28 research outputs found
A CONTINUUM-CHAINABLE APOSYNDETIC PLANE CONTINUUM
No full textABSTRACT. A continuum is said to be conthzuum-chainable provided, fbr each pair x,y of points and each e> 0, there exists a chain of subcontinua from x to y such that each link has diameter less than e. It is known that continuum-chainability and arcwise connectivity are equivalent in each of the following classes of continua: non-separating plane continua, continua irreducible about a finite set, atriodic continua and hereditarily unicoherent continua. In this paper, we construct a continuum-chainable aposyndetic plane continuum which is not arcwise connected. This answers a question by C. L. Hagopian and L. E. Rogers. I. Definitions and preliminary remarks. By a continuum we mean a compact connected metric space. A finite collection {C 1,C2,...,C n} of sets is a chain from x to y provided x belongs to C 1, y belongs to C n and, for i,j G ( 1....,n}, C i N Cj 4:0 if and only if li-jl •< 1. Let e 2> 0. By an e-continut•m-chain we mean a chain such that each link Cj is a subcontinuum of M with diameter less than e. Thus a continuum M is continuum chainable if and only if, for each pair of points x,y G M and each e 2> 0, there exists an e-continuum-chain from x to y. Continumn-chainability is a generalization of property S (see [4], page 20) which is equivalent to local connectedness (ibidem, page 23) for continua. It was introduced by Ira Rosenholtz. A continuum M is aposyndetic at x with respect to y (see [3]) provided there exists a subcontinuum H of M such that x C H ¸ (the interior of H) and y • H. If M is aposyndetic at each x G M with respect to each y C M\{x}, then we say that M is aposyndetic. A continuuln M is said to be semi-aposyndetic if for each pair of distinct points x and y of M, M is aposyndetic either at x with respect to y or at y with respect to x. In [1] (the author is indebted to C. L. Hagopian for tiffs reference), Hagopian showed that each semi-aposyndetic E-continuum (i.e., a plane continuum such that for each e> 0, there are at most a finite nmnber of complementary domains of M o
Every contractible fan is locally connected at its vertex
No full textWe prove that each contractible fan is locally connected at its vertex. It follows that every contractible fan is embeddable in the plane. This gives a solution to a problem raised by J. J. Charatonik and C. A. Eberhart.</p
Open and monotone fixed point free maps on uniquely arcwise connected continua
No full textIn this note we will construct uniquely arcwise connected continua admitting open and monotone fixed point free mappings, respectively. We will also show that each locally one-to-one map on a uniquely arcwise connected continuum has a fixed point.</p
Centerlines of regions in the sphere
No full textAbstractGiven a region U in the 2-sphere S such that the boundary of U contains at least two points, let D(U) be the collection of open circular disks (called maximal disks) in U whose boundary meets the boundary of U in at least two points and let U2 be the collection of all regions U⊂S such that for each D∈D(U), D meets the boundary of U in at most two points. In this paper we study geometric properties of regions U∈U2. We show for such U that the centerline (i.e., the set of centers of maximal disks) is always a smooth, connected 1-manifold. We also show that the boundary of U has at most two components and, if it has exactly two components, then the boundary is locally connected.These results are related the set of points E(X,Y) which are equidistant to two disjoint closed sets X and Y. In particular we investigate when the equidistant set is a 1-manifold
How sticky is the chaos/order boundary?
No full textIn dynamical systems with divided phase space, the vicinity of the boundary between regular and chaotic regions is often “sticky,” that is, trapping orbits from the chaotic region for long times. Here, we investigate the stickiness in the simplest mushroom billiard, which has a smooth such boundary, but surprisingly subtle behaviour. As a measure of stickiness, we investigate P(t), the probability of remaining in the mushroom cap for at least time t given uniform initial conditions in the chaotic part of the cap. The stickiness is sensitively dependent on the radius of the stem r via the Diophantine properties of ρ = (2/π) arccos r. Almost all ρ give rise to families of marginally unstable periodic orbits (MUPOs) where P(t) ∼ C/t, dominating the stickiness of the boundary. Here we consider the case where ρ is MUPO-free and has continued fraction expansion with bounded partial quotients. We show that t^2 P(t) is bounded, varying infinitely often between values whose ratio is at least 32/27. When ρ has an eventually periodic continued fraction expansion, that is, a quadratic irrational, t^2 P(t) converges to a log-periodic function. In general, we expect less regular behaviour, with upper and lower exponents lying between 1 and 2. The results may shed light on the parameter dependence of boundary stickiness in annular billiards and generic area preserving maps.<br/
Recurrent homeomorphisms on 𝑅² are periodic
No full textA homeomorphism
f
:
(
X
,
d
)
→
(
X
,
d
)
f:(X,d) \to (X,d)
of a metric space
(
X
,
d
)
(X,d)
onto
X
X
is recurrent provided that for each
ε
>
0
\varepsilon > 0
there exists a positive integer
n
n
such that
f
n
{f^n}
is
ε
\varepsilon
-close to the identity map on
X
X
. The notion of a recurrent homeomorphism is weaker than that of an almost periodic homeomorphism. The result announced in the title generalizes the theorem of Brechner for almost periodic homeomorphisms and answers a question of R. D. Edwards.</p
Fixed-point-free maps on tree-like continua
No full textAbstractA fixed-point-free homeomorphism on a tree-like continuum is described. The tree-like continuum is obtained as inverse limit of trees, and the homeomorphism is obtained as an induced map of the inverse limit space
