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

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.
Citation: Daniel Schnellmann. Typical points for one-parameter families of piecewise expanding maps of the interval. Discrete & Continuous Dynamical Systems - A, 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. 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. 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. 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. doi: 10.2307/1971367.

[5]

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

[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. doi: 10.1017/S0143385700009962.

[7]

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

[8]

P. Collet and J.-P. Eckmann, "Iterated Maps on the Interval as Dynamical Systems,", Birkhäuser, (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. 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. 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. 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, (1995).

[13]

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

[14]

V. A. Rohlin, Exact endomorphisms of a Lebesgue space,, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 25 (1961), 499.

[15]

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

[16]

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

[17]

G. Wagner, The ergodic behaviour of piecewise monotonic transformations,, Z. Wahrsch. Verw. Gebiete, 46 (1979), 317. 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. doi: 10.1090/S0002-9947-1978-0515555-9.

show all references

References:
[1]

V. Baladi, On the susceptibility function of piecewise expanding interval maps,, Comm. Math. Phys., 275 (2007), 839. 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. 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. 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. doi: 10.2307/1971367.

[5]

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

[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. doi: 10.1017/S0143385700009962.

[7]

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

[8]

P. Collet and J.-P. Eckmann, "Iterated Maps on the Interval as Dynamical Systems,", Birkhäuser, (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. 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. 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. 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, (1995).

[13]

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

[14]

V. A. Rohlin, Exact endomorphisms of a Lebesgue space,, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 25 (1961), 499.

[15]

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

[16]

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

[17]

G. Wagner, The ergodic behaviour of piecewise monotonic transformations,, Z. Wahrsch. Verw. Gebiete, 46 (1979), 317. 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. doi: 10.1090/S0002-9947-1978-0515555-9.

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