Elements of Differentiable Dynamics and Bifurcation Theory by David Ruelle

By David Ruelle

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For instance, close to the stable and unstable manifolds of a hyperbolic fixed point a for / , the graph transform method will produce a contracting and an expanding manifold for a perturbation / , and their intersection will be the new fixed point a. Similarly, a center (un)stable manifold will survive perturbation, but a strong stable or unstable manifold for a nonhyperbolic fixed point may not persist (the successive graph transforms walking away to infinity). The graph transform method extends to the situation where the fixed point a is replaced by orbits (nonperiodic in general) contained in a compact set K.

More precisely, for a,b e M, we write a y b if, for arbitrarily small ε > 0, there is an ε-pseudoorbit going from a to b. , it is reflexive (a y a) and transitive (a y b and bye imply a y c). , ifxyy and x —► a, y —► b then a y b. The chain-recurrent set is closed. The proof of these simple properties is left to the reader. 4. Basic Classes and Attractors Write a ~ b if a y b and b y a. Since >- is a preorder, ~ is an equivalence relation. We denote by [a] the equivalence class of a and write [a] ^ [b] when a y b.

1 are all equivalent. 50 Elements of Differentiable Dynamics and Bifurcation Theory 4. (Stable and unstable sets of general points). a dynamical system on M. For any a G M, we write W~ = {xeM : lim dist(/*a;, fa) t—>·οο Let (/*) be = 0}, and if / is bijective, W£ = {xeM : lim dist(/"*x, /"*a) = 0}. t—>oo We call >V~ the stable set of a, and W+ the unstable set of a. Show that, if a is chain recurrent, W~ y a y W+. Thus, if a, b are chain recurrent and W+ Π W6~ ^ 0, then a y b. If a - b, then W+ Π W6~ C [a].

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