By Luis Barreira

The thought of dynamical platforms is a vast and lively learn topic with connections to such a lot components of arithmetic. Dynamical platforms: An Introduction undertakes the tricky activity to supply a self-contained and compact advent.

Topics coated contain topological, low-dimensional, hyperbolic and symbolic dynamics, in addition to a quick advent to ergodic idea. particularly, the authors examine topological recurrence, topological entropy, homeomorphisms and diffeomorphisms of the circle, Sharkovski's ordering, the Poincaré-Bendixson conception, and the development of good manifolds, in addition to an creation to geodesic flows and the learn of hyperbolicity (the latter is usually absent in a primary introduction). additionally, the authors introduce the fundamentals of symbolic dynamics, the development of symbolic codings, invariant measures, Poincaré's recurrence theorem and Birkhoff's ergodic theorem.

The exposition is mathematically rigorous, concise and direct: all statements (except for a few effects from different components) are confirmed. even as, the textual content illustrates the speculation with many examples and one hundred forty workouts of variable degrees of hassle. the single necessities are a heritage in linear algebra, research and user-friendly topology.

This is a textbook basically designed for a one-semester or two-semesters path on the complex undergraduate or starting graduate degrees. it will possibly even be used for self-study and as a kick off point for extra complex topics.

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Extra info for Dynamical Systems: An Introduction

Sample text

Since each preimage Rα−n U is an open interval of length 2ε < 1/2 and the orbit of x is dense, there exists a sequence nk ∞ in N such that Rα−nk (x) → x + 1/2 when k → ∞. Thus, Rα−nk U ∩ U = ∅ for any sufficiently large k. This shows that the rotation Rα is not topologically mixing. 9 Now we consider the expanding map E2 : S 1 → S 1 . 4, there exists a point x ∈ S 1 whose positive semiorbit γ + (x) is dense in S 1 . 1 that the map E2 is topologically transitive. Now we show that E2 is also topologically mixing.

2 to the map g = f −1 . 2 Continuous Time Now we introduce the notions of α-limit set and ω-limit set for a dynamical system with continuous time. 4 Given a semiflow Φ = (ϕt )t≥0 of X, the ω-limit set of a point x ∈ X is defined by ω(x) = ωΦ (x) = ϕs (x) : s > t . t>0 Moreover, given a flow Φ = (ϕt )t∈R of X, the α-limit set of a point x ∈ X is defined by α(x) = αΦ (x) = ϕs (x) : s < t . 6 Consider the differential equation in polar coordinates r = r(r − 1)(r − 2), θ = 1. 7) We note that r > 0 for r ∈ (0, 1) ∪ (2, +∞) and that r < 0 for r ∈ (1, 2).

Therefore, E2−n U ∩ V ⊃ E2−n U ∩ I = ∅. This shows that the map E2 is topologically mixing. 10 Let TA : T2 → T2 be an automorphism of the torus T2 with |tr A| > 2. 12, we have det A = ±1. We also have det(A − λId) = λ2 − tr Aλ + det A, and since |tr A| > 2, the eigenvalues of the matrix A are the real numbers λ1 = tr A + (tr A)2 − 4 det A 2 and λ2 = tr A − (tr A)2 − 4 det A . 2 In particular, there exists a λ > 1 such that {|λ1 |, |λ2 |} = {λ, λ−1 }. Now we show that λ1 and λ2 are irrational. Clearly, λ1 and λ2 are rational if and only if m2 ± 4 = l 2 for some integer l ∈ N, where m = tr A.