RENDICONTI DEL SEMINARIO MATEMATICO

94 H. Koçak - K. Palmer - B. Coomesδ pseudo orbit {y k } +∞k=−∞ of Eq. (1) lying in S is ε-shadowed by a true orbit {x k} +∞Moreover, there is only one such orbit satisfyingk=−∞ .f (y k ) ∗ (x k − y k ) = 0f or k ∈ ZZ.In preparation for our second infinite shadowing result, we next introduce variousmathematical entities. Take a fixed pseudo orbit {y k } +∞k=−∞with associated times{h k } +∞k=−∞ . Let Y k be the subspace of IR n consisting of the vectors orthogonal to f (y k ).Then let Y be the Banach space of bounded sequences v = {v k } k∈ ZZ with v k ∈ Y k , andequip Y with the norm‖v‖ = sup ‖v k ‖.k∈ ZZAlso, let Ỹ be a similar Banach space except that v k ∈ Y k+1 . Then letbe the linear operator defined byL y : Y → Ỹ(L y v) k = v k+1 − P k+1 Dφ h k(y k )v k ,where P k : IR n → IR n is the orthogonal projection defined byP k v = v − f (y k) ∗ v‖ f (y k )‖ 2 f (y k).So L y is a linear operator associated with the derivative of the flow along the pseudoorbit, but restricted to the subspaces orthogonal to the vector field. This operator playsa key role in what follows. We assume that the operator is invertible with a boundedinverse:‖L −1y ‖ ≤ K.Next we define various constants. We begin withandM 0 = sup ‖ f (x)‖,x∈UM 1 = sup ‖D f (x)‖,x∈Uh min = infk∈ ZZ h k,h max = sup h k .k∈ ZZM 2 = sup ‖D 2 f (x)‖,x∈UNext, we choose a positive number ε 0 ≤ h min such that for all k and ‖x − y k ‖ ≤ ε 0 thesolution ϕ t (x) is defined and remains in U for 0 ≤ t ≤ h k + ε 0 . Continuing, we define = infk∈ ZZ ‖ f (y k)‖,¯M 0 = sup ‖ f (y k )‖,k∈ ZZ¯M 1 = sup ‖D f (y k )‖.k∈ ZZNow, we define the following constants in terms of the ones already given:{}C = max K, −1 (e M 1h maxK + 1) ,C = (1 − M 1 δC) −1 C,