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Introduction to Dynamical Systems Introduction to Genericity

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Introduction to Dynamical Systems Introduction to Genericity Students' Course: Introduction to Dynamical systems introduction to Genericity September 19, 2007 1 Genericity Generic property is a way to formulate typical property, one that characterize most functions in some function space. But before we can handle this formal de¯nition we need to discus few terms. De¯nition 1.1 (Topology) Let a set ­ and a collection T of subsets of ­ satisfying the following axioms: 1. ; 2 T , ­ 2 T . S 2. 8®; A® 2 T ! ® A® 2 T . (arbitrary union) 3. A; B 2 ­ ! A \ B 2 T . (¯nite intersection) The collection T is called a topology on ­, the elements of X are called points. Under this de¯nition, the sets in T are the open sets, and their complements in X are the closed sets. A topology induced by a metric, jj ¢ jj, is such, that every open set is a union of open balls de¯ned using the metric: Br(x) = fy 2 ­ : jjx ¡ yjj < rg (1.1) Hence, any metric space has a topology. Back to dynamical systems, vector ¯elds can be described as functions in the space Cr(M; N) (i.e., map f : M ! N, where f and its ¯rst r derivatives are continuous.). Two elements of Cr(M; N) are Ck "-close (k · r) if the functions and its k derivatives are "-close in some norm. Notice that if the manifolds N; M are Rn, this de¯nition is not well de¯ned as the domain is unbounded, and the behavior at in¯nity should be considered, which explain why most of the time we see a discussion on compact phase space. 1 De¯nition 1.2 (Cr-Topology) The topology on the Cr function space with a metric that takes into account the function and its ¯rst r derivatives is called Cr-topology. So in fact, we are looking at the function space as a metric space, however, we refer it as a topology since we are not interested in properties emerging from the metric itself, but rather in the topology induced by this metric. De¯nition 1.3 (Dense Set) A subset, E ½ F ½ ­ is called dense in F if it closure, E¹, satisfy E¹ \ F = F . When E is referred as dense, F = ­. The space Cr(M; R) is Baire space ([1]), hence any residual set is dense. This property of the function space is what makes generic property "typical". However, dense alone is not su±cient as we know in the reals, rationals are dense but also there compliment, the irrationals, is dense. Therefore, dense it not typical, as only one of them can be typical (and we know that there are much more irrationals than rationals.). So we turn to additional de¯nition. De¯nition 1.4 (Residual Set) T Let R ½ ­, it is called residual if there is a countable collection fGngn2N s.t., R = n2N Gn, where Gn are open and dense in ­. (It complement is called meager). De¯nition 1.5 (Baire Space) If ­ has a dense residual subset ­ is called Baire space. De¯nition 1.6 (Generic) A property is generic in Cr if the set holding this property contain a residual subset in the Cr-topology. Proposition 1.7 Cr(M; Rn) is a Bair space. Proof: First lets show that Cr(M; Rn) countable collection of open and dense subsets. Theorem 1.8 hyperbolic ¯xed points are generic. Applied meaning Why we should be interested in generic properties ? in applied math- ematics, especially in modelling, we build a mathematical model, but we have uncertainty regarding to the values of its parameters. Thus, we usually estimate the parameters from some real data, but estimates are only that, so if we know that a property we found in the analysis of the model is generic, we know that if we have some error in the parameter estimation the qualitative property will still hold. One last note, generic property can be eliminated when we have special symmetry. References [1] J. Palis and W. de Melo. Geometric Theory of Dynamics Systems. Springer-Verlag, 1980. 2.
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