scholarly journals Isometric Embeddings of Pro-Euclidean Spaces

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Barry Minemyer
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Metric Space ◽  
Inverse Limit ◽  
Isometric Embedding ◽  
Embedding Theorem ◽  
Euclidean Spaces ◽  
Compact Metric Space ◽  
Covering Dimension ◽  
Isometric Embeddings

Abstract In [12] Petrunin proves that a compact metric space X admits an intrinsic isometry into En if and only if X is a pro-Euclidean space of rank at most n, meaning that X can be written as a “nice” inverse limit of polyhedra. He also shows that either case implies that X has covering dimension at most n. The purpose of this paper is to extend these results to include both embeddings and spaces which are proper instead of compact. The main result of this paper is that any pro-Euclidean space of rank at most n is proper and admits an intrinsic isometric embedding into E2n+1. Since every n-dimensional Riemannian manifold is a pro-Euclidean space of rank at most n, this result is a partial generalization of (the C0 version of) the famous Nash isometric embedding theorem from [10].

scholarly journals Isometric embeddings of polyhedra into Euclidean space

2015 ◽  
Vol 07 (04) ◽  
pp. 677-692 ◽  
Author(s):  
Barry Minemyer
Keyword(s):  
Euclidean Space ◽  
Metric Space ◽  
Isometric Embedding ◽  
Piecewise Linear ◽  
Embedding Theorem ◽  
Embedding Theorems ◽  
Dimensional Simplex ◽  
Isometric Embeddings ◽  
Lipschitz Map ◽  
Hyperbolic Polyhedra

In this paper we consider piecewise linear (pl) isometric embeddings of Euclidean polyhedra into Euclidean space. A Euclidean polyhedron is just a metric space [Formula: see text] which admits a triangulation [Formula: see text] such that each n-dimensional simplex of [Formula: see text] is affinely isometric to a simplex in 𝔼n. We prove that any 1-Lipschitz map from an n-dimensional Euclidean polyhedron [Formula: see text] into 𝔼3n is ϵ-close to a pl isometric embedding for any ϵ > 0. If we remove the condition that the map be pl, then any 1-Lipschitz map into 𝔼2n + 1 can be approximated by a (continuous) isometric embedding. These results are extended to isometric embedding theorems of spherical and hyperbolic polyhedra into Euclidean space by the use of the Nash–Kuiper C1 isometric embedding theorem ([9] and [13]).

scholarly journals Sensitivity for topologically double ergodic dynamical systems

2021 ◽  
Vol 6 (10) ◽  
pp. 10495-10505
Author(s):  
Risong Li ◽  
◽  
Xiaofang Yang ◽  
Yongxi Jiang ◽  
Tianxiu Lu ◽  
...  
Keyword(s):  
Dynamical Systems ◽  
Metric Space ◽  
Inverse Limit ◽  
Compact Metric Space ◽  
Limit Space ◽  
Factor Map ◽  
Continuous Surjection ◽  
Inverse Limit Space

<abstract><p>As a stronger form of multi-sensitivity, the notion of ergodic multi-sensitivity (resp. strongly ergodically multi-sensitivity) is introduced. In particularly, it is proved that every topologically double ergodic continuous selfmap (resp. topologically double strongly ergodic selfmap) on a compact metric space is ergodically multi-sensitive (resp. strongly ergodically multi-sensitive). And for any given integer $ m\geq 2 $, $ f $ is ergodically multi-sensitive (resp. strongly ergodically multi-sensitive) if and only if so is $ f^{m} $. Also, it is shown that if $ f $ is a continuous surjection, then $ f $ is ergodically multi-sensitive (resp. strongly ergodically multi-sensitive) if and only if so is $ \sigma_{f} $, where $ \sigma_{f} $ is the shift selfmap on the inverse limit space $ \lim\limits_{\leftarrow}(X, f) $. Moreover, it is proved that if $ f:X\rightarrow X $ (resp. $ g:Y\rightarrow Y $) is a map on a nontrivial metric space $ (X, d) $ (resp. $ (Y, d') $), and $ \pi $ is a semiopen factor map between $ (X, f) $ and $ (Y, g) $, then the ergodic multi-sensitivity (resp. the strongly ergodic multi-sensitivity) of $ g $ implies the same property of $ f $.</p></abstract>

