scholarly journals Unique ergodicity of the horocycle flow on Riemannnian foliations

2018 ◽  
Vol 40 (6) ◽  
pp. 1459-1479
Author(s):  
F. ALCALDE CUESTA ◽  
F. DAL’BO ◽  
M. MARTÍNEZ ◽  
A. VERJOVSKY
Keyword(s):  
Main Tool ◽  
Hyperbolic Surface ◽  
Unique Ergodicity ◽  
Alternative Proof ◽  
Compact Metric Space ◽  
Horocycle Flow ◽  
Full Support ◽  
Classic Result ◽  
Minimal Foliations ◽  
Special Case

A classic result due to Furstenberg is the strict ergodicity of the horocycle flow for a compact hyperbolic surface. Strict ergodicity is unique ergodicity with respect to a measure of full support, and therefore it implies minimality. The horocycle flow has been previously studied on minimal foliations by hyperbolic surfaces on closed manifolds, where it is known not to be minimal in general. In this paper, we prove that for the special case of Riemannian foliations, strict ergodicity of the horocycle flow still holds. This, in particular, proves that this flow is minimal, which establishes a conjecture proposed by Matsumoto. The main tool is a theorem due to Coudène, which he presented as an alternative proof for the surface case. It applies to two continuous flows defining a measure-preserving action of the affine group of the line on a compact metric space, precisely matching the foliated setting. In addition, we briefly discuss the application of Coudène’s theorem to other kinds of foliations.

scholarly journals On equicontinuous factors of flows on locally path-connected compact spaces

2020 ◽  
pp. 1-18
Author(s):  
NIKOLAI EDEKO
Keyword(s):  
Lie Group ◽  
Metric Space ◽  
Betti Number ◽  
Alternative Proof ◽  
Simple Consequence ◽  
Compact Metric Space ◽  
Invariant Functions ◽  
Compact Spaces ◽  
Path Connected

Abstract We consider a locally path-connected compact metric space K with finite first Betti number $\textrm {b}_1(K)$ and a flow $(K, G)$ on K such that G is abelian and all G-invariant functions $f\,{\in}\, \text{\rm C}(K)$ are constant. We prove that every equicontinuous factor of the flow $(K, G)$ is isomorphic to a flow on a compact abelian Lie group of dimension less than ${\textrm {b}_1(K)}/{\textrm {b}_0(K)}$ . For this purpose, we use and provide a new proof for Theorem 2.12 of Hauser and Jäger [Monotonicity of maximal equicontinuous factors and an application to toral flows. Proc. Amer. Math. Soc.147 (2019), 4539–4554], which states that for a flow on a locally connected compact space the quotient map onto the maximal equicontinuous factor is monotone, i.e., has connected fibers. Our alternative proof is a simple consequence of a new characterization of the monotonicity of a quotient map $p\colon K\to L$ between locally connected compact spaces K and L that we obtain by characterizing the local connectedness of K in terms of the Banach lattice $\textrm {C}(K)$ .

scholarly journals Topological Infinite Gammoids, and a New Menger-Type Theorem for Infinite Graphs

10.37236/6083 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Johannes Carmesin
Keyword(s):  
Type Theorem ◽  
Main Tool ◽  
Infinite Graphs ◽  
Disjoint Paths ◽  
Natural Topology ◽  
Locally Finite ◽  
Finite Graphs ◽  
Locally Finite Graphs ◽  
Vertex Sets ◽  
Special Case

Answering a question of Diestel, we develop a topological notion of gammoids in infinite graphs which, unlike traditional infinite gammoids, always define a matroid.As our main tool, we prove for any infinite graph $G$ with vertex-sets $A$ and $B$, if every finite subset of $A$ is linked to $B$ by disjoint paths, then the whole of $A$ can be linked to the closure of $B$ by disjoint paths or rays in a natural topology on $G$ and its ends.This latter theorem implies the topological Menger theorem of Diestel for locally finite graphs. It also implies a special case of the infinite Menger theorem of Aharoni and Berger.

scholarly journals Best Lag Window for Spectrum Estimation of Law Order MA Process

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Ali Sami Rashid ◽  
Mohammed Jabber Hawas Allami ◽  
Ahmed Kareem Mutasher
Keyword(s):  
Computer Simulation ◽  
Spectral Density ◽  
Density Function ◽  
Moving Average ◽  
Main Tool ◽  
Spectral Density Function ◽  
Spectrum Estimation ◽  
Ma Process ◽  
Special Case

In this article, we investigate spectrum estimation of law order moving average (MA) process. The main tool is the lag window which is one of the important components of the consistent form to estimate spectral density function (SDF). We show, based on a computer simulation, that the Blackman window is the best lag window to estimate the SDF of MA1 and MA2 at the most values of parameters βi and series sizes n, except for a special case when β=−1 and n≥40 in MA1. In addition, the Hanning–Poisson window appears as the best to estimate the SDF of MA2 when β1=β2=−0.5 and n≥40.

scholarly journals OCNEANU'S TRACE AND STARKEY'S RULE

2003 ◽  
Vol 12 (07) ◽  
pp. 899-904 ◽  
Author(s):  
MEINOLF GECK ◽  
NICOLAS JACON
Keyword(s):  
Simple Proof ◽  
Hecke Algebras ◽  
Jones Polynomial ◽  
Main Tool ◽  
Type A ◽  
Character Tables ◽  
Special Case ◽  
Knots And Links

We give a new simple proof for the weights of Ocneanu's trace on Iwahori–Hecke algebras of type A. This trace is used in the construction of the HOMFLYPT-polynomial of knots and links (which includes the famous Jones polynomial as a special case). Our main tool is Starkey's rule concerning the character tables of Iwahori–Hecke algebras of type A.

