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August  2017, 37(7): 4019-4034. doi: 10.3934/dcds.2017170

Typical points and families of expanding interval mappings

Centre for Mathematical Sciences, Lund University, Box 118,221 00 Lund, Sweden

Received  November 2015 Revised  March 2017 Published  April 2017

Fund Project: The author thanks D. Schnellmann for useful comments.

We study parametrised families of piecewise expanding interval mappings $T_a \colon [0,1] \to [0,1]$ with absolutely continuous invariant measures $\mu_a$ and give sufficient conditions for a point $X(a)$ to be typical with respect to $(T_a, \mu_a)$ for almost all parameters a. This is similar to a result by D.Schnellmann, but with different assumptions.

Citation: 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
References:
[1]

M. Björklund and D. Schnellmann, Almost sure equidistribution in expansive families, Indag. Math. (N.S.), 20 (2009), 167-177.  doi: 10.1016/S0019-3577(09)00017-2.

[2]

Z. Kowalski, Invariant measure for piecewise monotonic transformation has a positive lower bound on its support, Bull. Acad. Polon. Sci. Sér. Sci. Math., 27 (1979), 53-57. 

[3]

C. Liverani, Decay of correlations for piecewise expanding maps, J. Statist. Phys., 78 (1995), 1111-1129.  doi: 10.1007/BF02183704.

[4]

M. Rychlik, Bounded variation and invariant measures, Studia Math., 76 (1983), 69-80. 

[5]

D. Schnellmann, Typical points for one-parameter families of piecewise expanding maps of the interval, Discrete Contin. Dyn. Syst., 31 (2011), 877-911.  doi: 10.3934/dcds.2011.31.877.

[6]

D. Schnellmann, Law of iterated logarithm and invariance principle for one-parameter families of interval maps, Probab. Theory Related Fields, 162 (2015), 365-409.  doi: 10.1007/s00440-014-0575-7.

[7]

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

[8]

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]

M. Björklund and D. Schnellmann, Almost sure equidistribution in expansive families, Indag. Math. (N.S.), 20 (2009), 167-177.  doi: 10.1016/S0019-3577(09)00017-2.

[2]

Z. Kowalski, Invariant measure for piecewise monotonic transformation has a positive lower bound on its support, Bull. Acad. Polon. Sci. Sér. Sci. Math., 27 (1979), 53-57. 

[3]

C. Liverani, Decay of correlations for piecewise expanding maps, J. Statist. Phys., 78 (1995), 1111-1129.  doi: 10.1007/BF02183704.

[4]

M. Rychlik, Bounded variation and invariant measures, Studia Math., 76 (1983), 69-80. 

[5]

D. Schnellmann, Typical points for one-parameter families of piecewise expanding maps of the interval, Discrete Contin. Dyn. Syst., 31 (2011), 877-911.  doi: 10.3934/dcds.2011.31.877.

[6]

D. Schnellmann, Law of iterated logarithm and invariance principle for one-parameter families of interval maps, Probab. Theory Related Fields, 162 (2015), 365-409.  doi: 10.1007/s00440-014-0575-7.

[7]

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

[8]

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.

Figure 1.  An example of a mapping T for which the assumptions in Corollary 1 are satisfied for $T_a (x) = T(ax)$, for all parameters in some interval $[1,a_1]$, $a_1 > 1$. Here we have taken $\delta = 2/5$. Assumption 6 is then that $\inf |T_a'| > 7/2$.
Figure 2.  An illustration of the action of $E_s$ with $s = 1.2$. The dashed lines show the original graph.
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