scholarly journals Fast and slow points of Birkhoff sums

2019 ◽  
Vol 40 (12) ◽  
pp. 3236-3256
Author(s):  
FRÉDÉRIC BAYART ◽  
ZOLTÁN BUCZOLICH ◽  
YANICK HEURTEAUX
Keyword(s):  
Growth Rate ◽  
Continuous Function ◽  
Unit Circle ◽  
General Context ◽  
Typical Element ◽  
Typical Part

We investigate the growth rate of the Birkhoff sums $S_{n,\unicode[STIX]{x1D6FC}}f(x)=\sum _{k=0}^{n-1}f(x+k\unicode[STIX]{x1D6FC})$, where $f$ is a continuous function with zero mean defined on the unit circle $\mathbb{T}$ and $(\unicode[STIX]{x1D6FC},x)$ is a ‘typical’ element of $\mathbb{T}^{2}$. The answer depends on the meaning given to the word ‘typical’. Part of the work will be done in a more general context.

On a System of Schröder Equations on the Circle

2003 ◽  
Vol 13 (07) ◽  
pp. 1883-1888 ◽  
Author(s):  
K. Ciepliński ◽  
M. C. Zdun
Keyword(s):  
Continuous Function ◽  
Unit Circle ◽  
Existence Of Solutions ◽  
Continuous Mappings

Let M be an arbitrary nonempty set and [Formula: see text] for t ∈ M be continuous mappings of the unit circle [Formula: see text]. The aim of this paper is to investigate the existence of solutions (Φ, c), where [Formula: see text] is a continuous function and [Formula: see text], of the following system of Schröder equations [Formula: see text] The particular case when card M = 1 is also considered.

Isomorphism Classes of Solenoidal Algebras I

1993 ◽  
Vol 36 (4) ◽  
pp. 414-418 ◽  
Author(s):  
Berndt Brenken
Keyword(s):  
Dynamical System ◽  
Growth Rate ◽  
Unit Circle ◽  
Compact Space ◽  
Periodic Points ◽  
Crossed Product ◽  
Roots Of Unity ◽  
Isomorphism Classes ◽  
Crossed Product Algebra ◽  
Product Algebra

AbstractEach g ∊ ℤ[x] defines a homeomorphism of a compact space We investigate the isomorphism classes of the C*-crossed product algebra Bg associated with the dynamical system An isomorphism invariant Eg of the algebra Bg is shown to determine the algebra Bg up to * or * anti-isomorphism if degree g ≤ 1 and 1 is not a root of g or if degree g = 2 and g is irreducible. It is also observed that the entropy of the dynamical system is equal to the growth rate of the periodic points if g has no roots of unity as zeros. This slightly extends the previously known equality of these two quantities under the assumption that g has no zeros on the unit circle.

scholarly journals Cyclicity in Dirichlet Spaces

2019 ◽  
Vol 62 (02) ◽  
pp. 247-257 ◽  
Author(s):  
Y. Elmadani ◽  
I. Labghail
Keyword(s):  
Continuous Function ◽  
Unit Circle ◽  
Borel Measure ◽  
Dirichlet Space ◽  
Sufficient Condition ◽  
Dirichlet Spaces ◽  
Closed Unit Disk ◽  
Finite Borel Measure ◽  
Weighted Dirichlet Space ◽  
Uniformly Continuous

AbstractLet $\unicode[STIX]{x1D707}$ be a positive finite Borel measure on the unit circle and ${\mathcal{D}}(\unicode[STIX]{x1D707})$ the associated harmonically weighted Dirichlet space. In this paper we show that for each closed subset $E$ of the unit circle with zero $c_{\unicode[STIX]{x1D707}}$ -capacity, there exists a function $f\in {\mathcal{D}}(\unicode[STIX]{x1D707})$ such that $f$ is cyclic (i.e., $\{pf:p\text{ is a polynomial}\}$ is dense in ${\mathcal{D}}(\unicode[STIX]{x1D707})$ ), $f$ vanishes on $E$ , and $f$ is uniformly continuous. Next, we provide a sufficient condition for a continuous function on the closed unit disk to be cyclic in ${\mathcal{D}}(\unicode[STIX]{x1D707})$ .