scholarly journals Homeomorphisms without the pseudo-orbit tracing property

1982 ◽  
Vol 88 ◽  
pp. 155-160 ◽  
Author(s):  
Nobuo Aoki
Keyword(s):  
Riemannian Manifold ◽  
Metric Space ◽  
Compact Riemannian Manifold ◽  
Compact Metric Space ◽  
Positive Dimension

Recently, A. Morimoto [5] proved that every isometry of a compact Riemannian manifold of positive dimension has not the pseudo-orbit tracing property, and that if a homeomorphism of a compact metric space has the pseudo-orbit tracing property then Eφ— 0φ(see § 1 for definition). The purpose of this paper is to show that every distal homeomorphism of a compact connected metric space has not the pseudo-orbit tracing property.

scholarly journals Crossed products and minimal dynamical systems

2018 ◽  
Vol 10 (02) ◽  
pp. 447-469 ◽  
Author(s):  
Huaxin Lin
Keyword(s):  
Metric Space ◽  
General Setting ◽  
Crossed Product ◽  
Compact Metric Space ◽  
Covering Dimension ◽  
Infinite Type ◽  
Mean Dimension ◽  
Tracial Rank ◽  
Minimal Dynamical Systems ◽  
Minimal Homeomorphisms

Let [Formula: see text] be an infinite compact metric space with finite covering dimension and let [Formula: see text] be two minimal homeomorphisms. We prove that the crossed product [Formula: see text]-algebras [Formula: see text] and [Formula: see text] are isomorphic if and only if they have isomorphic Elliott invariant. In a more general setting, we show that if [Formula: see text] is an infinite compact metric space and if [Formula: see text] is a minimal homeomorphism such that [Formula: see text] has mean dimension zero, then the tensor product of the crossed product with a UHF-algebra of infinite type has generalized tracial rank at most one. This implies that the crossed product is in a classifiable class of amenable simple [Formula: see text]-algebras.

On the global isometric embedding of pseudo-Riemannian manifolds

1970 ◽  
Vol 314 (1518) ◽  
pp. 417-428 ◽  
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Riemannian Manifolds ◽  
General Result ◽  
Isometric Embedding ◽  
Positive Definite ◽  
Space Time ◽  
Lower Dimension ◽  
Definite Case

It is shown that any pseudo-Riemannian manifold has (in Nash’s sense) a proper isometric embedding into a pseudo-Euclidean space, which can be made to be of arbitrarily high differentiability. The application of this to the positive definite case treated by Nash gives a new proof using a Euclidean space of substantially lower dimension. The general result is applied to the space-time of relativity, and the dimensions and signatures of the spaces needed to embed various cases are evaluated.

Decomposition Space Theory and the Bing Shrinking Criterion

2021 ◽  
pp. 62-76
Author(s):  
Christopher W. Davis ◽  
Boldizsár Kalmár ◽  
Min Hoon Kim ◽  
Henrik Rüping
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Embedding Theorem ◽  
Space Theory ◽  
Compact Metric Space ◽  
Decomposition Spaces ◽  
Compact Metric Spaces ◽  
Space Decomposition ◽  
Decomposition Space ◽  
Continuous Decomposition

‘Decomposition Space Theory and the Bing Shrinking Criterion’ gives a proof of the central Bing shrinking criterion and then provides an introduction to the key notions of the field of decomposition space theory. The chapter begins by proving the Bing shrinking criterion, which characterizes when a given map between compact metric spaces is approximable by homeomorphisms. Next, it develops the elements of the theory of decomposition spaces. A key fact is that a decomposition space associated with an upper semi-continuous decomposition of a compact metric space is again a compact metric space. Decomposition spaces are key in the proof of the disc embedding theorem.

