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Minor cycles for interval maps

Michał Misiurewicz — 1994

Fundamenta Mathematicae

For continuous maps of an interval into itself we consider cycles (periodic orbits) that are non-reducible in the sense that there is no non-trivial partition into blocks of consecutive points permuted by the map. Among them we identify the miror ones. They are those whose existence does not imply existence of other non-reducible cycles of the same period. Moreover, we find minor patterns of a given period with minimal entropy.

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...

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