The Ingram conjecture

Barge, Marcy¹ ; Bruin, Henk² ; Štimac, Sonja³

¹ Department of Mathematical Sciences, Montana State University, Bozeman, MT 59717, USA
² Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, 1090 Wien, Austria
³ Department of Mathematics, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia

Geometry & topology, Tome 16 (2012) no. 4, pp. 2481-2516.

Résumé

We prove the Ingram conjecture, ie we show that the inverse limit spaces of tent maps with different slopes in the interval $\left[1,2\right]$ are nonhomeomorphic. Based on the structure obtained from the proof, we also show that every self-homeomorphism of the inverse limit space of a tent map is pseudo-isotopic, on the core, to some power of the shift homeomorphism.

Détail

DOI : 10.2140/gt.2012.16.2481

Classification : 54H20, 37B45, 37E05
Keywords: tent map, inverse limit space, unimodal map, classification, pseudo-isotopy

Bibliographie

[1] J E Anderson, I F Putnam, Topological invariants for substitution tilings and their associated $C^{*}$–algebras, Ergodic Theory Dynam. Systems 18 (1998) 509

[2] M Barge, K M Brucks, B Diamond, Self-similarity in inverse limit spaces of the tent family, Proc. Amer. Math. Soc. 124 (1996) 3563

[3] M Barge, B Diamond, Homeomorphisms of inverse limit spaces of one-dimensional maps, Fund. Math. 146 (1995) 171

[4] M Barge, B Diamond, Inverse limit spaces of infinitely renormalizable maps, Topology Appl. 83 (1998) 103

[5] M Barge, B Diamond, Subcontinua of the closure of the unstable manifold at a homoclinic tangency, Ergodic Theory Dynam. Systems 19 (1999) 289

[6] M Barge, S Holte, Nearly one-dimensional Hénon attractors and inverse limits, Nonlinearity 8 (1995) 29

[7] M Barge, W T Ingram, Inverse limits on $[0,1]$ using logistic bonding maps, Topology Appl. 72 (1996) 159

[8] L Block, S Jakimovik, L Kailhofer, J Keesling, On the classification of inverse limits of tent maps, Fund. Math. 187 (2005) 171

[9] L Block, S Jakimovik, J Keesling, On Ingram's conjecture, Topology Proc. 30 (2006) 95

[10] L Block, J Keesling, B E Raines, S Štimac, Homeomorphisms of unimodal inverse limit spaces with a non-recurrent critical point, Topology Appl. 156 (2009) 2417

[11] K M Brucks, H Bruin, Subcontinua of inverse limit spaces of unimodal maps, Fund. Math. 160 (1999) 219

[12] K M Brucks, B Diamond, Monotonicity of auto-expansions, Phys. D 51 (1991) 39

[13] H Bruin, Planar embeddings of inverse limit spaces of unimodal maps, Topology Appl. 96 (1999) 191

[14] H Bruin, Inverse limit spaces of post-critically finite tent maps, Fund. Math. 165 (2000) 125

[15] H Bruin, Subcontinua of Fibonacci-like inverse limit spaces, Topology Proc. 31 (2007) 37

[16] A Douady, Topological entropy of unimodal maps: monotonicity for quadratic polynomials, from: "Real and complex dynamical systems" (editors B Branner, P Hjorth), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Kluwer Acad. Publ. (1995) 65

[17] M J Feigenbaum, Quantitative universality for a class of nonlinear transformations, J. Statist. Phys. 19 (1978) 25

[18] J Graczyk, G Światek, Generic hyperbolicity in the logistic family, Ann. of Math. 146 (1997) 1

[19] J Guckenheimer, Sensitive dependence to initial conditions for one-dimensional maps, Comm. Math. Phys. 70 (1979) 133

[20] F Hofbauer, P Raith, K Simon, Hausdorff dimension for some hyperbolic attractors with overlaps and without finite Markov partition, Ergodic Theory Dynam. Systems 27 (2007) 1143

[21] W T Ingram, Inverse limits on $[0,1]$ using tent maps and certain other piecewise linear bonding maps, from: "Continua" (editors H Cook, W T Ingram, K T Kuperberg, A Lelek, P Minc), Lecture Notes in Pure and Appl. Math. 170, Dekker (1995) 253

[22] L Kailhofer, A classification of inverse limit spaces of tent maps with periodic critical points, Fund. Math. 177 (2003) 95

[23] M Lyubich, Dynamics of quadratic polynomials. I, II, Acta Math. 178 (1997) 185, 247

[24] W De Melo, S Van Strien, One-dimensional dynamics, Ergeb. Math. Grenzgeb. 25, Springer (1993)

[25] J Milnor, W Thurston, On iterated maps of the interval, from: "Dynamical systems" (editor J C Alexander), Lecture Notes in Math. 1342, Springer (1988) 465

[26] M Misiurewicz, Embedding inverse limits of interval maps as attractors, Fund. Math. 125 (1985) 23

[27] M Misiurewicz, W Szlenk, Entropy of piecewise monotone mappings, Studia Math. 67 (1980) 45

[28] B E Raines, Inhomogeneities in non-hyperbolic one-dimensional invariant sets, Fund. Math. 182 (2004) 241

[29] B E Raines, S Štimac, A classification of inverse limit spaces of tent maps with a nonrecurrent critical point, Algebr. Geom. Topol. 9 (2009) 1049

[30] D Singer, Stable orbits and bifurcation of maps of the interval, SIAM J. Appl. Math. 35 (1978) 260

[31] S Štimac, A classification of inverse limit spaces of tent maps with finite critical orbit, Topology Appl. 154 (2007) 2265

[32] R Swanson, H Volkmer, Invariants of weak equivalence in primitive matrices, Ergodic Theory Dynam. Systems 20 (2000) 611

[33] W Szczechla, Inverse limits of certain interval mappings as attractors in two dimensions, Fund. Math. 133 (1989) 1

[34] C Tresser, P Coullet, Itérations d'endomorphismes et groupe de renormalisation, C. R. Acad. Sci. Paris Sér. A-B 287 (1978)

[35] R F Williams, One-dimensional non-wandering sets, Topology 6 (1967) 473

[36] R F Williams, Classification of one dimensional attractors, from: "Global Analysis", Proc. Symp. Pure Math. 14, Amer. Math. Soc. (1970) 341