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September  2014, 34(9): 3341-3352. doi: 10.3934/dcds.2014.34.3341

Density of fiberwise orbits in minimal iterated function systems on the circle

Pablo G. Barrientos 1, , Abbas Fakhari 2, and  Aliasghar Sarizadeh 3,

1. 

Instituto de Matemática e Estatística, UFF, Rua Mário Santos Braga s/n - Campus Valonguinhos, Niterói, Brazil
2. 

Department of Mathematics, Shahid Beheshti University, G.C.Tehran 19839, Iran
3. 

Department of Mathematics, Ilam University, P.O. Box 69315-516, Ilam, Iran

Received  August 2013 Revised  January 2014 Published  March 2014

We study the minimality of almost every orbital branch of minimal iterated function systems (IFSs). We prove that this kind of minimality holds for forward and backward minimal IFSs generated by orientation-preserving homeomorphisms of the circle. We provide new examples of iterated functions systems where this behavior persists under perturbation of the generators.
Keywords: Minimal IFS on the circle, minimality of orbital branches (fiberwise orbits)..
Mathematics Subject Classification: Primary: 37C05, 37C20; Secondary: 37E1.
Citation: Pablo G. Barrientos, Abbas Fakhari, Aliasghar Sarizadeh. Density of fiberwise orbits in minimal iterated function systems on the circle. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3341-3352. doi: 10.3934/dcds.2014.34.3341
References:
[1]

V. A. Antonov, Modeling Cyclic Evolution Processes: Synchronization by Means of Random Signal, Leingradskii Universitet Vestnik Matematika Mekhanika Astronomiia, 1984, 67-76.  Google Scholar

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P. G. Barrientos and A. Raibekas, Dynamics of iterated function systems on the circle close to rotations,, to appear in Ergodic Theory and Dynam. Systems, ().   Google Scholar

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C. Bonatti and N. Guelman, Smooth Conjugacy classes of circle diffeomorphisms with irrational rotation number,, preprint, ().   Google Scholar

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T. Golenishcheva-Kutuzova, A. S. Gorodetski, V. Kleptsyn and D. Volk, Translation numbers define generators of $F_k^+\to Homeo_+(\mathbbS^1)$,, preprint, ().   Google Scholar

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A. S. Gorodetski and Yu. S. Ilyashenko, Certain new robust properties of invariant sets and attractors of dynamical systems, Functional Analysis and Its Applications, 33 (1999), 95-105. doi: 10.1007/BF02465190.  Google Scholar

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A. S. Gorodetski and Yu. S. Ilyashenko, Some properties of skew products over a horseshoe and a solenoid, Tr. Mat. Inst. Steklova, 231 (2000), 96-112.  Google Scholar

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A. J. Homburg, Circle diffeomorphisms forced by expanding circle maps, Ergodic Theory and Dynam. Systems, 32 (2012), 2011-2024. doi: 10.1017/S014338571100068X.  Google Scholar

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A. J. Homburg, Synchronization in iterated function systems,, preprint, ().   Google Scholar

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J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747. doi: 10.1512/iumj.1981.30.30055.  Google Scholar

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V. A. Kleptsyn and M. B. Nalskii, Contraction of orbits in random dynamical systems on the circle, Functional Analysis and Its Applications, 38 (2004), 267-282. doi: 10.1007/s10688-005-0005-9.  Google Scholar

show all references

References:
[1]

V. A. Antonov, Modeling Cyclic Evolution Processes: Synchronization by Means of Random Signal, Leingradskii Universitet Vestnik Matematika Mekhanika Astronomiia, 1984, 67-76.  Google Scholar

[2]

A. Arbieto, A. Junqueira and B. Santiago, On weakly hyperbolic iterated functions systems,, preprint, ().   Google Scholar

[3]

M. F. Barnsley and K. Leśniak, The chaos game on a general iterated function system from a topological point of view,, preprint, ().   Google Scholar

[4]

M. F. Barnsley and A. Vince, The chaos game on a general iterated function system, Ergodic Theory and Dynam. Systems, 31 (2011), 1073-1079. doi: 10.1017/S0143385710000428.  Google Scholar

[5]

M. F. Barnsley and A. Vince, The conley attractor of an iterated function system, Bull. Aust. Math. Soc., 88 (2013), 267-279. doi: 10.1017/S0004972713000348.  Google Scholar

[6]

P. G. Barrientos and A. Raibekas, Dynamics of iterated function systems on the circle close to rotations,, to appear in Ergodic Theory and Dynam. Systems, ().   Google Scholar

[7]

C. Bonatti and N. Guelman, Smooth Conjugacy classes of circle diffeomorphisms with irrational rotation number,, preprint, ().   Google Scholar

[8]

T. Golenishcheva-Kutuzova, A. S. Gorodetski, V. Kleptsyn and D. Volk, Translation numbers define generators of $F_k^+\to Homeo_+(\mathbbS^1)$,, preprint, ().   Google Scholar

[9]

A. S. Gorodetski and Yu. S. Ilyashenko, Certain new robust properties of invariant sets and attractors of dynamical systems, Functional Analysis and Its Applications, 33 (1999), 95-105. doi: 10.1007/BF02465190.  Google Scholar

[10]

A. S. Gorodetski and Yu. S. Ilyashenko, Some properties of skew products over a horseshoe and a solenoid, Tr. Mat. Inst. Steklova, 231 (2000), 96-112.  Google Scholar

[11]

A. J. Homburg, Circle diffeomorphisms forced by expanding circle maps, Ergodic Theory and Dynam. Systems, 32 (2012), 2011-2024. doi: 10.1017/S014338571100068X.  Google Scholar

[12]

A. J. Homburg, Synchronization in iterated function systems,, preprint, ().   Google Scholar

[13]

J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747. doi: 10.1512/iumj.1981.30.30055.  Google Scholar

[14]

V. A. Kleptsyn and M. B. Nalskii, Contraction of orbits in random dynamical systems on the circle, Functional Analysis and Its Applications, 38 (2004), 267-282. doi: 10.1007/s10688-005-0005-9.  Google Scholar

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