Typical points for one-parameter families of piecewise expanding maps of the interval

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September 2011, 31(3): 877-911. doi: 10.3934/dcds.2011.31.877

Typical points for one-parameter families of piecewise expanding maps of the interval

Daniel Schnellmann ^1,

1.	Ecole Normale Supérieure, Départment de mathématiques et applications (DMA), 45 rue d’Ulm 75230 Paris cedex 05, France

Received March 2010 Revised July 2011 Published August 2011

Abstract
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For one-parameter families of piecewise expanding maps of the interval we establish sufficient conditions such that a given point in the interval is typical for the absolutely continuous invariant measure for a full Lebesgue measure set of parameters. In particular, we consider $C^{1,1}(L)$-versions of $\beta$-transformations, piecewise expanding unimodal maps, and Markov structure preserving one-parameter families. For families of piecewise expanding unimodal maps we show that the turning point is almost surely typical whenever the family is transversal.

Keywords: Typical points, absolutely continuous invariant measure., piecewise expanding maps of the interval.

Mathematics Subject Classification: Primary: 37E05, 37A05; Secondary: 37D2.

Citation: Daniel Schnellmann. Typical points for one-parameter families of piecewise expanding maps of the interval. Discrete and Continuous Dynamical Systems, 2011, 31 (3) : 877-911. doi: 10.3934/dcds.2011.31.877

References:

[1]	V. Baladi, On the susceptibility function of piecewise expanding interval maps, Comm. Math. Phys., 275 (2007), 839-859. doi: 10.1007/s00220-007-0320-5.
[2]	V. Baladi and D. Smania, Linear response formula for piecewise expanding unimodal maps, Nonlinearity, 21 (2008), 677-711. doi: 10.1088/0951-7715/21/4/003.
[3]	V. Baladi and D. Smania, Smooth deformation of piecewise expanding unimodal maps, Discrete Contin. Dyn. Syst., 23 (2009), 685-703. doi: 10.3934/dcds.2009.23.685.
[4]	M. Benedicks and L. Carleson, On iterations of $1-ax^2$ on $(-1,1)$, Ann. of Math. (2), 122 (1985), 1-25. doi: 10.2307/1971367.
[5]	M. Björklund and D. Schnellmann, Almost sure equidistribution in expansive families, Indag. Math. (N.S.), 20 (2009), 167-177.
[6]	K. Brucks and M. Misiurewicz, The trajectory of the turning point is dense for almost all tent maps, Ergodic Theory Dynam. Systems, 16 (1996), 1173-1183. doi: 10.1017/S0143385700009962.
[7]	H. Bruin, For almost every tent-map, the turning point is typical, Fund. Math., 155 (1998), 215-235.
[8]	P. Collet and J.-P. Eckmann, "Iterated Maps on the Interval as Dynamical Systems," Birkhäuser, Boston, 1980.
[9]	B. Faller and C.-E. Pfister, A point is normal for almost all maps $\beta x+\alpha\mod1$ or generalized $\beta$-transformations, Ergodic Theory Dynam. Systems, 29 (2009), 1529-1547. doi: 10.1017/S0143385708000874.
[10]	A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., 186 (1973), 481-488. doi: 10.1090/S0002-9947-1973-0335758-1.
[11]	T.-Y. Li and J. A. Yorke, Ergodic transformations from an interval into itself, Trans. Amer. Math. Soc., 235 (1978), 183-192. doi: 10.1090/S0002-9947-1978-0457679-0.
[12]	P. Mattila, "Geometry of Sets and Measures in Euclidean Spaces (Cambridge studies in advanced mathematics)," Cambridge University Press, Cambridge, 1995.
[13]	M. Misiurewicz and E. Visinescu, Kneading sequences of skew tent maps, Ann. Inst. H. Poincaré Probab. Statist., 27 (1991), 125-140.
[14]	V. A. Rohlin, Exact endomorphisms of a Lebesgue space, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 25 (1961), 499-530, (English) Amer. Math. Soc. Transl. Ser. 2, 39 (1964), 1-36.
[15]	J. Schmeling, Symbolic dynamics for $\beta$-shifts and self-normal numbers, Ergodic Theory Dynam. Systems, 17 (1997), 675-694. doi: 10.1017/S0143385797079182.
[16]	M. Tsujii, Lyapunov exponents in families of one-dimensional dynamical systems, Invent. Math., 111 (1993), 113-137. doi: 10.1007/BF01231282.
[17]	G. Wagner, The ergodic behaviour of piecewise monotonic transformations, Z. Wahrsch. Verw. Gebiete, 46 (1979), 317-324. doi: 10.1007/BF00538119.
[18]	S. Wong, Some metric properties of piecewise monotonic mappings of the unit interval, Trans. Amer. Math. Soc., 246 (1978), 493-500. doi: 10.1090/S0002-9947-1978-0515555-9.

