Measurable distal and topological distal systems

1999 ◽  
Vol 19 (4) ◽  
pp. 1063-1076 ◽  
Author(s):  
ELON LINDENSTRAUSS
Keyword(s):  
Continuous Function ◽  
Compact Group ◽  
Metric Space ◽  
Measurable Function ◽  
Borel Measure ◽  
Base Space ◽  
Compact Metric Space ◽  
Full Support

In this paper we prove that any ergodic measurably distal system can be realized as a minimal topologically distal system with an invariant Borel measure of full support. The proof depends upon a theorem stating that every measurable function from a measurable system with its base space being a compact metric space to a connected compact group is cohomologous to a continuous function.

scholarly journals Semi-smooth Points in Some Classical Function Spaces

2021 ◽  
Vol 77 (1) ◽  
Author(s):  
Beata Derȩgowska ◽  
Beata Gryszka ◽  
Karol Gryszka ◽  
Paweł Wójcik
Keyword(s):  
Continuous Function ◽  
Metric Space ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compact Metric Space ◽  
Spaces Of Continuous Functions ◽  
Classical Function ◽  
Necessary And Sufficient ◽  
Vector Valued

AbstractThe investigations of the smooth points in the spaces of continuous function were started by Banach in 1932 considering function space $$\mathcal {C}(\Omega )$$ C ( Ω ) . Singer and Sundaresan extended the result of Banach to the space of vector valued continuous functions $$\mathcal {C}(\mathcal {T},E)$$ C ( T , E ) , where $$\mathcal {T}$$ T is a compact metric space. The aim of this paper is to present a description of semi-smooth points in spaces of continuous functions $$\mathcal {C}_0(\mathcal {T},E)$$ C 0 ( T , E ) (instead of smooth points). Moreover, we also find necessary and sufficient condition for semi-smoothness in the general case.

Non-Extendability of Bounded Continuous Functions

1980 ◽  
Vol 32 (4) ◽  
pp. 867-879
Author(s):  
Ronnie Levy
Keyword(s):  
Continuous Function ◽  
Metric Space ◽  
Dense Subset ◽  
Continuous Functions ◽  
Dense Subspace ◽  
Compact Metric Space ◽  
Bounded Continuous Function ◽  
Bounded Functions ◽  
Bounded Continuous Functions ◽  
Completely Metrizable

If X is a dense subspace of Y, much is known about the question of when every bounded continuous real-valued function on X extends to a continuous function on Y. Indeed, this is one of the central topics of [5]. In this paper we are interested in the opposite question: When are there continuous bounded real-valued functions on X which extend to no point of Y – X? (Of course, we cannot hope that every function on X fails to extend since the restrictions to X of continuous functions on Y extend to Y.) In this paper, we show that if Y is a compact metric space and if X is a dense subset of Y, then X admits a bounded continuous function which extends to no point of Y – X if and only if X is completely metrizable. We also show that for certain spaces Y and dense subsets X, the set of bounded functions on X which extend to a point of Y – X form a first category subset of C*(X).

Study on Strong Sensitivity of Systems Satisfying the Large Deviations Theorem

2021 ◽  
Vol 31 (10) ◽  
pp. 2150151
Author(s):  
Risong Li ◽  
Tianxiu Lu ◽  
Xiaofang Yang ◽  
Yongxi Jiang
Keyword(s):  
Probability Measure ◽  
Characteristic Function ◽  
Large Deviations ◽  
Open Subset ◽  
Metric Space ◽  
Borel Probability Measure ◽  
Compact Metric Space ◽  
Full Support ◽  
Upper Density ◽  
Borel Probability

