Fibonacci-like unimodal inverse limit spaces and the core Ingram conjecture

Henk Bruin, Sonja Štimac



We study the structure of inverse limit space of so-called Fibonacci-like tent maps. The combinatorial constraints implied by the Fibonacci-like assumption allow us to introduce certain chains that enable a more detailed analysis of symmetric arcs within this space than is possible in the general case. We show that link-symmetric arcs are always symmetric or a well-understood concatenation of quasi-symmetric arcs. This leads to the proof of the Ingram Conjecture for cores of Fibonacci-like unimodal inverse limits.


Tent map; inverse limit space; Fibonacci unimodal map; structure of inverse limit spaces

Full Text:



A. Avila, M. Lyubich and W. de Melo, Regular or stochastic dynamics in real analytic families of unimodal maps, Invent. Math. 154 (2003), 451–550.

M. Barge, K. Brucks and B. Diamond, Self-similarity in inverse limit spaces of the tent family, Proc. Amer. Math. Soc. 124 (1996), 3563–3570.

M. Barge, H. Bruin and S. Stimac, The Ingram Conjecture, Geom. Topol. 16 (2012), 2481–2516.

L. Block, S. Jakimovik, J. Keesling and L. Kailhofer, On the classification of inverse limits of tent maps, Fund. Math. 187 (2005), 171–192.

K. Brucks and H. Bruin, Subcontinua of inverse limit spaces of unimodal maps, Fund. Math. 160 (1999), 219–246.

K. Brucks and H. Bruin, Topics in one-dimensional dynamics, London Math. Soc. Student Texts 62, Cambridge University Press 2004.

H. Bruin, Combinatorics of the kneading map, Int. Jour. of Bifur. and Chaos 5 (1995), 1339–1349.

H. Bruin, Topological conditions for the existence of absorbing Cantor sets, Trans. Amer. Math. Soc. 350 (1998), 2229–2263.

H. Bruin, Quasi-symmetry of conjugacies between interval maps, Nonlinearity 9 (1996), 1191–1207.

H. Bruin, Subcontinua of Fibonacci-like unimodal inverse limit spaces, Topology Proc. 31 (2007), 37–50.

H. Bruin, (Non)invertibility of Fibonacci-like unimodal maps restricted to their critical omega-limit sets, Topology Proc. 37 (2011), 459–480.

H. Bruin, G. Keller, T. Nowicki and S. van Strien, Wild Cantor Attractors exist, Ann. Math. (2) 143 (1996), 97–130.

H. Bruin, G. Keller and M. St.-Pierre, Adding machines and wild attractors, Ergodic Theory Dynam. Systems 17 (1997), 1267–1287.

F. Hofbauer, The topological entropy of a transformation x mapsto ax(1-x), Monath. Math. 90 (1980), 117–141.

F. Hofbauer and G. Keller, Some remarks on recent results about S-unimodal maps, Ann. Inst. Henri Poincar´e, Physique th´eorique, 53 (1990), 413–425.

L. Kailhofer, A classification of inverse limit spaces of tent maps with periodic critical points, Fund. Math. 177 (2003), 95–120.

G. Keller and T. Nowicki, Fibonacci maps re(al)-visited, Ergodic Theory Dynam. Systems 15 (1995), 99–120.

M. Lyubich, Combinatorics, geometry and attractors of quasi-quadratic maps, Ann. Math. (2) 140 (1994), 347–404.

M. Lyubich and J. Milnor, The Fibonacci unimodal map, J. Amer.Math. Soc. 6 (1993), 425–457.

J. Milnor, On the concept of attractor, Commun. Math. Phys. 99 (1985), 177–195.

B. Raines and S. Stimac, A classification of inverse limit spaces of tent maps with non-recurrent critical point, Algebr. Geom. Topol. 9 (2009), 1049–1088.

S. Stimac, Structure of inverse limit spaces of tent maps with finite critical orbit, Fund. Math. 191 (2006), 125–150.

S. Stimac, A classification of inverse limit spaces of tent maps with finite critical orbit, Topology Appl. 154 (2007), 2265–2281.


  • There are currently no refbacks.

Partnerzy platformy czasopism