# American Institute of Mathematical Sciences

September  2011, 31(3): 753-762. doi: 10.3934/dcds.2011.31.753

## On piecewise affine interval maps with countably many laps

 1 KM FSv ČVUT, Thákurova 7, 166 29 Praha 6, Czech Republic, Czech Republic

Received  March 2010 Revised  June 2011 Published  August 2011

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 $1/2$.
Citation: Jozef Bobok, Martin Soukenka. On piecewise affine interval maps with countably many laps. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 753-762. doi: 10.3934/dcds.2011.31.753
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##### References:
 [1] J. Bobok and M. Soukenka, Irreducibility, infinite level sets and small entropy,, to appear in Real Analysis Exchange, 36 (2011).   Google Scholar [2] E. M. Coven and M. C. Hidalgo, On the topological entropy of transitive maps of the interval,, Bull. Aust. Math. Soc., 44 (1991), 207.   Google Scholar [3] A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", Encyclopedia of Mathematics and its Applications, 54 (1995).   Google Scholar [4] M. Misiurewicz, Horseshoes for mappings of an interval,, Bull. Acad. Pol. Sci., 27 (1979), 167.   Google Scholar [5] M. Misiurewicz and P. Raith, Strict inequalities for the entropy of transitive piecewise monotone maps,, Discrete and Continuous Dynamical Systems, 13 (2005), 451.  doi: 10.3934/dcds.2005.13.451.  Google Scholar [6] J. Milnor and W. Thurston, On iterated maps of the interval,, in, 1342 (1988), 1986.   Google Scholar [7] W. Parry, Symbolic dynamics and transformations of the unit interval,, Trans. Amer. Math. Soc., 122 (1966), 368.  doi: 10.1090/S0002-9947-1966-0197683-5.  Google Scholar [8] P. Walters, "An Introduction to Ergodic Theory,", Graduate Texts in Mathematics, 79 (1982).   Google Scholar
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