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1
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Noninvertible minimal maps

100%
Kolyada S., Snoha L. u., Trofimchuk S.
Fundamenta Mathematicae
|
2001
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tom 168
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nr 2
141-163
EN
For a discrete dynamical system given by a compact Hausdorff space X and a continuous selfmap f of X the connection between minimality, invertibility and openness of f is investigated. It is shown that any minimal map is feebly open, i.e., sends open sets to sets with nonempty interiors (and if it is open then it is a homeomorphism). Further, it is shown that if f is minimal and A ⊆ X then both f(A) and $f^{-1}(A)$ share with A those topological properties which describe how large a set is. Using these results it is proved that any minimal map in a compact metric space is almost one-to-one and, moreover, when restricted to a suitable invariant residual set it becomes a minimal homeomorphism. Finally, two kinds of examples of noninvertible minimal maps on the torus are given-these are obtained either as a factor or as an extension of an appropriate minimal homeomorphism of the torus.
2
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Topological size of scrambled sets

100%
Blanchard F., Huang W., Snoha L. u.
Colloquium Mathematicum
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2008
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tom 110
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nr 2
293-361
EN
A subset S of a topological dynamical system (X,f) containing at least two points is called a scrambled set if for any x,y ∈ S with x ≠ y one has $lim inf_{n→∞} d(fⁿ(x),fⁿ(y)) = 0$ and $lim sup_{n→∞} d(fⁿ(x),fⁿ(y)) > 0$, d being the metric on X. The system (X,f) is called Li-Yorke chaotic if it has an uncountable scrambled set. These notions were developed in the context of interval maps, in which the existence of a two-point scrambled set implies Li-Yorke chaos and many other chaotic properties. In the present paper we address several questions about scrambled sets in the context of topological dynamics. There the assumption of Li-Yorke chaos, and also stronger ones like the existence of a residual scrambled set, or the fact that X itself is a scrambled set (in these cases the system is called residually scrambled or completely scrambled respectively), are not so highly significant. But they still provide valuable information. First, the following question arises naturally: is it true in general that a Li-Yorke chaotic system has a Cantor scrambled set, at least when the phase space is compact? This question is not answered completely but the answer is known to be yes when the system is weakly mixing or Devaney chaotic or has positive entropy, all properties implying Li-Yorke chaos; we show that the same is true for symbolic systems and systems without asymptotic pairs, which may not be Li-Yorke chaotic. More generally, there are severe restrictions on Li-Yorke chaotic dynamical systems without a Cantor scrambled set, if they exist. A second set of questions concerns the size of scrambled sets inside the space X itself. For which dynamical systems (X,f) do there exist first category, or second category, or residual scrambled sets, or a scrambled set which is equal to the whole space X? While reviewing existing results, we give examples of systems on arcwise connected continua in the plane having maximal scrambled sets with any prescribed cardinalities, in particular systems having at most finite or countable scrambled sets. We also give examples of Li-Yorke chaotic systems with at most first category scrambled sets. It is proved that minimal compact systems, graph maps and a large class of symbolic systems containing subshifts of finite type are never residually scrambled; assuming the Continuum Hypothesis, weakly mixing systems are shown to have second category scrambled sets. Various examples of residually scrambled systems are constructed. It is shown that for any minimal distal system there exists a non-disjoint completely scrambled system. Finally, various other questions are solved. For instance, a completely scrambled system may have a factor without any scrambled set, and a triangular map may have a scrambled set with non-empty interior.
3
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All solenoids of piecewise smooth maps are period doubling

100%
Alsedà L., López V. J., L'ubomír Snoha
Fundamenta Mathematicae
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1998
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tom 157
|
nr 2-3
121-138
EN
We show that piecewise smooth maps with a finite number of pieces of monotonicity and nowhere vanishing Lipschitz continuous derivative can have only period doubling solenoids. The proof is based on the fact that if $p_1 < ... < p_n$ is a periodic orbit of a continuous map f then there is a union set ${q_1,..., q_{n-1}}$ of some periodic orbits of f such that $p_i < q_i < p_{i+1}$ for any i.
4
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Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval

100%
Kolyada S., Misiurewicz M., L’ubomír Snoha
Fundamenta Mathematicae
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1999
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tom 160
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nr 2
161-181
EN
The topological entropy of a nonautonomous dynamical system given by a sequence of compact metric spaces $(X_i)^∞_{i = 1}$ and a sequence of continuous maps $(f_i)^∞_{i = 1}$, $f_i : X_i → X_{i+1}$, is defined. If all the spaces are compact real intervals and all the maps are piecewise monotone then, under some additional assumptions, a formula for the entropy of the system is obtained in terms of the number of pieces of monotonicity of $f_n ○... ○ f_2 ○ f_1$. As an application we construct a large class of smooth triangular maps of the square of type $2^∞$ and positive topological entropy.
5
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Minimal nonhomogeneous continua

100%
Bruin H., Kolyada S., Snoha L. u.
Colloquium Mathematicum
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2003
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tom 95
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nr 1
123-132
EN
We show that there are (1) nonhomogeneous metric continua that admit minimal noninvertible maps but have the fixed point property for homeomorphisms, and (2) nonhomogeneous metric continua that admit both minimal noninvertible maps and minimal homeomorphisms. The former continua are constructed as quotient spaces of the torus or as subsets of the torus, the latter are constructed as subsets of the torus.
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