Inverse limits on intervals using unimodal bonding maps having only periodic points whose periods are all the powers of two

• Autorzy: W. T. Ingram , Robert Roe
• Języki publikacji: EN

Abstrakty

EN

We derive several properties of unimodal maps having only periodic points whose period is a power of 2. We then consider inverse limits on intervals using a single strongly unimodal bonding map having periodic points whose only periods are all the powers of 2. One such mapping is the logistic map, $f_λ(x)$ = 4λx(1-x) on [f(λ),λ], at the Feigenbaum limit, λ ≈ 0.89249. It is known that this map produces an hereditarily decomposable inverse limit with only three topologically different subcontinua. Other examples of such maps are given and it is shown that any two strongly unimodal maps with periodic point whose only periods are all the powers of 2 produce homeomorphic inverse limits whenever each map has the additional property that the critical point lies in the closure of the orbit of the right endpoint of the interval.

Słowa kluczowe

EN

hereditarily decomposable continuum; logistic mapping; inverse limit

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 1999
• Tom: 81
• Numer: 1
• Strony: 51-61

Daty

wydano

1999

otrzymano

1998-08-27

poprawiono

1998-12-21

Twórcy

autor

W. T. Ingram

• Department of Mathematics and Statistics, University of Missouri-Rolla, Rolla, MO 65401, U.S.A.

autor

Robert Roe

• Department of Mathematics and Statistics, University of Missouri-Rolla, Rolla, MO 65401, U.S.A.

Bibliografia

• [1] M. Barge and W. T. Ingram, Inverse limits on [0,1] using logistic maps as bonding maps, Topology Appl. 72 (1996), 159-172.
• [2] P. Collet and J.-P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Birkhäuser, Basel, 1980.
• [3] R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Benjamin, Menlo Park, 1986.
• [4] W. T. Ingram, Periodicity and indecomposability, Proc. Amer. Math. Soc. 123 (1995), 1907-1916.
• [5] Z. Nitecki, Topological dynamics on the interval, in: Ergodic Theory and Dynamical Systems II, A. Katok (ed.), Birkhäuser, Boston, 1982, 1-73.