Open Access
2010/2011 Irreducibility, Infinite Level Sets and Small Entropy
Jozef Bobok, Martin Soukenka
Real Anal. Exchange 36(2): 449-462 (2010/2011).


We investigate continuous piecewise affine interval maps with countably many laps that preserve the Lebesgue measure. In particular, we construct such maps having knot points (a point \(x\) where Dini's derivatives satisfy \(D^{+}f(x)=D^{-}f(x)= \infty\) and \(D_{+}f(x)=D_{-}f(x)= -\infty\)) and estimate their topological entropy. Our main result is: for any \(\varepsilon>0\) we construct a continuous interval map \(g=g_{\varepsilon}\) such that (i) \(g\) preserves the Lebesgue measure; (ii) knot points of \(g\) are dense in \([0,1]\) and for a \(G_{\delta}\) dense set of \(z\)'s, the set \(g^{-1}(\{z\})\) is infinite; (iii) \(h_{\text{top}}(g)\le\log 2+\varepsilon\).


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Jozef Bobok. Martin Soukenka. "Irreducibility, Infinite Level Sets and Small Entropy." Real Anal. Exchange 36 (2) 449 - 462, 2010/2011.


Published: 2010/2011
First available in Project Euclid: 11 November 2011

zbMATH: 1271.37022
MathSciNet: MR3016728

Primary: 37B40
Secondary: 26A30

Keywords: interval map , knot point , Lebesgue measure , topological entropy

Rights: Copyright © 2010 Michigan State University Press

Vol.36 • No. 2 • 2010/2011
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