scholarly journals The structure of the C*-algebra of a locally injective surjection

2011 ◽  
Vol 32 (4) ◽  
pp. 1226-1248 ◽  
Author(s):  
TOKE MEIER CARLSEN ◽  
KLAUS THOMSEN
Keyword(s):  
Metric Space ◽  
Matrix Algebra ◽  
Finite Covering ◽  
Compact Metric Space ◽  
Covering Dimension ◽  
Classification Result ◽  
Full Matrix ◽  
C Algebra ◽  
Simple Quotient ◽  
The Ideal

AbstractIn this paper we investigate the ideal structure of the C*-algebra of a locally injective surjection introduced by the second-named author. Our main result is that a simple quotient of the C*-algebra of a locally injective surjection on a compact metric space of finite covering dimension is either a full matrix algebra, a crossed product of a minimal homeomorphism of a compact metric space of finite covering dimension, or it is purely infinite and hence covered by the classification result of Kirchberg and Phillips. It follows in particular that if the C*-algebra of a locally injective surjection on a compact metric space of finite covering dimension is simple, then it is automatically purely infinite, unless the map in question is a homeomorphism. A corollary of this result is that if the C*-algebra of a one-sided subshift is simple, then it is also purely infinite.

A COMPUTATIONAL ERGODIC THEOREM FOR INFINITE ITERATED FUNCTION SYSTEMS

2008 ◽  
Vol 08 (03) ◽  
pp. 365-381 ◽  
Author(s):  
NGUYEN DINH CONG ◽  
DOAN THAI SON ◽  
STEFAN SIEGMUND
Keyword(s):  
Dynamical Systems ◽  
Euclidean Space ◽  
Ergodic Theorem ◽  
Error Bound ◽  
Metric Space ◽  
Iterated Function Systems ◽  
Random Dynamical Systems ◽  
Compact Metric Space ◽  
Iterated Function ◽  
Time Average

Iterated function systems are examples of random dynamical systems and became popular as generators of fractals like the Sierpinski Gasket and the Barnsley Fern. In this paper we prove an ergodic theorem for iterated function systems which consist of countably many functions and which are contractive on average on an arbitrary compact metric space and we provide a computational version of this ergodic theorem in Euclidean space which allows to numerically approximate the time average together with an explicit error bound. The results are applied to an explicit example.

Minimal entropy and Mostow's rigidity theorems

1996 ◽  
Vol 16 (4) ◽  
pp. 623-649 ◽  
Author(s):  
Gérard Besson ◽  
Gilles Courtois ◽  
Sylvestre Gallot
Keyword(s):  
Riemannian Manifold ◽  
Topological Entropy ◽  
Metric Space ◽  
Universal Cover ◽  
Compact Metric Space ◽  
Entropy Functional ◽  
Volume Entropy ◽  
Image Position ◽  
A Minor ◽  
Special Metrics

Let (Y, g) be a compact connected n-dimensional Riemannian manifold and let () be its universal cover endowed with the pulled-back metric. If y ∈ , we definewhere B(y, R) denotes the ball of radius R around y in . It is a well known fact that this limit exists and does not depend on y ([Man]). The invariant h(g) is called the volume entropy of the metric g but, for the sake of simplicity, we shall use the term entropy. The idea of recognizing special metrics in terms of this invariant looks at first glance very optimistic. First the entropy, which behaves like the inverse of a distance, is sensitive to changes of scale which makes it a bad invariant: however, this is a minor drawback that can be circumvented by looking at the behaviour of the entropy functional on the space of metrics with fixed volume (equal to one for example). Nevertheless, it seems very unlikely that two numbers, the entropy and the volume, might characterize any metric. The very first person to consider such a possibility was Katok ([Kat1]). In this article the entropy is thought of as a dynamical invariant which actually is suggested by its name. More precisely, let us define this dynamical invariant, which is called the topological entropy: let (M, d) be a compact metric space and ψt, a flow on it, we define.

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