# Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval

Fundamenta Mathematicae (1999)

• Volume: 160, Issue: 2, page 161-181
• ISSN: 0016-2736

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## Abstract

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The topological entropy of a nonautonomous dynamical system given by a sequence of compact metric spaces ${\left({X}_{i}\right)}_{i=1}^{\infty }$ and a sequence of continuous maps ${\left({f}_{i}\right)}_{i=1}^{\infty }$, ${f}_{i}:{X}_{i}\to {X}_{i+1}$, 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 ${2}^{\infty }$ and positive topological entropy.

## How to cite

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Kolyada, Sergiĭ, Misiurewicz, Michał, and Snoha, L’ubomír. "Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval." Fundamenta Mathematicae 160.2 (1999): 161-181. <http://eudml.org/doc/212386>.

abstract = {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.},
author = {Kolyada, Sergiĭ, Misiurewicz, Michał, Snoha, L’ubomír},
journal = {Fundamenta Mathematicae},
keywords = {nonautonomous dynamical system; topological entropy; triangular maps; piecewise monotone maps; $C^∞$ maps; nonautonomous systems; discrete dynamical system; piecewise monotone dynamical systems},
language = {eng},
number = {2},
pages = {161-181},
title = {Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval},
url = {http://eudml.org/doc/212386},
volume = {160},
year = {1999},
}

TY - JOUR
AU - Misiurewicz, Michał
AU - Snoha, L’ubomír
TI - Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval
JO - Fundamenta Mathematicae
PY - 1999
VL - 160
IS - 2
SP - 161
EP - 181
AB - 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.
LA - eng
KW - nonautonomous dynamical system; topological entropy; triangular maps; piecewise monotone maps; $C^∞$ maps; nonautonomous systems; discrete dynamical system; piecewise monotone dynamical systems
UR - http://eudml.org/doc/212386
ER -

## References

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