scholarly journals Optimal location of resources and Steiner symmetry in a population dynamics model in heterogeneous environments

2022 ◽  
Vol 47 (1) ◽  
pp. 305-324
Author(s):  
Claudia Anedda ◽  
Fabrizio Cuccu
Keyword(s):  
Growth Rate ◽  
Sufficient Conditions ◽  
Spatial Location ◽  
Spatial Arrangement ◽  
Optimal Choice ◽  
Dirichlet Boundary Conditions ◽  
General Context ◽  
Heterogeneous Environments ◽  
Local Growth ◽  
Total Size

The subject of this paper is inspired by Cantrell and Cosner (1989) and Cosner, Cuccu and Porru (2013). Cantrell and Cosner (1989) investigate the dynamics of a population in heterogeneous environments by means of diffusive logistic equations. An important part of their study consists in finding sufficient conditions which guarantee the survival of the species. Mathematically, this task leads to the weighted eigenvalue problem \(-\Delta u =\lambda m u \) in a bounded smooth domain \(\Omega\subset \mathbb{R}^N\), \(N\geq 1\), under homogeneous Dirichlet boundary conditions, where \(\lambda \in \mathbb{R}\) and \(m\in L^\infty(\Omega)\). The domain \(\Omega\) represents the environment and \(m(x)\), called the local growth rate, says where the favourable and unfavourable habitats are located. Then, Cantrell and Cosner (1989) consider a class of weights \(m(x)\) corresponding to environments where the total sizes of favourable and unfavourable habitats are fixed, but their spatial arrangement is allowed to change; they determine the best choice among them for the population to survive. In our work we consider a sort of refinement of the result above. We write the weight \(m(x)\) as sum of two (or more) terms, i.e. \(m(x)=f_1(x)+f_2(x)\), where \(f_1(x)\) and \(f_2(x)\) represent the spatial densities of the two resources which contribute to form the local growth rate \(m(x)\). Then, we fix the total size of each resource allowing its spatial location to vary. As our first main result, we show that there exists an optimal choice of \(f_1(x)\) and \(f_2(x)\) and find the form of the optimizers. Our proof relies on some results in Cosner, Cuccu and Porru (2013) and on a new property (to our knowledge) about the classes of rearrangements of functions. Moreover, we show that if \(\Omega\) is Steiner symmetric, then the best arrangement of the resources inherits the same kind of symmetry. (Actually, this is proved in the more general context of the classes of rearrangements of measurable functions.

Bounding smooth solutions of Bezout equations

2017 ◽  
Vol 121 (1) ◽  
pp. 121
Author(s):  
Nikolai Nikolski
Keyword(s):  
Growth Rate ◽  
Unit Circle ◽  
Spectral Parameter ◽  
Upper Bound ◽  
Topological Algebra ◽  
Corona Theorem ◽  
Corona Problem ◽  
Smooth Solutions ◽  
Smooth Functions ◽  
Bezout Equation

Given data $f=(f_1,f_2,\dots ,f_n)$ in the holomorphic part $ A= F_+$ of a symmetric Banach\slash topological algebra $ F$ on the unit circle $\mathbb{T}$, we estimate solutions $ g_j\in A$ to the corresponding Bezout equation $\sum _{j=1}^ng_jf_j=1$ in terms of the lower spectral parameter δ, $0< \delta \leq |f(z)|$, and an inversion controlling function $c_1(\delta ,F)$ for the algebra $F$. A scheme developed issues from an analysis of the famous Uchiyama-Wolff proof to the Carleson corona theorem and includes examples of algebras of “smooth” functions, as Beurling-Sobolev, Lipschitz, or Wiener-Dirichlet algebras. There is no real “corona problem” in this setting, the issue is in the growth rate of the upper bound for $\|g\|_{A^n}$ as $\delta \to 0$ and in numerical values of the quantities that occur, which are determined as accurately as possible.