Let [Formula: see text] be a nontrivial compact metric space with metric [Formula: see text] and [Formula: see text] be a continuous self-map, [Formula: see text] be the sigma-algebra of Borel subsets of [Formula: see text], and [Formula: see text] be a Borel probability measure on [Formula: see text] with [Formula: see text] for any open subset [Formula: see text] of [Formula: see text]. This paper proves the following results : (1) If the pair [Formula: see text] has the property that for any [Formula: see text], there is [Formula: see text] such that [Formula: see text] for any open subset [Formula: see text] of [Formula: see text] and all [Formula: see text] sufficiently large (where [Formula: see text] is the characteristic function of the set [Formula: see text]), then the following hold : (a) The map [Formula: see text] is topologically ergodic. (b) The upper density [Formula: see text] of [Formula: see text] is positive for any open subset [Formula: see text] of [Formula: see text], where [Formula: see text]. (c) There is a [Formula: see text]-invariant Borel probability measure [Formula: see text] having full support (i.e. [Formula: see text]). (d) Sensitivity of the map [Formula: see text] implies positive lower density sensitivity, hence ergodical sensitivity. (2) If [Formula: see text] for any two nonempty open subsets [Formula: see text], then there exists [Formula: see text] satisfying [Formula: see text] for any nonempty open subset [Formula: see text], where [Formula: see text] there exist [Formula: see text] with [Formula: see text].

scholarly journals A note on Riemann integrability

1978 ◽  
Vol 1 (1) ◽  
pp. 69-73
Author(s):  
G. A. Beer
Keyword(s):  
Metric Space ◽  
Borel Measure ◽  
Measure Zero ◽  
Bounded Function ◽  
Closed Ball ◽  
Open Ball ◽  
Compact Metric Space ◽  
Finite Borel Measure ◽  
Set Of Points ◽  
Points Of Discontinuity

In this note we define Riemann integrabillty for real valued functions defined on a compact metric space accompanied by a finite Borel measure. If the measure of each open ball equals the measure of its corresponding closed ball, then a bounded function is Riemann integrable if and only if its set of points of discontinuity has measure zero.

scholarly journals Pointwise measurable functions

2004 ◽  
Vol 95 (2) ◽  
pp. 305
Author(s):  
Herman Render ◽  
Lothar Rogge
Keyword(s):  
Large Class ◽  
Metric Space ◽  
Measurable Function ◽  
Polish Space ◽  
Topological Spaces ◽  
Range Space ◽  
Compact Metric Space ◽  
Delta Set ◽  
Measurable Functions ◽  
Borel Measurable

We introduce the new concept of pointwise measurability. It is shown in this paper that a measurable function is measurable at each point and that for a large class of topological spaces the converse also holds. Moreover it can be seen that a function which is continuous at a point is Borel-measurable at this point too. Furthermore the set of measurability points is considered. If the range space is a $\sigma$-compact metric space, then this set is a $G_{\delta}$-set; if the range space is only a Polish space this is in general not true any longer.

When Scott is weak on the top

1997 ◽  
Vol 7 (5) ◽  
pp. 401-417 ◽  
Author(s):  
ABBAS EDALAT
Keyword(s):  
Continuous Function ◽  
Probability Measure ◽  
Metric Space ◽  
Expected Value ◽  
Maximal Elements ◽  
Compact Metric Space ◽  
Hölder Continuous ◽  
Power Domain ◽  
Route To Chaos ◽  
Unique Invariant Measure

We construct an approximating chain of simple valuations on the upper space of a compact metric space whose lub is a given probability measure on the metric space. We show that whenever a separable metric space is homeomorphic to a Gδ subset of an ω-continuous dcpo equipped with its Scott topology, the space of probability measures of the metric space equipped with the weak topology is homeomorphic with a subset of the maximal elements of the probabilistic power domain of the ω-continuous dcpo. Given an effective approximation of a probability measure by an increasing chain of normalised valuations on the upper space of a compact metric space, we show that the expected value of any Hölder continuous function on the space can be obtained up to any given accuracy. We present a novel application in computing integrals in dynamical systems. We obtain an algorithm to compute the expected value of any Hölder continuous function with respect to the unique invariant measure of the Feigenbaum map in the periodic doubling route to chaos.

