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

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, 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 $1/2$.
Keywords: topological entropy., Interval map, knot point, topological conjugacy.
Mathematics Subject Classification: Primary: 37E05, 37B4.
Citation: Jozef Bobok, Martin Soukenka. On piecewise affine interval maps with countably many laps. Discrete & Continuous Dynamical Systems, 2011, 31 (3) : 753-762. doi: 10.3934/dcds.2011.31.753
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), 207-213.  Google Scholar

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M. Misiurewicz and P. Raith, Strict inequalities for the entropy of transitive piecewise monotone maps, Discrete and Continuous Dynamical Systems, 13 (2005), 451-468. doi: 10.3934/dcds.2005.13.451.  Google Scholar

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J. Milnor and W. Thurston, On iterated maps of the interval, in "Dynamical Systems" (College Park, MD, 1986-1987), Lecture Notes in Math., 1342, Springer, Berlin, (1988), 465-563.  Google Scholar

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W. Parry, Symbolic dynamics and transformations of the unit interval, Trans. Amer. Math. Soc., 122 (1966), 368-378. doi: 10.1090/S0002-9947-1966-0197683-5.  Google Scholar

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P. Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

show all references

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), 207-213.  Google Scholar

[3]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.  Google Scholar

[4]

M. Misiurewicz, Horseshoes for mappings of an interval, Bull. Acad. Pol. Sci., Sér. Sci. Math., 27 (1979), 167-169. 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), 451-468. doi: 10.3934/dcds.2005.13.451.  Google Scholar

[6]

J. Milnor and W. Thurston, On iterated maps of the interval, in "Dynamical Systems" (College Park, MD, 1986-1987), Lecture Notes in Math., 1342, Springer, Berlin, (1988), 465-563.  Google Scholar

[7]

W. Parry, Symbolic dynamics and transformations of the unit interval, Trans. Amer. Math. Soc., 122 (1966), 368-378. doi: 10.1090/S0002-9947-1966-0197683-5.  Google Scholar

[8]

P. Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

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