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We establish necessary and sufficient conditions for a dynamical system to be topologically conjugate to the Morse minimal set, the shift orbit closure of the Morse sequence. Conditions for topological conjugacy to the closely related Toeplitz minimal set are also derived.
A combinatorial model for a property of continuous self-maps of a compact interval is a self-map \pi of a finite ordered set such that every continuous \pi-weakly monotone self-map of a compact interval has that property. We identify the minimal combinatorial models for the property ‘the set of periods is a given set’. Here the word minimal refers to the number of points in the domain of the model. We also identify the minimal permutation models and, in appropriate cases, the minimal combinatorial models for properties involving ‘horseshoes’.
We prove a multidimensional version of the theorem that every shift of finite type has a power that can be realized as the same power of a tiling system. We also show that the set of entropies of tiling systems equals the set of entropies of shifts of finite type.
The topological entropy of a continuous map of the interval is the supremum of the topological entropies of the piecewise linear maps associated to its finite invariant sets. We show that for transitive maps, this supremum is attained at some finite invariant set if and only if the map is piecewise monotone and the set contains the endpoints of the interval and the turning points of the map.
Let f denote a continuous map of a compact interval to itself, P(f) the set of periodic points of f and Λ(f) the set of ω-limit points of f. Sarkovskǐi has shown that Λ(f) is closed, and hence ⊆Λ(f), and Nitecki has shown that if f is piecewise monotone, then Λ(f)=. We prove that if x∈Λ(f)−, then the set of ω-limit points of x is an infinite minimal set. This result provides the inspiration for the construction of a map f for which Λ(f)≠.
We study the dynamics of continuous maps of the circle with periodic points. We show that the centre is the closure of the periodic points and that the depth of the centre is at most two. We also characterize the property that every power is transitive in terms of transitivity of a single power and some periodic data.
For continuous maps ƒ of the circle to itself, we show: (A) the set of nonwandering points of ƒ coincides with that of ƒn for every odd n; (B) ƒ has a horseshoe if and only if it has a non-wandering homoclinic point; (C) if the set of periodic points is closed and non-empty, then every non-wandering point is periodic.
For each n≥2, we find the minimum value of the topological entropies of all continuous self-maps of the circle having a fixed point and a point of least period n, and we exhibit a map with this minimal entropy.
We show that, for maps of the interval, the non-wandering set of the map coincides with the non-wandering set of each of its odd powers, while the nonwandering set of any of its even powers can be strictly smaller.
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