Cardinal invariants of monotone and porous sets

2012 ◽  
Vol 77 (1) ◽  
pp. 159-173 ◽  
Author(s):  
Michael Hrušák ◽  
Ondřej Zindulka
Keyword(s):  
Metric Space ◽  
Linear Order ◽  
Cantor Set ◽  
Strong Connection ◽  
Cardinal Invariants ◽  
Porous Sets

AbstractA metric space (X, d) ismonotoneif there is a linear order < onXand a constantcsuch thatd(x, y)≤c d(x, z)for allx<y<zinX. We investigate cardinal invariants of theσ-idealMongenerated by monotone subsets of the plane. Since there is a strong connection between monotone sets in the plane and porous subsets of the line, plane and the Cantor set, cardinal invariants of these ideals are also investigated. In particular, we show that non(Mon) ≥mσ-linked, but non(Mon) <mσ-centeredis consistent. Also cov(Mon) <cand cof (N) < cov(Mon) are consistent.

scholarly journals Cardinal invariants of strongly porous sets

2017 ◽  
pp. 1-16
Author(s):  
Osvaldo Guzmán ◽  
Michael Hrušák ◽  
Arturo Martinez-Celis
Keyword(s):  
Cardinal Invariants ◽  
Porous Sets

On the comparison of measure functions

1975 ◽  
Vol 78 (3) ◽  
pp. 483-491 ◽  
Author(s):  
John Hawkes
Keyword(s):  
Metric Space ◽  
Real Line ◽  
Short Proof ◽  
Cartesian Product ◽  
Cantor Set ◽  
Partial Answer ◽  
Hausdorff Measures ◽  
Cartesian Product Sets ◽  
Image Position ◽  
Product Sets

Suppose that f and g are measure functions and that µf and µg are the corresponding Hausdorff measures. We are interested in how relationships between f and g are reflected in relationships between µf and µg and vice versa. For example, ifRogers ((8), p. 80) has shown that there exists a metric space Ω which has subsets A and B with µf(A) = µg(B) = 0 and, µf(B) = µg(A) = ∞. We are more interested in what happens when the metric space in question is the real line. To get the above result we need to assume that both f and g are starshaped. This is just about a best possible result since if we assume only that f is a power of t the conclusion fails. Further-more we show that there exist measure functions f and g satisfying (1) such that µf and, µg are identical. We also consider related questions: when are, µf and, µg equivalent measures and when are they identical?The proofs depend on the construction in Theorem 3.1 of a Cantor set having pre-scribed measure properties. Although the construction is not difficult it turns out to be quite useful to appeal to the existence of such a set. We illustrate this remark by giving a short proof of a known theorem on cartesian product sets. We also make use of these ideas in section 5 where we discuss some properties of a class of net meastues and give a partial answer to a problem posed by Billingsley.

Homeomorphisms with the whole compacta being scrambled sets

2001 ◽  
Vol 21 (1) ◽  
pp. 77-91 ◽  
Author(s):  
WEN HUANG ◽  
XIANGDONG YE
Keyword(s):  
Metric Space ◽  
Arbitrary Dimension ◽  
Cantor Set ◽  
Basic Properties

A homeomorphism on a metric space (X,d) is completely scrambled if for each x\not= y\in X, \lim sup_{n\longrightarrow +\infty} d(f^n(x),f^n(y))>0 and \lim inf_{n\ longrightarrow +\infty}d(f^n(x),f^n(y))=0. We study the basic properties of completely scrambled homeomorphisms on compacta and show that there are ‘many’ compacta admitting completely scrambled homeomorphisms, which include some countable compacta (we give a characterization), the Cantor set and continua of arbitrary dimension.

scholarly journals Dynamics of induced systems

2016 ◽  
Vol 37 (7) ◽  
pp. 2034-2059 ◽  
Author(s):  
ETHAN AKIN ◽  
JOSEPH AUSLANDER ◽  
ANIMA NAGAR
Keyword(s):  
Phase Space ◽  
Uniform Convergence ◽  
Metric Space ◽  
Cantor Set ◽  
Main Theme ◽  
Hausdorff Topology ◽  
Continuous Maps ◽  
Dynamical Properties ◽  
Compact Subsets

In this paper we study the dynamical properties of actions on the space of compact subsets of the phase space. More precisely, if$X$is a metric space, let$2^{X}$denote the space of non-empty compact subsets of$X$provided with the Hausdorff topology. If$f$is a continuous self-map on$X$, there is a naturally induced continuous self-map$f_{\ast }$on$2^{X}$. Our main theme is the interrelation between the dynamics of$f$and$f_{\ast }$. For such a study, it is useful to consider the space${\mathcal{C}}(K,X)$of continuous maps from a Cantor set$K$to$X$provided with the topology of uniform convergence, and$f_{\ast }$induced on${\mathcal{C}}(K,X)$by composition of maps. We mainly study the properties of transitive points of the induced system$(2^{X},f_{\ast })$both topologically and dynamically, and give some examples. We also look into some more properties of the system$(2^{X},f_{\ast })$.

