Wyniki wyszukiwania

1

On the Lyapunov numbers

100%

Kolyada S., Rybak O.

Colloquium Mathematicum

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2013

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tom 131

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nr 2

209-218

EN

We introduce and study the Lyapunov numbers-quantitative measures of the sensitivity of a dynamical system (X,f) given by a compact metric space X and a continuous map f: X → X. In particular, we prove that for a minimal topologically weakly mixing system all Lyapunov numbers are the same.

2

Noninvertible minimal maps

81%

Kolyada S., Snoha L. u., Trofimchuk S.

Fundamenta Mathematicae

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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.

3

Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval

81%

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.

Ograniczanie wyników

2 Fundamenta Mathematicae

1 Colloquium Mathematicum

3 Kolyada S.

1 L’ubomír Snoha

1 Misiurewicz M.

1 Rybak O.

1 Snoha L. u.

1 Trofimchuk S.

1 2013

1 2001

1 1999

On the Lyapunov numbers

Noninvertible minimal maps

Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval