# 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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topIngram, 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

top- [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] P. Collet and J.-P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Birkhäuser, Basel, 1980. Zbl0458.58002
- [3] R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Benjamin, Menlo Park, 1986. Zbl0632.58005
- [4] W. T. Ingram, Periodicity and indecomposability, Proc. Amer. Math. Soc. 123 (1995), 1907-1916. Zbl0851.54036
- [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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