Perfect Images of Zero-Dimensional Separable Metric Spaces

1982 ◽  
Vol 25 (1) ◽  
pp. 41-47 ◽  
Author(s):  
Jan Van Mill ◽  
R. Grant Woods
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Dense Subset ◽  
Cantor Set ◽  
Locally Compact ◽  
Compact Metric Space ◽  
Continuous Surjection

AbstractLet Q denote the rationals, P the irrationals, C the Cantor set and L the space C − {p} (where p ∈ C). Let f : X → Y be a perfect continuous surjection. We show: (1) If X ∈ {Q, P, Q × P}, or if f is irreducible and X ∈ {C, L}, then Y is homeomorphic to X if Y is zero-dimensional. (2) If X ∈ {P, C, L} and f is irreducible, then there is a dense subset S of Y such that f|f ← [S] is a homeomorphism onto S. However, if Z is any σ-compact nowhere locally compact metric space then there is a perfect irreducible continuous surjection from Q × C onto Z such that each fibre of the map is homeomorphic to C.

scholarly journals A theorem on homeomorphism groups and products of spaces

1969 ◽  
Vol 1 (1) ◽  
pp. 137-141 ◽  
Author(s):  
A. R. Vobach
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Structure Theorem ◽  
Cantor Set ◽  
Compact Metric Space ◽  
Compact Metric Spaces ◽  
Group Of Homeomorphisms ◽  
Homeomorphism Groups

Let H(C) be the group of homeomorphisms of the Cantor set, C, onto itself. Let p: C → M be a map of C onto a compact metric space M, and let G(p, M) be is a group.The map p: C → M is standard, if for each (x, y) ∈ C × C such that p(x) = p(y), there is a sequence and a sequence such that xn → x and hn (xn) → y Standard maps and their associated groups characterize compact metric spaces in the sense that: Two such spaces, M and N, are homeomorphic if and only if, given p standard from C onto M, there is a standard q from C onto N for which G(p, M) = h−1G(q, N)h, for some h ∈ H(C) The present paper exhibits a structure theorem connecting these characterizing subgroups of H(C) and products of spaces: Let M1 and M2 be compact metric spaces. Then there are standard maps p: C → M1 × M2 and pi: C → Mi, i = 1, 2, such that G(p, M1 × M2) = G(p1, M1) ∩ G(p2, M2).

One point intersections of middle-α Cantor sets

1994 ◽  
Vol 14 (3) ◽  
pp. 537-549
Author(s):  
Roger L. Kraft
Keyword(s):  
Equivalence Relation ◽  
Linear Order ◽  
Single Point ◽  
Order Relation ◽  
Cantor Set ◽  
Equivalence Classes ◽  
Cantor Sets ◽  
Invariant Subset ◽  
The One

AbstractFor α ∈ (0, 1), let Γα denote the middle-α Cantor set contained in the interval [0,1]. Let denote the set of all t∈[−1,1] such that Γα ∩(Γα+t)is a single point. Both a geometric and a symbolic description of each is presented. The symbolic description of each will be as a shift invariant subset of the one-sided two shift that is determined by inequalities. An equivalence relation is defined on the one-sided two shift and then a linear order relation is defined on the equivalence classes. This order relation on the equivalence classes describes how the sets shrink as a decreases from ⅓ to 0.

Weakly Confluent Mappings and Atriodic Suslinian Curves

1978 ◽  
Vol 30 (01) ◽  
pp. 32-44 ◽  
Author(s):  
H. Cook ◽  
A. Lelek
Keyword(s):  
Continuous Function ◽  
Branch Point ◽  
Metric Space ◽  
Related Result ◽  
Cantor Set ◽  
Topological Spaces ◽  
Compact Metric Space ◽  
Covering Theorem ◽  
The Given

There are theorems in which some classes of topological spaces are characterized by means of properties of mappings of these spaces into a single space. For example, it is well known that a compactum X is at most n-dimensional if and only if no mapping of X irto an (n + l)-cube has a stable value [5, Theorems VI. 1-2, pp. 75-77]. Also, a curve X is tree-like if and only if no mapping of X into a figure eight is homotopically essential [1, Theorem 1, pp. 74-75; 8, p. 91]. By a curve we mean any at most 1-dimensional continuum; a continuum is a connected compactum; a compactum is a compact metric space, and a mapping is a continuous function. The aim of the present paper is to prove another theorem of this type. We distinguish a class of curves and show that it is characterized by imposing the condition that no weakly confluent mapping [13] can transform the given curve onto a simple triod (see 2.4). A related result is applied to a generalized branch-point covering theorem (see 3.2). In addition, two results are obtained in which we establish some characterizations of weakly confluent images and preimages of the product of the Cantor set and an arc (see 1.1 and 2.2). Continua that are such images turn out to be identical with regular curves (see 1.3).

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