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References:

[1]	V. Baladi, On the susceptibility function of piecewise expanding interval maps, Comm. Math. Phys., 275 (2007), 839-859. doi: 10.1007/s00220-007-0320-5.
[2]	V. Baladi and D. Smania, Linear response formula for piecewise expanding unimodal maps, Nonlinearity, 21 (2008), 677-711. doi: 10.1088/0951-7715/21/4/003.
[3]	V. Baladi and D. Smania, Smooth deformation of piecewise expanding unimodal maps, Discrete Contin. Dyn. Syst., 23 (2009), 685-703. doi: 10.3934/dcds.2009.23.685.
[4]	M. Benedicks and L. Carleson, On iterations of $1-ax^2$ on $(-1,1)$, Ann. of Math. (2), 122 (1985), 1-25. doi: 10.2307/1971367.
[5]	M. Björklund and D. Schnellmann, Almost sure equidistribution in expansive families, Indag. Math. (N.S.), 20 (2009), 167-177.
[6]	K. Brucks and M. Misiurewicz, The trajectory of the turning point is dense for almost all tent maps, Ergodic Theory Dynam. Systems, 16 (1996), 1173-1183. doi: 10.1017/S0143385700009962.
[7]	H. Bruin, For almost every tent-map, the turning point is typical, Fund. Math., 155 (1998), 215-235.
[8]	P. Collet and J.-P. Eckmann, "Iterated Maps on the Interval as Dynamical Systems," Birkhäuser, Boston, 1980.
[9]	B. Faller and C.-E. Pfister, A point is normal for almost all maps $\beta x+\alpha\mod1$ or generalized $\beta$-transformations, Ergodic Theory Dynam. Systems, 29 (2009), 1529-1547. doi: 10.1017/S0143385708000874.
[10]	A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., 186 (1973), 481-488. doi: 10.1090/S0002-9947-1973-0335758-1.
[11]	T.-Y. Li and J. A. Yorke, Ergodic transformations from an interval into itself, Trans. Amer. Math. Soc., 235 (1978), 183-192. doi: 10.1090/S0002-9947-1978-0457679-0.
[12]	P. Mattila, "Geometry of Sets and Measures in Euclidean Spaces (Cambridge studies in advanced mathematics)," Cambridge University Press, Cambridge, 1995.
[13]	M. Misiurewicz and E. Visinescu, Kneading sequences of skew tent maps, Ann. Inst. H. Poincaré Probab. Statist., 27 (1991), 125-140.
[14]	V. A. Rohlin, Exact endomorphisms of a Lebesgue space, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 25 (1961), 499-530, (English) Amer. Math. Soc. Transl. Ser. 2, 39 (1964), 1-36.
[15]	J. Schmeling, Symbolic dynamics for $\beta$-shifts and self-normal numbers, Ergodic Theory Dynam. Systems, 17 (1997), 675-694. doi: 10.1017/S0143385797079182.
[16]	M. Tsujii, Lyapunov exponents in families of one-dimensional dynamical systems, Invent. Math., 111 (1993), 113-137. doi: 10.1007/BF01231282.
[17]	G. Wagner, The ergodic behaviour of piecewise monotonic transformations, Z. Wahrsch. Verw. Gebiete, 46 (1979), 317-324. doi: 10.1007/BF00538119.
[18]	S. Wong, Some metric properties of piecewise monotonic mappings of the unit interval, Trans. Amer. Math. Soc., 246 (1978), 493-500. doi: 10.1090/S0002-9947-1978-0515555-9.

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