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May  2018, 38(5): 2527-2539. doi: 10.3934/dcds.2018105

Stability of the distribution function for piecewise monotonic maps on the interval

Michal Málek 1, and  Peter Raith 2,

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

Received  August 2017 Revised  November 2017 Published  March 2018

Fund Project: The research was partially supported by the projects 42p11 and 38p10 of AKTION Česká republika – Österreich, and by RVO funding for IČ47813059.

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.

Keywords: Distributional chaos, piecewise monotonic map, distribution function, interval map, perturbation, basic set.
Mathematics Subject Classification: 37E05, 37D45, 37C75.
Citation: Michal Málek, Peter Raith. Stability of the distribution function for piecewise monotonic maps on the interval. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2527-2539. doi: 10.3934/dcds.2018105
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.

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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