Typical points and families of expanding interval mappings

Tomas Persson

doi:10.3934/dcds.2017170

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2017, Volume 37, Issue 7: 4019-4034. Doi: 10.3934/dcds.2017170

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Typical points and families of expanding interval mappings

Tomas Persson

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

Received: October 31, 2015

Revised: February 28, 2017

Published: August 2017

The author thanks D. Schnellmann for useful comments

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Abstract

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.

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Mathematics Subject Classification: Primary: 37E05, 37A10; Secondary: 37D20.

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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 $\inf |T_a'| > 7/2$.

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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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References

	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.
	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.
	C. Liverani , Decay of correlations for piecewise expanding maps, J. Statist. Phys., 78 (1995) , 1111-1129. doi: 10.1007/BF02183704.
	M. Rychlik , Bounded variation and invariant measures, Studia Math., 76 (1983) , 69-80.
	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.
	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.
	G. Wagner , The ergodic behaviour of piecewise monotonic transformations, Z. Wahrsch. Verw. Gebiete, 46 (1979) , 317-324. doi: 10.1007/BF00538119.
	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.