On piecewise affine interval maps with countably many laps

Jozef Bobok; Martin Soukenka

doi:10.3934/dcds.2011.31.753

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2011, Volume 31, Issue 3: 753-762. Doi: 10.3934/dcds.2011.31.753

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On piecewise affine interval maps with countably many laps

Jozef Bobok^1, and
Martin Soukenka^1,

1.
KM FSv ČVUT, Thákurova 7, 166 29 Praha 6

Received: March 2010

Revised: June 2011

Published: September 2011

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Abstract

We study a special conjugacy class $\mathcal F$ of continuous piecewise monotone interval maps: with countably many laps, which are locally eventually onto and have common topological entropy $\log9$. We show that $\mathcal F$ contains a piecewise affine map $f_{\lambda}$ with a constant slope $\lambda$ if and only if $\lambda\ge 9$. Our result specifies the known fact that for piecewise affine interval leo maps with countably many pieces of monotonicity and a constant slope $\pm\lambda$, the topological (measure-theoretical) entropy is not determined by $\lambda$. We also consider maps from the class $\mathcal F$ preserving the Lebesgue measure. We show that some of them have a knot point (a point $x$ where Dini's derivatives satisfy $D^{+}f(x)=D^{-}f(x)= \infty$ and $D_{+}f(x)=D_{-}f(x)= -\infty$) in its fixed point $1/2$.

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Mathematics Subject Classification: Primary: 37E05, 37B40.

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References

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