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 and Continuous Dynamical Systems, 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.

[2]

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

[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. 

[4]

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

[5]

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

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

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.

[2]

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

[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. 

[4]

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

[5]

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

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

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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