Embedding Smooth Dendroids in Hyperspaces

1979 ◽  
Vol 31 (1) ◽  
pp. 130-138 ◽  
Author(s):  
J. Grispolakis ◽  
E. D. Tymchatyn
Keyword(s):  
Continuous Function ◽  
Partial Order ◽  
Metric Space ◽  
Hausdorff Metric ◽  
Vietoris Topology ◽  
Compact Metric Space ◽  
Partially Ordered ◽  
Arcwise Connected ◽  
Ordered Space ◽  
Image Position

A continuum will be a connected, compact, metric space. By a mapping we mean a continuous function. By a partially ordered space X we mean a continuum X together with a partial order which is closed when regarded as a subset of X × X. We let 2x (resp. C(X)) denote the hyperspace of closed subsets (resp. subcontinua) of X with the Vietoris topology which coincides with the topology induced by the Hausdorff metric. The hyperspaces 2X and C(X) are arcwise connected metric continua (see [3, Theorem 2.7]). If A ⊂ X we let C(A) denote the subspace of subcontinua of X which lie in A.If X is a partially ordered space we define two functions L, M : X → 2X by setting for each x ∊ X

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).

scholarly journals KMS states on the crossed product -algebra of a homeomorphism

2021 ◽  
pp. 1-42
Author(s):  
JOHANNES CHRISTENSEN ◽  
KLAUS THOMSEN
Keyword(s):  
Continuous Function ◽  
Ergodic Theory ◽  
Metric Space ◽  
Crossed Product ◽  
Compact Metric Space ◽  
Kms States ◽  
Parameter Group ◽  
Intimate Relation ◽  
Crossed Product Algebra ◽  
Product Algebra

Abstract Let $\phi :X\to X$ be a homeomorphism of a compact metric space X. For any continuous function $F:X\to \mathbb {R}$ there is a one-parameter group $\alpha ^{F}$ of automorphisms (or a flow) on the crossed product $C^*$ -algebra $C(X)\rtimes _{\phi }\mathbb {Z}$ defined such that $\alpha ^{F}_{t}(fU)=fUe^{-itF}$ when $f \in C(X)$ and U is the canonical unitary in the construction of the crossed product. In this paper we study the Kubo--Martin--Schwinger (KMS) states for these flows by developing an intimate relation to the ergodic theory of non-singular transformations and show that the structure of KMS states can be very rich and complicated. Our results are complete concerning the set of possible inverse temperatures; in particular, we show that when $C(X) \rtimes _{\phi } \mathbb Z$ is simple this set is either $\{0\}$ or the whole line $\mathbb R$ .

scholarly journals Strong submeasures and applications to non-compact dynamical systems

2020 ◽  
pp. 1-23
Author(s):  
TUYEN TRUNG TRUONG
Keyword(s):  
Metric Space ◽  
Bounded Operator ◽  
Continuous Functions ◽  
Open Dense Subset ◽  
Compact Metric Space ◽  
Basic Properties ◽  
Banach Theorem ◽  
Complex Varieties ◽  
Definition Of ◽  
Meromorphic Maps

Abstract A strong submeasure on a compact metric space X is a sub-linear and bounded operator on the space of continuous functions on X. A strong submeasure is positive if it is non-decreasing. By the Hahn–Banach theorem, a positive strong submeasure is the supremum of a non-empty collection of measures whose masses are uniformly bounded from above. There are many natural examples of continuous maps of the form $f:U\rightarrow X$ , where X is a compact metric space and $U\subset X$ is an open-dense subset, where f cannot extend to a reasonable function on X. We can mention cases such as transcendental maps of $\mathbb {C}$ , meromorphic maps on compact complex varieties, or continuous self-maps $f:U\rightarrow U$ of a dense open subset $U\subset X$ where X is a compact metric space. For the aforementioned mentioned the use of measures is not sufficient to establish the basic properties of ergodic theory, such as the existence of invariant measures or a reasonable definition of measure-theoretic entropy and topological entropy. In this paper we show that strong submeasures can be used to completely resolve the issue and establish these basic properties. In another paper we apply strong submeasures to the intersection of positive closed $(1,1)$ currents on compact Kähler manifolds.

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