On piecewise affine interval maps with countably many laps

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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

Abstract
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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 and 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).
[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.
[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.
[4]	M. Misiurewicz, Horseshoes for mappings of an interval, Bull. Acad. Pol. Sci., Sér. Sci. Math., 27 (1979), 167-169.
[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.
[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.
[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.
[8]	P. Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.

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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).
[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.
[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.
[4]	M. Misiurewicz, Horseshoes for mappings of an interval, Bull. Acad. Pol. Sci., Sér. Sci. Math., 27 (1979), 167-169.
[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.
[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.
[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.
[8]	P. Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.

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