1,720,966 research outputs found
A function for which the ω-limit points are not contained in the closure of periodic points,
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Attractors for iterated function schemes on [0,1]^N are exceptional
No full textLet X = [0,1]^N with S = {S_1,...,S_M} a finite set of contraction maps from X to itself. A nonempty compact subset E of X is an attractor for the iterated function scheme S if E = cup_{i=1}^M S_i(E) = S(E). We show that the typical closed set in [0, 1]^N is not an attractor of any iterated function scheme
Prevalence and structure of adding machines for cellular automata
No full textAbstractConsider the collection of left permutive cellular automata Φ with no memory, defined on the space S of all doubly infinite sequences from a finite alphabet. There exists Z˜, a dense subset of S, such that Φ:cl{Φn(x):n⩾0}→cl{Φn(x):n⩾0} is topologically conjugate to an odometer for all x∈Z˜ so long as Φm is not the identity map for any m. Moreover, Φ generates the same odometer for all x∈Z˜. The set Z˜ is a dense Gδ subset with full measure of a particular subspace of S
Asymptotically stable sets and the stability of ω-limit sets
No full textAbstractLet C be the collection of continuous self-maps of the unit interval I=[0,1] to itself. For f∈C and x∈I, let ω(x,f) be the ω-limit set of f generated by x, and following Block and Coppel, we take Q(x,f) to be the intersection of all the asymptotically stable sets of f containing ω(x,f). We show that Q(x,f) tells us quite a bit about the stability of ω(x,f) subject to perturbations of either x or f, or both. For example, a chain recurrent point y is contained in Q(x,f) if and only if there are arbitrarily small perturbations of f to a new function g that give us y as a point of ω(x,g). We also study the structure of the map Q taking (x,f)∈I×C to Q(x,f). We prove that Q is upper semicontinuous and a Baire 1 function, hence continuous on a residual subset of I×C. We also consider the map Qf:I→K given by x↦Q(x,f), and find that this map is continuous if and only if it is a constant map; that is, only when the set Q(f)={Q(x,f):x∈I} is a singleton
Stability in the family of ω-limit sets of alternating systems
No full textAbstractLet f and g be elements of C(I) with x∈I=[0,1]. We study the ω-limit sets ω(x,[f,g]) generated by alternating trajectories of the form γ(x,[f,g])={x,f(x),g(f(x)),f(g(f(x))),…}, as well as the sets Λ([f,g])=⋃x∈Iω(x,[f,g]) and L([f,g])={ω(x,[f,g]):x∈I}. In particular, we show that(1)If g is constant on no interval J⊆I, then there exists a residual set S⊆C(I) so that the maps Λ:C(I)×C(I)→K and L:C(I)×C(I)→K⋆ taking (f,g) to Λ([f,g]) and L([f,g]), respectively, are both continuous at (f,g) whenever f∈S.(2)The map ω:I×C(I)×C(I)→K taking (x,f,g) to ω(x,[f,g]) is in the second class of Baire, and for any g∈C(I) there exists a residual set T⊆I×C(I) so that ω is continuous at (x,f,g) whenever (x,f)∈T.(3)If f is constant on no interval J⊆I, then there exists a residual set D⊆I×C(I) so that ω(x,[f,g])=ω(x,g∘f)∪ω(f(x),f∘g), where both ω(x,g∘f) and ω(f(x),f∘g) are adding machines of type ∞, whenever (x,g)∈D
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