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Inverse Limit Spaces Satisfying a Poincaré Inequality

Jeff Cheeger, Bruce Kleiner (2015)

Analysis and Geometry in Metric Spaces

We give conditions on Gromov-Hausdorff convergent inverse systems of metric measure graphs which imply that the measured Gromov-Hausdorff limit (equivalently, the inverse limit) is a PI space i.e., it satisfies a doubling condition and a Poincaré inequality in the sense of Heinonen-Koskela [12]. The Poincaré inequality is actually of type (1, 1). We also give a systematic construction of examples for which our conditions are satisfied. Included are known examples of PI spaces, such as Laakso spaces,...

Inverse Limits, Economics, and Backward Dynamics.

Judy Kennedy (2008)

RACSAM

We survey recent papers on the problem of backward dynamics in economics, providing along the way a glimpse at the economics perspective, a discussion of the economic models and mathematical tools involved, and a list of applicable literature in both mathematics and economics.

Inverse limits of tentlike maps on trees

Stewart Baldwin (2010)

Fundamenta Mathematicae

We investigate generalizations of Ingram's Conjecture involving maps on trees. We show that for a class of tentlike maps on the k-star with periodic critical orbit, different maps in the class have distinct inverse limit spaces. We do this by showing that such maps satisfy the conclusion of the Pseudo-isotopy Conjecture, i.e., if h is a homeomorphism of the inverse limit space, then there is an integer N such that h and σ̂^N switch composants in the same way, where σ̂ is the standard shift map of...

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

W. Ingram, Robert Roe (1999)

Colloquium Mathematicae

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...

Inverse sequences with proper bonding maps

Tomás Fernández-Bayort, Antonio Quintero (2010)

Colloquium Mathematicae

Some topological properties of inverse limits of sequences with proper bonding maps are studied. We show that (non-empty) limits of euclidean half-lines are one-ended generalized continua. We also prove the non-existence of a universal object for such limits with respect to closed embeddings. A further result states that limits of end-preserving sequences of euclidean lines are two-ended generalized continua.

Irreducibility of inverse limits on intervals

David Ryden (2000)

Fundamenta Mathematicae

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.

Isolated points and redundancy

P. Alirio J. Peña, Jorge E. Vielma (2011)

Commentationes Mathematicae Universitatis Carolinae

We describe the isolated points of an arbitrary topological space ( X , τ ) . If the τ -specialization pre-order on X has enough maximal elements, then a point x X is an isolated point in ( X , τ ) if and only if x is both an isolated point in the subspaces of τ -kerneled points of X and in the τ -closure of { x } (a special case of this result is proved in Mehrvarz A.A., Samei K., On commutative Gelfand rings, J. Sci. Islam. Repub. Iran 10 (1999), no. 3, 193–196). This result is applied to an arbitrary subspace of the prime...

I-weight of compact and locally compact LOTS

Brad Bailey (2007)

Commentationes Mathematicae Universitatis Carolinae

Ram’ırez-Páramo proved that under GCH for the class of compact Hausdorff spaces i-weight reflects all cardinals [A reflection theorem for i-weight, Topology Proc. 28 (2004), no. 1, 277–281]. We show that in ZFC i-weight reflects all cardinals for the class of compact LOTS. We define local i-weight, then calculate i-weight of locally compact LOTS and paracompact spaces in terms of the extent of the space and the i-weight of open sets or the local i-weight. For locally compact LOTS we find a necessary...

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