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On the dimension of the compact invariant sets of certain non-linear maps. (English) Zbl 0544.58014
Dynamical systems and turbulence, Proc. Symp., Coventry 1980, Lect. Notes Math. 898, 230-242 (1981).
[For the entire collection see Zbl 0465.00017.]
Let E be a Banach space, $$U\subset E$$ be an open set and f:$$U\to E$$ be a $$C^ 1$$-map. It is shown that if $$\Lambda \subset E$$ is a compact set such that $$f(\Lambda)\supset \Lambda$$ and for every $$x\in \Lambda$$ the derivative $$D_ xf$$ can be decomposed as a sum of a compact map and a contraction, then the limit capacity (and moreover the Hausdorff dimension) of $$\Lambda$$ is finite.
Reviewer: I.U.Bronshtejn

##### MSC:
 37C70 Attractors and repellers of smooth dynamical systems and their topological structure 34K30 Functional-differential equations in abstract spaces
##### Keywords:
Banach space; limit capacity; Hausdorff dimension
Zbl 0465.00017