scholarly journals Normal Light Interior Functions Defined in the Unit Disk

1970 ◽  
Vol 39 ◽  
pp. 149-155 ◽  
Author(s):  
J.H. Mathews
Keyword(s):  
Continuous Function ◽  
Unit Circle ◽  
Unit Disk ◽  
Riemann Sphere ◽  
Hyperbolic Metric ◽  
Chordal Metric ◽  
Normal Light ◽  
Uniformly Continuous

Let D be the unit disk, C the unit circle, and f a continuous function from D into the Riemann sphere W. We say that f is normal if f is uniformly continuous with respect to the non-Euclidean hyperbolic metric in D and the chordal metric in W.

scholarly journals Dirichlet integral and Picard principle

1982 ◽  
Vol 86 ◽  
pp. 85-99
Author(s):  
Mitsuru Nakai ◽  
Toshimasa Tada
Keyword(s):  
Continuous Function ◽  
Unit Circle ◽  
Unit Disk ◽  
Ideal Boundary ◽  
Dirichlet Integral ◽  
Hölder Continuous ◽  
Relative Boundary ◽  
The Family ◽  
The Ideal

A density P on the punctured unit disk Ω:0 < |z| <1 is a 2-form P(z)dxdy whose coefficient P(z) is a real valued nonnegative locally Hölder continuous function on the closed punctured unit disk Ω:0< |z| <≦1. Here we consider Ω as an end of the punctured sphere 0 < |z| ≦ + + so that the point z = 0 is viewed as the ideal boundary δΣ of Σ and the unit circle |z| = 1 as the relative boundary δΣ of Σ. We denote by D = D(Σ) the family of densities on Σ.

scholarly journals Maximizing points and coboundaries for an irrational rotation on a circle

2012 ◽  
Vol 33 (1) ◽  
pp. 24-48 ◽  
Author(s):  
JULIEN BRÉMONT ◽  
ZOLTÁN BUCZOLICH
Keyword(s):  
Dynamical System ◽  
Continuous Function ◽  
Unit Circle ◽  
Continuous Functions ◽  
Irrational Rotation ◽  
Additive Constant ◽  
Hölder Continuous

AbstractConsider an irrational rotation of a unit circle and a real continuous function. A point is declared ‘maximizing’ if the growth of the ergodic sums at this point is maximal up to an additive constant. In the case of two-sided ergodic sums, the existence of a maximizing point for a continuous function implies that it is the coboundary of a continuous function. In contrast, we build, for the ‘usual’ one-sided ergodic sums, examples in Hölder or smooth classes, indicating that all kinds of behaviour of the function with respect to the dynamical system are possible. We also show that generic continuous functions are without maximizing points, not only for rotations, but also for the transformation 2x mod 1. For this latter transformation, it is known that any Hölder continuous function has a maximizing point.

The threshold energy for atom displacement in fcc metals

1990 ◽  
Vol 48 (4) ◽  
pp. 530-531
Author(s):  
Wilfried Sigle ◽  
Matthias Hohenstein ◽  
Alfred Seeger
Keyword(s):  
Growth Rate ◽  
Elevated Temperatures ◽  
Threshold Energy ◽  
Dislocation Loops ◽  
Absolute Minimum ◽  
Voltage Electron ◽  
Atom Displacement ◽  
Electron Microscopes ◽  
Fcc Metal

Prolonged electron irradiation of metals at elevated temperatures usually leads to the formation of large interstitial-type dislocation loops. The growth rate of the loops is proportional to the total cross-section for atom displacement,which is implicitly connected with the threshold energy for atom displacement, Ed . Thus, by measuring the growth rate as a function of the electron energy and the orientation of the specimen with respect to the electron beam, the anisotropy of Ed can be determined rather precisely. We have performed such experiments in situ in high-voltage electron microscopes on Ag and Au at 473K as a function of the orientation and on Au as a function of temperature at several fixed orientations.Whereas in Ag minima of Ed are found close to <100>,<110>, and <210> (13-18eV), (Fig.1) atom displacement in Au requires least energy along <100>(15-19eV) (Fig.2). Au is thus the first fcc metal in which the absolute minimum of the threshold energy has been established not to lie in or close to the <110> direction.

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