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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.

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##### 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.
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 $\inf |T_a'| > 7/2$.
An illustration of the action of $E_s$ with $s = 1.2$. The dashed lines show the original graph.
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