scholarly journals Mixing for suspension flows over skew-translations and time-changes of quasi-abelian filiform nilflows

2018 ◽  
Vol 39 (12) ◽  
pp. 3407-3436 ◽  
Author(s):  
DAVIDE RAVOTTI
Keyword(s):  
Fourier Coefficients ◽  
Continuous Functions ◽  
Dense Subspace ◽  
Dimensional Torus ◽  
Smooth Functions ◽  
Suspension Flows ◽  
Time Changes ◽  
Uniquely Ergodic ◽  
Dense Set ◽  
Roof Function

We consider suspension flows over uniquely ergodic skew-translations on a $d$-dimensional torus $\mathbb{T}^{d}$ for $d\geq 2$. We prove that there exists a set $\mathscr{R}$ of smooth functions, which is dense in the space $\mathscr{C}(\mathbb{T}^{d})$ of continuous functions, such that every roof function in $\mathscr{R}$ which is not cohomologous to a constant induces a mixing suspension flow. We also construct a dense set of mixing examples which is explicitly described in terms of their Fourier coefficients. In the language of nilflows on nilmanifolds, our result implies that, for every uniquely ergodic nilflow on a quasi-abelian filiform nilmanifold, there exists a dense subspace of smooth time-changes in which mixing occurs if and only if the time-change is not cohomologous to a constant. This generalizes a theorem by Avila, Forni and Ulcigrai [Mixing for time-changes of Heisenberg nilflows. J. Differential Geom.89(3) (2011), 369–410] for the classical Heisenberg group.

Generic rotation sets

2020 ◽  
pp. 1-13
Author(s):  
SEBASTIÁN PAVEZ-MOLINA
Keyword(s):  
Dynamical System ◽  
Convex Body ◽  
Probability Measures ◽  
Topological Dynamical System ◽  
Continuous Vector ◽  
Rotation Set ◽  
Vector Valued Function ◽  
Vector Valued ◽  
Uniquely Ergodic ◽  
Dense Set

Abstract Let $(X,T)$ be a topological dynamical system. Given a continuous vector-valued function $F \in C(X, \mathbb {R}^{d})$ called a potential, we define its rotation set $R(F)$ as the set of integrals of F with respect to all T-invariant probability measures, which is a convex body of $\mathbb {R}^{d}$ . In this paper we study the geometry of rotation sets. We prove that if T is a non-uniquely ergodic topological dynamical system with a dense set of periodic measures, then the map $R(\cdot )$ is open with respect to the uniform topologies. As a consequence, we obtain that the rotation set of a generic potential is strictly convex and has $C^{1}$ boundary. Furthermore, we prove that the map $R(\cdot )$ is surjective, extending a result of Kucherenko and Wolf.

scholarly journals Restriction of Families of Conversion Operators in Lp Spaces

2021 ◽  
Vol 27 (3) ◽  
pp. 449-460
Author(s):  
A. D. Nakhman
Keyword(s):  
Fourier Coefficients ◽  
Continuous Functions ◽  
Parameter Family ◽  
Maximal Operators ◽  
Limiting Behavior ◽  
Lp Spaces ◽  
Almost Everywhere ◽  
Spaces Of Continuous Functions

We study a one-parameter family of convolutional operators acting in Lebesgue Lp spaces. The case of integral kernels given by the Fourier coefficients is considered. It is established that the condition of the coefficients being quasiconvex ensures the boundedness of the corresponding maximal operators. The limiting behavior of families in the metrics of spaces of continuous functions and Lp, p ≥ 1, classes is studied, and their convergence is obtained almost everywhere. The ways of possible generalizations and distributions are indicated.

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).

scholarly journals Yang–Mills Measure on the Two-Dimensional Torus as a Random Distribution

2019 ◽  
Vol 372 (3) ◽  
pp. 1027-1058
Author(s):  
Ilya Chevyrev
Keyword(s):  
Random Distribution ◽  
Random Variable ◽  
Continuous Functions ◽  
Small Scale ◽  
Dimensional Torus ◽  
Hölder Continuous ◽  
Line Segments ◽  
Yang Mills ◽  
Hölder Continuous Functions ◽  
On Line

Abstract We introduce a space of distributional 1-forms $$\Omega ^1_\alpha $$Ωα1 on the torus $$\mathbf {T}^2$$T2 for which holonomies along axis paths are well-defined and induce Hölder continuous functions on line segments. We show that there exists an $$\Omega ^1_\alpha $$Ωα1-valued random variable A for which Wilson loop observables of axis paths coincide in law with the corresponding observables under the Yang–Mills measure in the sense of Lévy (Mem Am Math Soc 166(790), 2003). It holds furthermore that $$\Omega ^1_\alpha $$Ωα1 embeds into the Hölder–Besov space $$\mathcal {C}^{\alpha -1}$$Cα-1 for all $$\alpha \in (0,1)$$α∈(0,1), so that A has the correct small scale regularity expected from perturbation theory. Our method is based on a Landau-type gauge applied to lattice approximations.

scholarly journals Well-posedness of the difference schemes of the high order of accuracy for elliptic equations

2006 ◽  
Vol 2006 ◽  
pp. 1-12 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Pavel E. Sobolevskiĭ
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Continuous Functions ◽  
High Order ◽  
Difference Schemes ◽  
Well Posedness ◽  
Smooth Functions ◽  
Order Of Accuracy ◽  
High Order Of Accuracy ◽  
The Difference

It is well known the differential equation−u″(t)+Au(t)=f(t)(−∞<t<∞)in a general Banach spaceEwith the positive operatorAis ill-posed in the Banach spaceC(E)=C((−∞,∞),E)of the bounded continuous functionsϕ(t)defined on the whole real line with norm‖ϕ‖C(E)=sup⁡−∞<t<∞‖ϕ(t)‖E. In the present paper we consider the high order of accuracy two-step difference schemes generated by an exact difference scheme or by Taylor's decomposition on three points for the approximate solutions of this differential equation. The well-posedness of these difference schemes in the difference analogy of the smooth functions is obtained. The exact almost coercive inequality for solutions inC(τ,E)of these difference schemes is established.

scholarly journals Rates in the strong invariance principle for ergodic automorphisms of the torus

2014 ◽  
Vol 14 (02) ◽  
pp. 1350021
Author(s):  
Jérôme Dedecker ◽  
Florence Merlevède ◽  
Françoise Pène
Keyword(s):  
Invariance Principle ◽  
Fourier Coefficients ◽  
Rates Of Convergence ◽  
Partial Sums ◽  
Dimensional Torus ◽  
Strong Invariance ◽  
Ergodic Automorphism ◽  
Strong Invariance Principle ◽  
Natural Way

Let T be an ergodic automorphism of the d-dimensional torus 𝕋d. In the spirit of Le Borgne, we give conditions on the Fourier coefficients of a function f from 𝕋d to ℝ under which the partial sums f ◦ T + f ◦ T2 + ⋯ + f ◦ Tn satisfy a strong invariance principle. Next, reinforcing the condition on the Fourier coefficients in a natural way, we obtain explicit rates of convergence in the strong invariance principle, up to n1/4 log n.

Fourier coefficients of continuous functions

2012 ◽  
Vol 91 (5-6) ◽  
pp. 645-656 ◽  
Author(s):  
L. Gogoladze ◽  
V. Tsagareishvili
Keyword(s):  
Fourier Coefficients ◽  
Continuous Functions
Sign in / Sign up

Export Citation Format

Share Document