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].

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.

Another Proof of the Contraction Mapping Principle

1968 ◽  
Vol 11 (4) ◽  
pp. 605-606
Author(s):  
D.W. Boyd ◽  
J. S. W. Wong
Keyword(s):  
Metric Space ◽  
Contraction Mapping ◽  
Contraction Mapping Principle ◽  
Alternative Proof ◽  
Intersection Theorem ◽  
Cantor’S Theorem ◽  
Special Case

In a recent note of Kolodner [2], the Cantor Intersection Theorem is used to give an alternative proof of the well known Contraction Mapping Principle. Kolodner applied Cantor's theorem first to a bounded metric space and then reduced the general case to this special case. Sometime ago, we found a somewhat different proof of the Contraction Mapping Principle using Cantor's theorem. Since our proof seems somewhat more direct we propose to present it here.

Logical Distinction Between Diagrammatic and Cohen–Lyndon Asphericity

2003 ◽  
Vol 13 (03) ◽  
pp. 241-253
Author(s):  
Igor Biskup
Keyword(s):  
Main Tool ◽  
Alternative Proof ◽  
Two Dimensional ◽  
Dimensional Model ◽  
Group Presentation ◽  
Combinatorial Properties ◽  
Group Presentations ◽  
Two Dimensional Model ◽  
The Status ◽  
Reverse Implication

We resolve the status of the implication CLA ⇒ DA in the negative by showing that the two-dimensional model of the group presentation (a, b : a, b-1 aba-1 b-1) is DA but not CLA. This settles a question that has been addressed in [2, 7, 10, 11]. In 1941, Whitehead posed the question whether asphericity is a hereditary property for two-dimensional CW complexes. This question remains unanswered. Out of its study developed the formulation of several combinatorial properties for group presentations that are sufficient (but not necessary) for asphericity of the associated two-dimensional model. The logical relationships between these flavors of asphericity are just partially understood. In this article we show that two of these flavors of asphericity are in fact distinct. As a consequence, all of the known flavors are distinct. An argument of Lyndon and Schupp [5, III Property 10.6] shows that if a two-dimensional CW complex K is Cohen–Lyndon aspherical (CLA), then K is also diagrammatically aspherical (DA). The status of the reverse implication had been open prior to this writing. We will present an alternative proof for the implication CLA ⇒ DA and demonstrate that every spherical picture over the presentation (a, b : a, b-2 aba-1) can be reduced without insertions of dipoles, thus concluding that the presentation is DA. A straightforward argument shows that the presentation cannot be CLA. Our main tool is the theory of spherical pictures and picture moves of which we will give a short survey.

A condition for unique ergodicity of minimal symbolic flows

1992 ◽  
Vol 12 (3) ◽  
pp. 425-428 ◽  
Author(s):  
Michael D. Boshernitzan
Keyword(s):  
Unique Ergodicity ◽  
Interval Exchange ◽  
Sufficient Condition ◽  
Interval Exchange Transformations ◽  
Special Case

AbstractA sufficient condition for unique ergodicity of symbolic flows is provided. In an important but special case of interval exchange transformations, the condition has already been validated by W. Veech.

scholarly journals The chain recurrent set, attractors, and explosions

1985 ◽  
Vol 5 (3) ◽  
pp. 321-327 ◽  
Author(s):  
Louis Block ◽  
John E. Franke
Keyword(s):  
Compact Space ◽  
Metric Space ◽  
Periodic Points ◽  
Limit Set ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Special Case ◽  
Equivalent Conditions ◽  
Chain Recurrent Set ◽  
Recurrent Set

AbstractCharles Conley has shown that for a flow on a compact metric space, a point x is chain recurrent if and only if any attractor which contains the & ω-limit set of x also contains x. In this paper we show that the same statement holds for a continuous map of a compact metric space to itself, and additional equivalent conditions can be given. A stronger result is obtained if the space is locally connected.It follows, as a special case, that if a map of the circle to itself has no periodic points then every point is chain recurrent. Also, for any homeomorphism of the circle to itself, the chain recurrent set is either the set of periodic points or the entire circle. Finally, we use the equivalent conditions mentioned above to show that for any continuous map f of a compact space to itself, if the non-wandering set equals the chain recurrent set then f does not permit Ω-explosions. The converse holds on manifolds.

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