 By N. Biggs

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Consider the cells M 2 (i) ∈ M2+ and M 1 (j) so that M 2 (i) ⊆ M 1 (j) ∈ C1 . It is suﬃcient to proof that M 1 (j) is non-leaving. It follows from the deﬁnition of M2+ that there exists an inﬁnite path i1 , i2 , . . on G2 with the initial vertex i1 = i. An inﬁnite path j1 , j2 , . . on G1 could be obtained as follows. Let jp be a vertex which corresponds to the cell M 1 (jp ) ∈ C1 so that M 1 (jp ) ⊇ M 2 (ip ), whereby j1 = j and M 2 (ip ) ∈ C2 . The constructed path is admissible on G1 . So the cell M 1 (j) is non-leaving.

F (x0 )|| be small enough. If detF (x0 ) = 0 then there exists the inverse matrix (F (x0 ))−1 . So we can compute K = (F (x0 ))−1 and R = (F (x0 ))−1 F (x0 ) . e. x0 is a good approximation to the solution. The Lipschitz constant L can be estimated by means of the second derivative of F . 1) holds. Let f be a diﬀeomorphism deﬁned on a manifold M and {x1 , x2 , . . , xp } be a p-periodic ε-trajectory of f . As M is a manifold, there are neighbourhoods V (xi ) ≡ Vi which we identify with balls of radii ai .

Algorithm Step 1. An initial covering C of a compact K is determined. For C a symbolic image G of a dynamical system is constructed. Step 2. All non-leaving vertices of G are detected. The neighborhood U = { M (i) : i is a non−leaving vertex} of the positive invariant set is obtained. Step 3. Cells corresponding to non-leaving vertices are subdivided, while cells corresponding to leaving vertices are excluded. Step 4. For the collection of cells obtained the new symbolic image is constructed. Step 5.

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