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

Colloquium Mathematicae (1999)

• Volume: 81, Issue: 1, page 51-61
• ISSN: 0010-1354

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## Abstract

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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, ${f}_{\lambda }\left(x\right)$ = 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.

## How to cite

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Ingram, W., and Roe, Robert. "Inverse limits on intervals using unimodal bonding maps having only periodic points whose periods are all the powers of two." Colloquium Mathematicae 81.1 (1999): 51-61. <http://eudml.org/doc/210729>.

@article{Ingram1999,
abstract = {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.},
author = {Ingram, W., Roe, Robert},
journal = {Colloquium Mathematicae},
keywords = {hereditarily decomposable continuum; logistic mapping; inverse limit; logistic map; Feigenbaum limit; hereditarily decomposable inverse limit},
language = {eng},
number = {1},
pages = {51-61},
title = {Inverse limits on intervals using unimodal bonding maps having only periodic points whose periods are all the powers of two},
url = {http://eudml.org/doc/210729},
volume = {81},
year = {1999},
}

TY - JOUR
AU - Ingram, W.
AU - Roe, Robert
TI - Inverse limits on intervals using unimodal bonding maps having only periodic points whose periods are all the powers of two
JO - Colloquium Mathematicae
PY - 1999
VL - 81
IS - 1
SP - 51
EP - 61
AB - 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.
LA - eng
KW - hereditarily decomposable continuum; logistic mapping; inverse limit; logistic map; Feigenbaum limit; hereditarily decomposable inverse limit
UR - http://eudml.org/doc/210729
ER -

## References

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1. [1] M. Barge and W. T. Ingram, Inverse limits on [0,1] using logistic maps as bonding maps, Topology Appl. 72 (1996), 159-172. Zbl0859.54030
2. [2] P. Collet and J.-P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Birkhäuser, Basel, 1980. Zbl0458.58002
3. [3] R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Benjamin, Menlo Park, 1986. Zbl0632.58005
4. [4] W. T. Ingram, Periodicity and indecomposability, Proc. Amer. Math. Soc. 123 (1995), 1907-1916. Zbl0851.54036
5. [5] Z. Nitecki, Topological dynamics on the interval, in: Ergodic Theory and Dynamical Systems II, A. Katok (ed.), Birkhäuser, Boston, 1982, 1-73.

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