\varphi,{\mathbb Z})\) if there exists a continuous surjection \(p\) from \(X'\) onto \(X\) satisfying \(p \circ \varphi' = \varphi \circ p\). The authors study a special kind of extension, called a tame extension, introduced by \textit{J. Peters} and \textit{T. Pennings} in [Proc. Am. Math. Soc. 105, No. 1, 80-86 (1989; Zbl 0674.43003) and J. Math. Anal. Appl. 159, No. 2, 345-360 (1991; Zbl 0734.54024)]. Roughly speaking, a tame extension arises by adding to the commutative \(C^*\)-algebra \(C(X)\) of continuous functions on \(X\) a family of bounded functions on \(X\) having not too many discontinuities and then passing to the spectrum \(X'\) of this new commutative \(C^*\)-algebra. The homeomorphism \(\varphi\) of \(X\) is naturally extended to \(X'\). The authors study how several properties of dynamical systems are or are not preserved through tame extensions. They study source, sink and saddle points, the shadowing property and distallity.
scientific article; zbMATH DE number 1472030