×

The Ingram conjecture. (English) Zbl 1285.54030

Continua as inverse limits have been studied for a long time. One reason for such intense research in this area is the fact that inverse sequences with very simple spaces and simple bonding maps can produce as inverse limits extremely complicated continua. This may happen even in the case where all the spaces are unit intervals \([0,1]\) and all the bonding maps are the same. The famous Brouwer-Janiszewski-Knaster continuum, that can be presented as the inverse limit of the inverse sequence of closed unit intervals \([0,1]\) with the standard tent bonding map \(T_{2}:[0,1]\to [0,1]\), \(T_2(t)=\min\{2t,2(1-t)\}\) is an example. One of the most famous problems in the area is the one about classifying the inverse limits of closed unit intervals \([0,1]\) with a single tent map \(T_s\), \(1\leq s\leq 2\), \(T_s(t)=\min\{st,s(1-t)\}\), on \([0,1]\) as the bonding map.
In 1991, W.T. Ingram stated the following conjecture for tent maps:
If \(1\leq s<s'\leq 2\), then the corresponding inverse limit spaces \(\displaystyle \lim_{\leftarrow}([0,1],T_s)\) and \(\displaystyle \lim_{\leftarrow}([0,1],T_{s'})\) are nonhomeomorphic.
This conjecture became known as the Ingram conjecture and it has received a large amount of attention in the last 23 years. It generated a large number of articles, in which certain special cases of the conjecture were proved. Finally, the conjecture was proved in 2009 by the authors of the paper under review. Their proof is presented in this paper.
The following is the main result of the paper.
{ Theorem 1.1.} If \(1\leq s<s'\leq 2\), then the corresponding inverse limit spaces \(\displaystyle \lim_{\leftarrow}([0,1],T_s)\) and \(\displaystyle \lim_{\leftarrow}([0,1],T_{s'})\) are nonhomeomorphic.

MSC:

