Covering relations for multidimensional dynamical systems
A form of topological hyperbolicity is introduced. It gives a sufficient condition for positive topological entropy (one form of ``chaos'') and existence of infinitely many periodic orbits. This condition can be checked by finite precision computations. It is a generalization, in the presence of contracting behavior, of the basic fact that if \(f\) is a continuous map and \(B_1,\dots, B_n,B_{n+1}:=B_n\) are compact topological balls with \(f(B_{i})\supset B_{i+1}\), then there exists \(x\in B_1\) with \(f^n(x)=x\) and \(f^{i-1}(x)\in B_i\) for \(1\leq i\leq n\). Let us state the condition replacing \(f(B_i)\supset B_{i+1}\). \(B_i,B_{i+1}\) are identified with standard rectangles \(I_u\times I_s\) (\(I_u,I_s\) are disks in Euclidean spaces) such that, in this identification, the map \(f| B_i\) is homotopic to an affine map with image \(\Delta\) satisfying \(\{0\}\times I_u\subset\Delta\subset\{0\}\times \hat I_u\) with \(\hat I_u\) a larger disk. Additionally, the homotopy must respect boundaries in the following sense: it does not intersect \(I_u\times\partial I_s\) and it maps \(I_s\times\partial I_u\) away from \(I_u\times I_s\).
scientific article; zbMATH DE number 2092983
scientific article