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