American Institute of Mathematical Sciences

Advanced
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 & Continuous Dynamical Systems - A, 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.   Google Scholar

[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.  Google Scholar

[3]

F. Balibrea, J. Smítal and M. Štefánková, The three versions of distributional chaos, Chaos Solitons Fractals, 23 (2005), 1581-1583.   Google Scholar

[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.  Google Scholar

[5]

F. Hofbauer, Piecewise invertible dynamical systems, Probab. Theory Related Fields, 72 (1986), 359-386.  doi: 10.1007/BF00334191.  Google Scholar

[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.  Google Scholar

[7]

P. Raith, Continuity of the Hausdorff dimension for piecewise monotonic maps, Israel J. Math., 80 (1992), 97-133.  doi: 10.1007/BF02808156.  Google Scholar

[8]

P. Raith, Continuity of the Hausdorff dimension for invariant subsets of interval maps, Acta Math. Univ. Comenian., 63 (1994), 39-53.   Google Scholar

[9]

P. Raith, The behaviour of the nonwandering set of a piecewise monotonic interval map under small perturbations, Math. Bohem., 122 (1997), 37-55.   Google Scholar

[10]

P. Raith, The dynamics of piecewise monotonic maps under small perturbations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 783-811.   Google Scholar

[11]

B. Schweizer, A. Sklar and J. Smítal, Distributional (and other) chaos and its measurement, Real Anal. Exchange, 26 (2000/2001), 495-524.   Google Scholar

[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.  Google Scholar

show all references

References:
[1]

M. Babilonová, Distributional chaos for triangular maps, Ann. Math. Sil., 13 (1999), 33-38.   Google Scholar

[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.  Google Scholar

[3]

F. Balibrea, J. Smítal and M. Štefánková, The three versions of distributional chaos, Chaos Solitons Fractals, 23 (2005), 1581-1583.   Google Scholar

[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.  Google Scholar

[5]

F. Hofbauer, Piecewise invertible dynamical systems, Probab. Theory Related Fields, 72 (1986), 359-386.  doi: 10.1007/BF00334191.  Google Scholar

[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.  Google Scholar

[7]

P. Raith, Continuity of the Hausdorff dimension for piecewise monotonic maps, Israel J. Math., 80 (1992), 97-133.  doi: 10.1007/BF02808156.  Google Scholar

[8]

P. Raith, Continuity of the Hausdorff dimension for invariant subsets of interval maps, Acta Math. Univ. Comenian., 63 (1994), 39-53.   Google Scholar

[9]

P. Raith, The behaviour of the nonwandering set of a piecewise monotonic interval map under small perturbations, Math. Bohem., 122 (1997), 37-55.   Google Scholar

[10]

P. Raith, The dynamics of piecewise monotonic maps under small perturbations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 783-811.   Google Scholar

[11]

B. Schweizer, A. Sklar and J. Smítal, Distributional (and other) chaos and its measurement, Real Anal. Exchange, 26 (2000/2001), 495-524.   Google Scholar

[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.  Google Scholar

[1]

Christian Wolf. A shift map with a discontinuous entropy function. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 319-329. doi: 10.3934/dcds.2020012
[2]

James W. Cannon, Mark H. Meilstrup, Andreas Zastrow. The period set of a map from the Cantor set to itself. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2667-2679. doi: 10.3934/dcds.2013.33.2667
[3]

Frédéric Faure. Prequantum chaos: Resonances of the prequantum cat map. Journal of Modern Dynamics, 2007, 1 (2) : 255-285. doi: 10.3934/jmd.2007.1.255
[4]

C. Bonanno, G. Menconi. Computational information for the logistic map at the chaos threshold. Discrete & Continuous Dynamical Systems - B, 2002, 2 (3) : 415-431. doi: 10.3934/dcdsb.2002.2.415
[5]

Yuanhong Chen, Chao Ma, Jun Wu. Moving recurrent properties for the doubling map on the unit interval. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 2969-2979. doi: 10.3934/dcds.2016.36.2969
[6]

Wacław Marzantowicz, Justyna Signerska. Firing map of an almost periodic input function. Conference Publications, 2011, 2011 (Special) : 1032-1041. doi: 10.3934/proc.2011.2011.1032
[7]

Hun Ki Baek, Younghae Do. Dangerous Border-Collision bifurcations of a piecewise-smooth map. Communications on Pure & Applied Analysis, 2006, 5 (3) : 493-503. doi: 10.3934/cpaa.2006.5.493
[8]

Zhiying Qin, Jichen Yang, Soumitro Banerjee, Guirong Jiang. Border-collision bifurcations in a generalized piecewise linear-power map. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 547-567. doi: 10.3934/dcdsb.2011.16.547
[9]

David Burguet. Examples of $\mathcal{C}^r$ interval map with large symbolic extension entropy. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 873-899. doi: 10.3934/dcds.2010.26.873
[10]

Sébastien Gouëzel. An interval map with a spectral gap on Lipschitz functions, but not on bounded variation functions. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1205-1208. doi: 10.3934/dcds.2009.24.1205
[11]

Denis Volk. Almost every interval translation map of three intervals is finite type. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2307-2314. doi: 10.3934/dcds.2014.34.2307
[12]

Johan Matheus Tuwankotta, Eric Harjanto. Strange attractors in a predator–prey system with non-monotonic response function and periodic perturbation. Journal of Computational Dynamics, 2019, 6 (2) : 469-483. doi: 10.3934/jcd.2019024
[13]

Piotr Oprocha. Specification properties and dense distributional chaos. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 821-833. doi: 10.3934/dcds.2007.17.821
[14]

Piotr Oprocha, Pawel Wilczynski. Distributional chaos via isolating segments. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 347-356. doi: 10.3934/dcdsb.2007.8.347
[15]

Lidong Wang, Xiang Wang, Fengchun Lei, Heng Liu. Mixing invariant extremal distributional chaos. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6533-6538. doi: 10.3934/dcds.2016082
[16]

C. R. Chen, S. J. Li. Semicontinuity of the solution set map to a set-valued weak vector variational inequality. Journal of Industrial & Management Optimization, 2007, 3 (3) : 519-528. doi: 10.3934/jimo.2007.3.519
[17]

Steven M. Pederson. Non-turning Poincaré map and homoclinic tangencies in interval maps with non-constant topological entropy. Conference Publications, 2001, 2001 (Special) : 295-302. doi: 10.3934/proc.2001.2001.295
[18]

Angela A. Albanese, Xavier Barrachina, Elisabetta M. Mangino, Alfredo Peris. Distributional chaos for strongly continuous semigroups of operators. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2069-2082. doi: 10.3934/cpaa.2013.12.2069
[19]

Yu-Hao Liang, Wan-Rou Wu, Jonq Juang. Fastest synchronized network and synchrony on the Julia set of complex-valued coupled map lattices. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 173-184. doi: 10.3934/dcdsb.2016.21.173
[20]

Daniel Fusca. The Madelung transform as a momentum map. Journal of Geometric Mechanics, 2017, 9 (2) : 157-165. doi: 10.3934/jgm.2017006

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (80)
  • HTML views (209)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences