October  2017, 37(10): 5253-5269. doi: 10.3934/dcds.2017227

Examples of minimal set for IFSs

Universidad de La República. Facultad de Ingenieria. IMERL, Julio Herrera y Reissig 565. C.P. 11300, Montevideo, Uruguay

 

Received  December 2016 Revised  April 2017 Published  June 2017

We exhibit different examples of minimal sets for an IFS of homeomorphisms with rational rotation number. It is proved that these examples are, from a topological point of view, the unique possible cases.

Citation: Nancy Guelman, Jorge Iglesias, Aldo Portela. Examples of minimal set for IFSs. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5253-5269. doi: 10.3934/dcds.2017227
References:
[1]

P. G. Barrientos and A. Raibekas, Dynamics of iterated function systems on the circle close to rotations, Ergodic Theory Dynam. Systems, 35 (2015), 1345-1368.  doi: 10.1017/etds.2013.112.  Google Scholar

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A. Portela, Regular Interval Cantor sets of S1 and minimality, Bulletin of the Brazilian Mathematical Society, New Series, 40 (2009), 53-75.  doi: 10.1007/s00574-009-0002-3.  Google Scholar

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K. Shinohara, Some examples of minimal Cantor sets for iterated function systems with overlap, Tokyo J. Math., 37 (2014), 225-236.  doi: 10.3836/tjm/1406552441.  Google Scholar

show all references

References:
[1]

P. G. Barrientos and A. Raibekas, Dynamics of iterated function systems on the circle close to rotations, Ergodic Theory Dynam. Systems, 35 (2015), 1345-1368.  doi: 10.1017/etds.2013.112.  Google Scholar

[2]

M. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math., 49 (1979), 5-233.   Google Scholar

[3]

J. A. Guthrie and J. E. Nymann, The topological structure of the set of subsums of an infinite series, Colloq. Math., 55 (1988), 323-327.   Google Scholar

[4]

A. N. Kercheval, Denjoy minimal sets are far from affine, Ergod. Th. & Dynam. Sys., 22 (2002), 1803-1812.  doi: 10.1017/S0143385702000512.  Google Scholar

[5]

B. Kra and J. Schmeling, Diophantine classes, dimension and Denjoy maps, Acta Arith., 105 (2002), 323-340.  doi: 10.4064/aa105-4-2.  Google Scholar

[6]

D. McDuff, $C^1$-minimal subset of the circle, Ann. Inst. Fourier, Grenoble, 31 (1981), 177-193.  doi: 10.5802/aif.822.  Google Scholar

[7]

P. Mendes and F. Olivera, On the topological structure of the arithmetic sum of two Cantor sets, Nonlinearity, 7 (1994), 329-343.  doi: 10.1088/0951-7715/7/2/002.  Google Scholar

[8]

A. Navas, Grupos de Difeomorfismos Del Círculo. (Spanish) [Groups of Diffeomorphisms of the Circle] Ensaios Matemáticos [Mathematical Surveys], 13. Sociedade Brasileira de Matemática, Rio de Janeiro, 2007.  Google Scholar

[9]

Z. Nitecki, Cantorvals and subsum sets of null sequences, Amer. Math. Monthly, 122 (2015), 862-870.  doi: 10.4169/amer.math.monthly.122.9.862.  Google Scholar

[10]

A. Portela, Regular Interval Cantor sets of S1 and minimality, Bulletin of the Brazilian Mathematical Society, New Series, 40 (2009), 53-75.  doi: 10.1007/s00574-009-0002-3.  Google Scholar

[11]

K. Shinohara, On the minimality of semigroups action on the interval which are C1-close to identity, Proc. London Math. Soc., 109 (2014), 1175-1202.  doi: 10.1112/plms/pdu032.  Google Scholar

[12]

K. Shinohara, Some examples of minimal Cantor sets for iterated function systems with overlap, Tokyo J. Math., 37 (2014), 225-236.  doi: 10.3836/tjm/1406552441.  Google Scholar

Figure 3.  These figures correspond to example 2.
Figure 4.  This figures correspond to example 3.
Figure 5.  This figure corresponds to example 4.
Figure 6.  These figures correspond to example 5.
Figure 7.  This figure corresponds to example 6.
Figure 9.  These figures correspond to example 7.
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