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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

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 & 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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M. Björklund and D. Schnellmann, Almost sure equidistribution in expansive families, Indag. Math. (N.S.), 20 (2009), 167-177.  Google Scholar

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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.  Google Scholar

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H. Bruin, For almost every tent-map, the turning point is typical, Fund. Math., 155 (1998), 215-235.  Google Scholar

[8]

P. Collet and J.-P. Eckmann, "Iterated Maps on the Interval as Dynamical Systems," Birkhäuser, Boston, 1980. Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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P. Mattila, "Geometry of Sets and Measures in Euclidean Spaces (Cambridge studies in advanced mathematics)," Cambridge University Press, Cambridge, 1995. Google Scholar

[13]

M. Misiurewicz and E. Visinescu, Kneading sequences of skew tent maps, Ann. Inst. H. Poincaré Probab. Statist., 27 (1991), 125-140.  Google Scholar

[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.  Google Scholar

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J. Schmeling, Symbolic dynamics for $\beta$-shifts and self-normal numbers, Ergodic Theory Dynam. Systems, 17 (1997), 675-694. doi: 10.1017/S0143385797079182.  Google Scholar

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M. Tsujii, Lyapunov exponents in families of one-dimensional dynamical systems, Invent. Math., 111 (1993), 113-137. doi: 10.1007/BF01231282.  Google Scholar

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G. Wagner, The ergodic behaviour of piecewise monotonic transformations, Z. Wahrsch. Verw. Gebiete, 46 (1979), 317-324. doi: 10.1007/BF00538119.  Google Scholar

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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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

M. Björklund and D. Schnellmann, Almost sure equidistribution in expansive families, Indag. Math. (N.S.), 20 (2009), 167-177.  Google Scholar

[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.  Google Scholar

[7]

H. Bruin, For almost every tent-map, the turning point is typical, Fund. Math., 155 (1998), 215-235.  Google Scholar

[8]

P. Collet and J.-P. Eckmann, "Iterated Maps on the Interval as Dynamical Systems," Birkhäuser, Boston, 1980. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

P. Mattila, "Geometry of Sets and Measures in Euclidean Spaces (Cambridge studies in advanced mathematics)," Cambridge University Press, Cambridge, 1995. Google Scholar

[13]

M. Misiurewicz and E. Visinescu, Kneading sequences of skew tent maps, Ann. Inst. H. Poincaré Probab. Statist., 27 (1991), 125-140.  Google Scholar

[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.  Google Scholar

[15]

J. Schmeling, Symbolic dynamics for $\beta$-shifts and self-normal numbers, Ergodic Theory Dynam. Systems, 17 (1997), 675-694. doi: 10.1017/S0143385797079182.  Google Scholar

[16]

M. Tsujii, Lyapunov exponents in families of one-dimensional dynamical systems, Invent. Math., 111 (1993), 113-137. doi: 10.1007/BF01231282.  Google Scholar

[17]

G. Wagner, The ergodic behaviour of piecewise monotonic transformations, Z. Wahrsch. Verw. Gebiete, 46 (1979), 317-324. doi: 10.1007/BF00538119.  Google Scholar

[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.  Google Scholar

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