54H20 Topological dynamics (MSC2010)
37B45 Continua theory in dynamics
37E05 Dynamical systems involving maps of the interval
54F15 Continua and generalizations
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] J E Anderson, I F Putnam, Topological invariants for substitution tilings and their associated \(C^*\)-algebras, Ergodic Theory Dynam. Systems 18 (1998) 509 · Zbl 1053.46520
[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 · Zbl 0917.54041
[3] M Barge, B Diamond, Homeomorphisms of inverse limit spaces of one-dimensional maps, Fund. Math. 146 (1995) 171 · Zbl 0851.54037
[4] M Barge, B Diamond, Inverse limit spaces of infinitely renormalizable maps, Topology Appl. 83 (1998) 103 · Zbl 0967.54031
[5] M Barge, B Diamond, Subcontinua of the closure of the unstable manifold at a homoclinic tangency, Ergodic Theory Dynam. Systems 19 (1999) 289 · Zbl 1073.37504
[6] M Barge, S Holte, Nearly one-dimensional Hénon attractors and inverse limits, Nonlinearity 8 (1995) 29 · Zbl 0813.58015
[7] M Barge, W T Ingram, Inverse limits on \([0,1]\) using logistic bonding maps, Topology Appl. 72 (1996) 159 · Zbl 0859.54030
[8] L Block, S Jakimovik, L Kailhofer, J Keesling, On the classification of inverse limits of tent maps, Fund. Math. 187 (2005) 171 · Zbl 1092.54011
[9] L Block, S Jakimovik, J Keesling, On Ingram’s conjecture, Topology Proc. 30 (2006) 95 · Zbl 1135.54016
[10] L Block, J Keesling, B E Raines, S \vStimac, Homeomorphisms of unimodal inverse limit spaces with a non-recurrent critical point, Topology Appl. 156 (2009) 2417 · Zbl 1181.37015
[11] K M Brucks, H Bruin, Subcontinua of inverse limit spaces of unimodal maps, Fund. Math. 160 (1999) 219 · Zbl 0953.54032
[12] K M Brucks, B Diamond, Monotonicity of auto-expansions, Phys. D 51 (1991) 39 · Zbl 0752.41030
[13] H Bruin, Planar embeddings of inverse limit spaces of unimodal maps, Topology Appl. 96 (1999) 191 · Zbl 0954.54019
[14] H Bruin, Inverse limit spaces of post-critically finite tent maps, Fund. Math. 165 (2000) 125 · Zbl 0973.37011
[15] H Bruin, Subcontinua of Fibonacci-like inverse limit spaces, Topology Proc. 31 (2007) 37 · Zbl 1133.37005
[16] A Douady, Topological entropy of unimodal maps: monotonicity for quadratic polynomials (editors B Branner, P Hjorth), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Kluwer Acad. Publ. (1995) 65 · Zbl 0923.58018
[17] M J Feigenbaum, Quantitative universality for a class of nonlinear transformations, J. Statist. Phys. 19 (1978) 25 · Zbl 0509.58037
[18] J Graczyk, G Światek, Generic hyperbolicity in the logistic family, Ann. of Math. 146 (1997) 1 · Zbl 0936.37015
[19] J Guckenheimer, Sensitive dependence to initial conditions for one-dimensional maps, Comm. Math. Phys. 70 (1979) 133 · Zbl 0429.58012
[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 · Zbl 1131.37031
[21] W T Ingram, Inverse limits on \([0,1]\) using tent maps and certain other piecewise linear bonding maps (editors H Cook, W T Ingram, K T Kuperberg, A Lelek, P Minc), Lecture Notes in Pure and Appl. Math. 170, Dekker (1995) 253 · Zbl 0819.54029
[22] L Kailhofer, A classification of inverse limit spaces of tent maps with periodic critical points, Fund. Math. 177 (2003) 95 · Zbl 1028.54038
[23] M Lyubich, Dynamics of quadratic polynomials. I, II, Acta Math. 178 (1997) 185, 247 · Zbl 0908.58053
[24] W de Melo, S van Strien, One-dimensional dynamics, Ergeb. Math. Grenzgeb. 25, Springer (1993) · Zbl 0791.58003
[25] J Milnor, W Thurston, On iterated maps of the interval (editor J C Alexander), Lecture Notes in Math. 1342, Springer (1988) 465 · Zbl 0664.58015
[26] M Misiurewicz, Embedding inverse limits of interval maps as attractors, Fund. Math. 125 (1985) 23 · Zbl 0587.58032
[27] M Misiurewicz, W Szlenk, Entropy of piecewise monotone mappings, Studia Math. 67 (1980) 45 · Zbl 0445.54007
[28] B E Raines, Inhomogeneities in non-hyperbolic one-dimensional invariant sets, Fund. Math. 182 (2004) 241 · Zbl 1053.37005
[29] B E Raines, S \vStimac, A classification of inverse limit spaces of tent maps with a nonrecurrent critical point, Algebr. Geom. Topol. 9 (2009) 1049 · Zbl 1231.37009
[30] D Singer, Stable orbits and bifurcation of maps of the interval, SIAM J. Appl. Math. 35 (1978) 260 · Zbl 0391.58014
[31] S \vStimac, A classification of inverse limit spaces of tent maps with finite critical orbit, Topology Appl. 154 (2007) 2265 · Zbl 1115.37043
[32] R Swanson, H Volkmer, Invariants of weak equivalence in primitive matrices, Ergodic Theory Dynam. Systems 20 (2000) 611 · Zbl 0984.37019
[33] W Szczechla, Inverse limits of certain interval mappings as attractors in two dimensions, Fund. Math. 133 (1989) 1 · Zbl 0708.58012
[34] C Tresser, P Coullet, Itérations d’endomorphismes et groupe de renormalisation, C. R. Acad. Sci. Paris Sér. A-B 287 (1978) · Zbl 0402.54046
[35] R F Williams, One-dimensional non-wandering sets, Topology 6 (1967) 473 · Zbl 0159.53702
[36] R F Williams, Classification of one dimensional attractors, Proc. Symp. Pure Math. 14, Amer. Math. Soc. (1970) 341 · Zbl 0213.50401
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.