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

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 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.

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.

[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.

[1]

Simon Lloyd, Edson Vargas. Critical covering maps without absolutely continuous invariant probability measure. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2393-2412. doi: 10.3934/dcds.2019101

[2]

Tomas Persson. Typical points and families of expanding interval mappings. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 4019-4034. doi: 10.3934/dcds.2017170

[3]

Jiu Ding, Aihui Zhou. Absolutely continuous invariant measures for piecewise $C^2$ and expanding mappings in higher dimensions. Discrete and Continuous Dynamical Systems, 2000, 6 (2) : 451-458. doi: 10.3934/dcds.2000.6.451

[4]

Jawad Al-Khal, Henk Bruin, Michael Jakobson. New examples of S-unimodal maps with a sigma-finite absolutely continuous invariant measure. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 35-61. doi: 10.3934/dcds.2008.22.35

[5]

Lucia D. Simonelli. Absolutely continuous spectrum for parabolic flows/maps. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 263-292. doi: 10.3934/dcds.2018013

[6]

Viviane Baladi, Daniel Smania. Smooth deformations of piecewise expanding unimodal maps. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 685-703. doi: 10.3934/dcds.2009.23.685

[7]

Damien Thomine. A spectral gap for transfer operators of piecewise expanding maps. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 917-944. doi: 10.3934/dcds.2011.30.917

[8]

Magnus Aspenberg, Viviane Baladi, Juho Leppänen, Tomas Persson. On the fractional susceptibility function of piecewise expanding maps. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 679-706. doi: 10.3934/dcds.2021133

[9]

Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002

[10]

Michal Málek, Peter Raith. Stability of the distribution function for piecewise monotonic maps on the interval. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2527-2539. doi: 10.3934/dcds.2018105

[11]

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

[12]

Nigel P. Byott, Mark Holland, Yiwei Zhang. On the mixing properties of piecewise expanding maps under composition with permutations. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3365-3390. doi: 10.3934/dcds.2013.33.3365

[13]

Peyman Eslami. Inducing schemes for multi-dimensional piecewise expanding maps. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 353-368. doi: 10.3934/dcds.2021120

[14]

Oliver Butterley. An alternative approach to generalised BV and the application to expanding interval maps. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3355-3363. doi: 10.3934/dcds.2013.33.3355

[15]

Xavier Bressaud. Expanding interval maps with intermittent behaviour, physical measures and time scales. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 517-546. doi: 10.3934/dcds.2004.11.517

[16]

Fawwaz Batayneh, Cecilia González-Tokman. On the number of invariant measures for random expanding maps in higher dimensions. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5887-5914. doi: 10.3934/dcds.2021100

[17]

Yan Wang, Guanggan Chen. Invariant measure of stochastic fractional Burgers equation with degenerate noise on a bounded interval. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3121-3135. doi: 10.3934/cpaa.2019140

[18]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217

[19]

Jérôme Buzzi, Sylvie Ruette. Large entropy implies existence of a maximal entropy measure for interval maps. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 673-688. doi: 10.3934/dcds.2006.14.673

[20]

Adrian Tudorascu. On absolutely continuous curves of probabilities on the line. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5105-5124. doi: 10.3934/dcds.2019207

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (67)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]