I have a question on the basin of attraction: Does the dynamic flow on every bounded region inside of a basin of attraction has volume preserving property?
2026-02-23 03:04:20.1771815860
Does dynamical system has volume preserving property in basin of attraction?
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Well, every point in a basin of attraction will get arbitrarily close to the attractor, that's what is meant by attraction. Hence every finite region within a basin of attraction will shrink to a point.
However, in a complex dynamical system there might be Siegel discs or Hermann rings that are conjugate to a rotation of the unit disc, see for example
https://en.wikipedia.org/wiki/Classification_of_Fatou_components
As the dynamic is basically a rotation of the usit disc, restricting the mapping to such domains is volume preserving, but points therein do not converge to an attractor.