Pierre Collet's Concepts and Results in Chaotic Dynamics: A Short Course PDF

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Extra info for Concepts and Results in Chaotic Dynamics: A Short Course

Example text

Show that there is a constant D > 1 such that for any vector v ∈ T Mx0 D−1 (v, v) ≤ v, v ≤ D(v, v) . Hint: write v = ξ + η with ξ ∈ E u and η ∈ E s . 13). Again, we show the proof only for M = R d . , conjugating with a translation) we can assume that x0 = 0. If we denote L = D0 f , we have f (x) = Lx + Q(x) with D0 Q = 0; that is, Q denotes terms of higher order (near 0). Since L is hyperbolic, we can decompose as before E = E u ⊕ E s , and if we denote by Lu the restriction of L to the invariant subspace E u (respectively Ls , the restriction of L to the invariant subspace E s ), there is a number β > 1 such that the spectrum of Lu is outside the disk of radius β in the complex plane, centered at the origin, while the spectrum of Ls is inside the disk of radius β −1 .

3 General Theory of Topological Entropy This subsection is mostly after (Denker, Grillenberger, and Sigmund 1976). Open covers. Let U = {Uα } be a covering of a compact space X by open sets Uα , where α is in some countable index set, that is ∪α Uα = X. This is called an open cover. Let U = {Uβ } be another such cover. Then one says that U is finer than U, written as U ≥ U, if for each α there is a β for which Uβ ⊂ Uα . One calls a cover U a subcover of U, written as U ⊂ U, if Uβ ∈ U implies that also Uβ ∈ U.

Finally, one defines H(U) = log N (U) . 9. H(U) ≥ H(f −1 (U)) . 10. H(U ∨ U ) ≤ H(U) + H(U ) . One next defines (U)n0 ≡ U ∨ f −1 (U) ∨ · · · ∨ f −n (U) , is just the smallest cardinality of a family of n-tuples of eleand then N (U)n−1 0 ments of U such that for any x ∈ X there exists an n-tuple (Uα0 , . . , Uαn−1 ) so that f k (x) ∈ Uαk for k = 0, . . , n − 1 . 11. 3) exists. H(U, f ) is called the topological entropy of the map f for the open cover U. 12. The expression htop (f ) = sup{H(U, f ) | U an open cover of X} , is called the topological entropy of f .

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