Indecomposable continua with one and two composants
Induced almost continuous functions on hyperspaces
For a metric continuum X, let C(X) (resp., ) be the hyperspace of subcontinua (resp., nonempty closed subsets) of X. Let f: X → Y be an almost continuous function. Let C(f): C(X) → C(Y) and be the induced functions given by and . In this paper, we prove that: • If is almost continuous, then f is continuous. • If C(f) is almost continuous and X is locally connected, then f is continuous. • If X is not locally connected, then there exists an almost continuous function f: X → [0,1] such that...
Inhomogeneities in non-hyperbolic one-dimensional invariant sets
The topology of one-dimensional invariant sets (attractors) is of great interest. R. F. Williams [20] demonstrated that hyperbolic one-dimensional non-wandering sets can be represented as inverse limits of graphs with bonding maps that satisfy certain strong dynamical properties. These spaces have "homogeneous neighborhoods" in the sense that small open sets are homeomorphic to the product of a Cantor set and an arc. In this paper we examine inverse limits of graphs with more complicated bonding...
Inverse Limit Spaces Satisfying a Poincaré Inequality
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.
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 smooth continua
Inverse limits of tentlike maps on trees
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
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, = 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
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 and indecomposability in inverse limits
Irreducibility of inverse limits on intervals
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.
Irreducible continua of higher dimension
Irreducible continua with degenerate end-tranches and arcwise accessibility in hyperspaces
Irreducible mapping and the lightness of open mappings.
Irreducible mappings and HU-terminal continua.
Irreducibly confluent mappings
Krasinkiewicz maps from compacta to polyhedra
We prove that the set of all Krasinkiewicz maps from a compact metric space to a polyhedron (or a 1-dimensional locally connected continuum, or an n-dimensional Menger manifold, n ≥ 1) is a dense -subset of the space of all maps. We also investigate the existence of surjective Krasinkiewicz maps from continua to polyhedra.
Large free set
Lelek fan from a projective Fraïssé limit
We show that a natural quotient of the projective Fraïssé limit of a family that consists of finite rooted trees is the Lelek fan. Using this construction, we study properties of the Lelek fan and of its homeomorphism group. We show that the Lelek fan is projectively universal and projectively ultrahomogeneous in the class of smooth fans. We further show that the homeomorphism group of the Lelek fan is totally disconnected, generated by every neighbourhood of the identity, has a dense conjugacy...
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