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A non-homogeneous boundary value problem of the sixth order Boussinesq equation in a quarter plane
Stability of the distribution function for piecewise monotonic maps on the interval
1. | Mathematical Institute, Silesian University in Opava, Na Rybníčku 1, 746 01 Opava, Czech Republic |
2. | Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria |
For piecewise monotonic maps the notion of approximating distribution function is introduced. It is shown that for a mixing basic set it coincides with the usual distribution function. Moreover, it is proved that the approximating distribution function is upper semi-continuous under small perturbations of the map.
References:
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M. Babilonová,
Distributional chaos for triangular maps, Ann. Math. Sil., 13 (1999), 33-38.
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F. Balibrea, B. Schweizer, A. Sklar and J. Smítal,
Generalized specification property and distributional chaos, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 1683-1694.
doi: 10.1142/S0218127403007539. |
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F. Balibrea, J. Smítal and M. Štefánková,
The three versions of distributional chaos, Chaos Solitons Fractals, 23 (2005), 1581-1583.
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A. Blokh, The 'spectral' decomposition for one-dimensional maps, in Dynamics Reported,
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Springer, Berlin, 4 (1995), 1-59. |
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F. Hofbauer,
Piecewise invertible dynamical systems, Probab. Theory Related Fields, 72 (1986), 359-386.
doi: 10.1007/BF00334191. |
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R. Hric and M. Málek,
Omega limit sets and distributional chaos on graphs, Topology
Appl., 153 (2006), 2469-2475.
doi: 10.1016/j.topol.2005.09.007. |
[7] |
P. Raith,
Continuity of the Hausdorff dimension for piecewise monotonic maps, Israel J.
Math., 80 (1992), 97-133.
doi: 10.1007/BF02808156. |
[8] |
P. Raith,
Continuity of the Hausdorff dimension for invariant subsets of interval maps, Acta
Math. Univ. Comenian., 63 (1994), 39-53.
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[9] |
P. Raith,
The behaviour of the nonwandering set of a piecewise monotonic interval map under small perturbations, Math. Bohem., 122 (1997), 37-55.
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[10] |
P. Raith,
The dynamics of piecewise monotonic maps under small perturbations, Ann. Scuola
Norm. Sup. Pisa Cl. Sci., 24 (1997), 783-811.
|
[11] |
B. Schweizer, A. Sklar and J. Smítal,
Distributional (and other) chaos and its measurement, Real Anal. Exchange, 26 (2000/2001), 495-524.
|
[12] |
B. Schweizer and J. Smítal,
Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc., 344 (1994), 737-754.
doi: 10.1090/S0002-9947-1994-1227094-X. |
show all references
References:
[1] |
M. Babilonová,
Distributional chaos for triangular maps, Ann. Math. Sil., 13 (1999), 33-38.
|
[2] |
F. Balibrea, B. Schweizer, A. Sklar and J. Smítal,
Generalized specification property and distributional chaos, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 1683-1694.
doi: 10.1142/S0218127403007539. |
[3] |
F. Balibrea, J. Smítal and M. Štefánková,
The three versions of distributional chaos, Chaos Solitons Fractals, 23 (2005), 1581-1583.
|
[4] |
A. Blokh, The 'spectral' decomposition for one-dimensional maps, in Dynamics Reported,
Expositions in Dynamical Systems, (eds. : C. K. R. T. Jones, U. Kirchgraber, H. O. Walther),
Springer, Berlin, 4 (1995), 1-59. |
[5] |
F. Hofbauer,
Piecewise invertible dynamical systems, Probab. Theory Related Fields, 72 (1986), 359-386.
doi: 10.1007/BF00334191. |
[6] |
R. Hric and M. Málek,
Omega limit sets and distributional chaos on graphs, Topology
Appl., 153 (2006), 2469-2475.
doi: 10.1016/j.topol.2005.09.007. |
[7] |
P. Raith,
Continuity of the Hausdorff dimension for piecewise monotonic maps, Israel J.
Math., 80 (1992), 97-133.
doi: 10.1007/BF02808156. |
[8] |
P. Raith,
Continuity of the Hausdorff dimension for invariant subsets of interval maps, Acta
Math. Univ. Comenian., 63 (1994), 39-53.
|
[9] |
P. Raith,
The behaviour of the nonwandering set of a piecewise monotonic interval map under small perturbations, Math. Bohem., 122 (1997), 37-55.
|
[10] |
P. Raith,
The dynamics of piecewise monotonic maps under small perturbations, Ann. Scuola
Norm. Sup. Pisa Cl. Sci., 24 (1997), 783-811.
|
[11] |
B. Schweizer, A. Sklar and J. Smítal,
Distributional (and other) chaos and its measurement, Real Anal. Exchange, 26 (2000/2001), 495-524.
|
[12] |
B. Schweizer and J. Smítal,
Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc., 344 (1994), 737-754.
doi: 10.1090/S0002-9947-1994-1227094-X. |
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