Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval

Sergiĭ Kolyada; Michał Misiurewicz; L’ubomír Snoha

Fundamenta Mathematicae (1999)

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

Abstract

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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.

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>.

@article{Kolyada1999,
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 - Kolyada, Sergiĭ
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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  12. [KS] S. Kolyada and L'. Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput. Dynam. 4 (1996), 205-233. Zbl0909.54012
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  14. [M] M. Misiurewicz, Attracting set of positive measure for a C ∞ map of an interval, Ergodic Theory Dynam. Systems 2 (1982), 405-415. Zbl0522.58032
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