Cone conditions and covering relations for topologically normally hyperbolic invariant manifolds

Combinatorial approach to detection of fixed points, periodic orbits, and symbolic dynamics

“…, pM k q n u. (7) exit set Further, we divide each D j into small cubes tpC j q β u β1,...,pNj q n , of side 1{N j ,…”

“…. , k, determine a coarse decomposition C α of pr0, 1s n q k as in (7). By Theorem 4.1, there exists a cube C α¦ pC 1 q α ¦ 1 ¢pC 2 q α ¦ 2 ¢.…”

“…The method in this paper has potential applications to computer assisted proofs in dynamical systems. In this area, the method of correctly aligned windows has been used extensively to prove the existence of fixed points, periodic orbits, symbolic dynamics, and much more, when combined with other tools; some references include [24,11,20,22,2,7,6,17,3]. As mentioned earlier, the correct alinement of windows requires that the crossing of the windows is topologically non-trivial.…”

Strict localization of eigenvectors and eigenvalues

“…The notion of cone condition originally appeared in the late 60's in the works of Alekseev, Anosov, Moser and Sinai [22]. These techniques are used in the examination of differential equations [4,9,14,19,25]. In Section 3 we show the basic properties of operations.…”

“…To calculate the approximation of eigenvectors and eigenvalues one can use any numerical method 4. By P −1 we denote such an interval matrix that Q −1 ∈ P −1 for every Q ∈ P .…”