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On the Lyapunov numbers

Sergiĭ KolyadaOleksandr Rybak — 2013

Colloquium Mathematicae

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.

Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval

Sergiĭ KolyadaMichał MisiurewiczL’ubomír Snoha — 1999

Fundamenta Mathematicae

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...

Minimal nonhomogeneous continua

Henk BruinSergiǐ KolyadaL'ubomír Snoha — 2003

Colloquium Mathematicae

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.

Noninvertible minimal maps

Sergiĭ KolyadaL'ubomír SnohaSergeĭ Trofimchuk — 2001

Fundamenta Mathematicae

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...

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