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

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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

Abstract
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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 and 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.
[2]	A. Arbieto, A. Junqueira and B. Santiago, On weakly hyperbolic iterated functions systems,, preprint, ().
[3]	M. F. Barnsley and K. Leśniak, The chaos game on a general iterated function system from a topological point of view,, preprint, ().
[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.
[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.
[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, ().
[7]	C. Bonatti and N. Guelman, Smooth Conjugacy classes of circle diffeomorphisms with irrational rotation number,, preprint, ().
[8]	T. Golenishcheva-Kutuzova, A. S. Gorodetski, V. Kleptsyn and D. Volk, Translation numbers define generators of $F_k^+\to Homeo_+(\mathbbS^1)$,, preprint, ().
[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.
[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.
[11]	A. J. Homburg, Circle diffeomorphisms forced by expanding circle maps, Ergodic Theory and Dynam. Systems, 32 (2012), 2011-2024. doi: 10.1017/S014338571100068X.
[12]	A. J. Homburg, Synchronization in iterated function systems,, preprint, ().
[13]	J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747. doi: 10.1512/iumj.1981.30.30055.
[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.

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.
[2]	A. Arbieto, A. Junqueira and B. Santiago, On weakly hyperbolic iterated functions systems,, preprint, ().
[3]	M. F. Barnsley and K. Leśniak, The chaos game on a general iterated function system from a topological point of view,, preprint, ().
[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.
[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.
[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, ().
[7]	C. Bonatti and N. Guelman, Smooth Conjugacy classes of circle diffeomorphisms with irrational rotation number,, preprint, ().
[8]	T. Golenishcheva-Kutuzova, A. S. Gorodetski, V. Kleptsyn and D. Volk, Translation numbers define generators of $F_k^+\to Homeo_+(\mathbbS^1)$,, preprint, ().
[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.
[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.
[11]	A. J. Homburg, Circle diffeomorphisms forced by expanding circle maps, Ergodic Theory and Dynam. Systems, 32 (2012), 2011-2024. doi: 10.1017/S014338571100068X.
[12]	A. J. Homburg, Synchronization in iterated function systems,, preprint, ().
[13]	J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747. doi: 10.1512/iumj.1981.30.30055.
[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.

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