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

100%

Ingram W. T., Roe R.

Colloquium Mathematicum

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1999

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tom 81

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nr 1

51-61

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.

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On approximate inverse systems and resolutions

75%

Mardešić S.

Fundamenta Mathematicae

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1993

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tom 142

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nr 3

241-255

EN

Recently, L. R. Rubin, T. Watanabe and the author have introduced approximate inverse systems and approximate resolutions, a new tool designed to study topological spaces. These systems differ from the usual inverse systems in that the bonding maps $p_{aa'}$ are not subject to the commutativity requirement $p_{aa' p_{a'a''} = p_{aa''}, a ≤ a' ≤ a''$. Instead, the mappings $p_{aa'}p_{a'a''}$ and $p_{aa''}$ are allowed to differ in a way controlled by coverings $U_a$, called meshes, which are associated with the members $X_a$ of the system. The purpose of this paper is to consider a more general and much simpler notion of approximate system and approximate resolution, which does not require meshes. The main result is a construction which associates with any approximate resolution in the new sense an approximate resolution in the previous sense in such a way that previously obtained results remain valid in the present more general setting.

3

Irreducibility of inverse limits on intervals

63%

Ryden D. J.

Fundamenta Mathematicae

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2000

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tom 165

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nr 1

29-53

EN

A procedure for obtaining points of irreducibility for an inverse limit on intervals is developed. In connection with this, the following are included. A semiatriodic continuum is defined to be a continuum that contains no triod with interior. Characterizations of semiatriodic and unicoherent continua are given, as well as necessary and sufficient conditions for a subcontinuum of a semiatriodic and unicoherent continuum M to lie within the interior of a proper subcontinuum of M.

4

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Hyperspace retractions for curves

53%

Charatonik J. J., Charatonik W. J., Omiljanowski K., Prajs J. R.

Instytut Matematyczny Polskiej Akademii Nauk

EN

CONTENTS 1. Introduction...................................5 2. Preliminaries.................................7 3. Hyperspace retractions.................9 4. Applications to selections............15 5. Applications to means.................18 References.....................................32

Ograniczanie wyników

2 Fundamenta Mathematicae

1 Colloquium Mathematicum

1 Charatonik J. J.

1 Charatonik W. J.

1 Ingram W. T.

1 Mardešić S.

1 Omiljanowski K.

1 Prajs J. R.

1 Roe R.

1 Ryden D. J.

1 2000

1 1999

1 1997

1 1993

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

On approximate inverse systems and resolutions

Irreducibility of inverse limits on intervals

Hyperspace retractions for curves