On the minimality of semigroup actions on the interval which are \(C^{1}\)-close to the identity. (English) Zbl 1312.37024
This paper is concerned with semigroup actions on the interval \([0,1]\), which are generated by two attracting maps. If the generators are sufficiently close in the \(C^2\)-norm, then it is known that the minimal set coincides with the entire interval. By constructing a counterexample it is illustrated that this result does not hold under the \(C^1\)-topology. In detail, it is shown that a sequence \((f_n,g_n)\) of \(C^1\)-diffeomorphisms on their images satisfying \(f_n'(0),g_n'(0)\in(0,1)\), which converges to \((\mathrm{id},\mathrm{id})\) in the \(C^1\)-topology, might have a minimal set not equal to \([0,1]\).
Reviewer: Christian Pötzsche (Klagenfurt)
MSC:
| 37C70 | Attractors and repellers of smooth dynamical systems and their topological structure |
| 37D30 | Partially hyperbolic systems and dominated splittings |
| 37E05 | Dynamical systems involving maps of the interval |
| 37H20 | Bifurcation theory for random and stochastic dynamical systems |
| 57R30 | Foliations in differential topology; geometric theory |
References:
| [1] | A.Avila, J.Bochi, J.Yoccoz, ‘Uniformly hyperbolic finite‐valued \(\text{SL} ( 2 , \mathbb{R} )\)‐cocycles’, Comment. Math. Helv., 85 (2010) 813-884. · Zbl 1201.37032 |
| [2] | P.Barrientos, Y.Ki, A.Raibekas, ‘Symbolic blender‐horseshoes and applications’, Preprint, 2012, http://arxiv.org/abs/1211.7088. · Zbl 1336.37025 |
| [3] | P.Barrientos, A.Raibekas, ‘Dynamics of iterated function systems on the circle close to rotations’, Preprint, http://arxiv.org/abs/1303.2521. · Zbl 1352.37121 |
| [4] | B.Deroin, V.Kleptsyn, A.Navas, ‘Sur la dynamique unidimensionnelle en régularité intermédiaire’, Acta Math., 199 (2007) 199-262. · Zbl 1139.37025 |
| [5] | L. J.Díaz, K.Gelfert, ‘Porcupine‐like horseshoes: transitivity, Lyapunov spectrum, and phase transitions’, Fund. Math., 216 (2012) 55-100. · Zbl 1273.37027 |
| [6] | L. J.Díaz, K.Gelfert, M.Rams, ‘Rich phase transitions in step skew products’, Nonlinearity, 24 (2011) 3391-3412. · Zbl 1263.37049 |
| [7] | G. Duminy, Unpublished manuscript, Université de Lille, 1977. |
| [8] | G.Hardy, E.Wright, An introduction to the theory of numbers, 6th edn (Oxford University Press, Oxford, 2008) xxii+621 pp. · Zbl 1159.11001 |
| [9] | M.Herman, ‘Sur la conjugaison différentiable des difféomorphismes du cercle á des rotations’, Publ. Math. Inst. Hautes Études Sci., 49 (1979) 5-233. · Zbl 0448.58019 |
| [10] | Y.Ilyashenko, A.Negut, ‘Invisible parts of attractors’, Nonlinearity, 23 (2010) 1199-1219. · Zbl 1204.37017 |
| [11] | V.Kleptsyn, D.Volk, ‘Physical measures for nonlinear random walks on interval’, Mosc. Math. J., 14 (2014) 339-365. · Zbl 1295.82011 |
| [12] | C.Moreira, ‘Stable intersections of Cantor sets and homoclinic bifurcations’, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996) 741-781. · Zbl 0865.58035 |
| [13] | C.Moreira, ‘There are no \(C^1\)‐stable intersections of regular Cantor sets’, Acta Math., 206 (2011) 311-323. · Zbl 1251.37032 |
| [14] | A.Navas, ‘Sur les groupes de difféomorphismes du cercle engendrés par des éléments proches des rotations’, Enseign. Math., (2)50 (2004) 29-68. · Zbl 1085.37034 |
| [15] | A.Navas, Groups of circle diffeomorphisms, Translation of the 2007 Spanish edition, Chicago Lectures in Mathematics 99 (University of Chicago Press, Chicago, 2011) xviii+290 pp. · Zbl 1236.37002 |
| [16] | A.Navas, ‘Sur les rapprochements par conjugaison en dimension 1 et classe \(C^1\)’, Preprint, 2012, http://arxiv.org/abs/1208.4815. |
| [17] | J.Palis, F.Takens, ‘Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations’, Fractal dimensions and infinitely many attractors, Cambridge Studies in Advanced Mathematics 35 (Cambridge University Press, Cambridge, 1993) x+234 pp. · Zbl 0790.58014 |
| [18] | A.Raibekas, ‘\(C^2\)‐iterated function systems on the circle and symbolic blender‐like’, PhD Thesis, http://www.preprint.impa.br/Shadows/SERIE_C/2011/146.html. |
| [19] | K.Shinohara, ‘Some examples of minimal Cantor set for IFSs with overlap’, Preprint, http://arxiv.org/abs/1304.5831. |
| [20] | T.Tsuboi, ‘Homological and dynamical study on certain groups of Lipschitz homeomorphisms of the circle’, J. Math. Soc. Japan, 47 (1995) 1-30. · Zbl 0852.57031 |
| [21] | M.Tsujii, ‘Physical measures for partially hyperbolic surface endomorphisms’, Acta Math., 194 (2005) 37-132. · Zbl 1105.37022 |
| [22] | R.Ures, ‘Abundance of hyperbolicity in the \(C^1\) topology’, Ann. Sci. École Norm. Sup., (4)28 (1995) 747-760. · Zbl 0989.37013 |
| [23] | M.Viana, J.Yang, ‘Physical measure and absolute continuity for one‐dimensional center direction’, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013) 845-877. · Zbl 1417.37105 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
