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Periodic orbits of piecewise monotone maps

Indiana University-Purdue University Indianapolis (IUPUI) / Much is known about periodic orbits in dynamical systems of
continuous interval maps. Of note is the theorem of Sharkovsky.
In 1964 he proved that, for a continuous map $f$ on $\mathbb{R}$,
the existence of periodic orbits of certain periods force the
existence of periodic orbits of certain other periods. Unfortunately
there is currently no analogue of this theorem for maps of $\mathbb{R}$
which are not continuous. Here we consider discontinuous interval maps
of a particular variety, namely piecewise monotone interval maps.
We observe how the presence of a given periodic orbit forces
other periodic orbits, as well as the direct
analogue of Sharkovsky's theorem in special families of
piecewise monotone maps. We conclude by investigating the entropy of
piecewise linear maps. Among particular one parameter families of
piecewise linear maps, entropy remains constant even as the parameter varies.
We provide a simple geometric explanation of this phenomenon known as
entropy locking.
Date23 April 2018
CreatorsCosper, David
ContributorsMisiurewicz, Michal
Source SetsIndiana University-Purdue University Indianapolis
Detected LanguageEnglish

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