scholarly journals Local Properties of Riesz Minimal Energy Configurations and Equilibrium Measures

2017 ◽  
Vol 2019 (16) ◽  
pp. 5066-5086 ◽  
Author(s):  
D P Hardin ◽  
A Reznikov ◽  
E B Saff ◽  
A Volberg
Keyword(s):  
Equilibrium Measure ◽  
Regularity Property ◽  
Compact Set ◽  
Dimensional Manifold ◽  
Smoothness Assumption ◽  
Equilibrium Measures ◽  
Point Configurations ◽  
Local Properties ◽  
Energy Configurations ◽  
Separation Results

Abstract We investigate separation properties of $N$-point configurations that minimize discrete Riesz $s$-energy on a compact set $A\subset \mathbb{R}^p$. When $A$ is a smooth $(p-1)$-dimensional manifold without boundary and $s\in [p-2, p-1)$, we prove that the order of separation (as $N\to \infty$) is the best possible. The same conclusions hold for the points that are a fixed positive distance from the boundary of $A$ whenever $A$ is any $p$-dimensional set. These estimates extend a result of Dahlberg for certain smooth $(p-1)$-dimensional surfaces when $s=p-2$ (the harmonic case). Furthermore, we obtain the same separation results for “greedy” $s$-energy points. We deduce our results from an upper regularity property of the $s$-equilibrium measure (i.e., the measure that solves the continuous minimal Riesz $s$-energy problem), and we show that this property holds under a local smoothness assumption on the set $A$.

scholarly journals Invariance of green equilibrium measure on the domain

Filomat ◽  
2013 ◽  
Vol 27 (4) ◽  
pp. 593-600 ◽  
Author(s):  
Stamatis Pouliasis
Keyword(s):  
Level Set ◽  
Wide Class ◽  
Equilibrium Measure ◽  
The Other ◽  
Equilibrium Potential ◽  
Compact Set ◽  
Compact Sets ◽  
Equilibrium Measures ◽  
The One

We prove that the Green equilibrium measure and the Green equilibrium energy of a compact set K relative to the domains D and ? are the same if and only if D is nearly equal to ?, for a wide class of compact sets K. Also, we prove that equality of Green equilibrium measures arises if and only if the one domain is related with a level set of the Green equilibrium potential of K relative to the other domain.

scholarly journals BERGMAN KERNELS AND EQUILIBRIUM MEASURES FOR POLARIZED PSEUDO-CONCAVE DOMAINS

2010 ◽  
Vol 21 (01) ◽  
pp. 77-115 ◽  
Author(s):  
ROBERT J. BERMAN
Keyword(s):  
Toeplitz Operators ◽  
Equilibrium Measure ◽  
Markov Property ◽  
General Setting ◽  
Space Structure ◽  
Tensor Power ◽  
Bergman Kernels ◽  
Equilibrium Measures ◽  
Holomorphic Sections ◽  
Positive Line Bundle

Let X be a domain in a closed polarized complex manifold (Y,L), where L is a (semi-) positive line bundle over Y. Any given Hermitian metric on L induces by restriction to X a Hilbert space structure on the space of global holomorphic sections on Y with values in the k-th tensor power of L (also using a volume form ωn on X. In this paper the leading large k asymptotics for the corresponding Bergman kernels and metrics are obtained in the case when X is a pseudo-concave domain with smooth boundary (under a certain compatibility assumption). The asymptotics are expressed in terms of the curvature of L and the boundary of X. The convergence of the Bergman metrics is obtained in a more general setting where (X,ωn) is replaced by any measure satisfying a Bernstein–Markov property. As an application the (generalized) equilibrium measure of the polarized pseudo-concave domain X is computed explicitly. Applications to the zero and mass distribution of random holomorphic sections and the eigenvalue distribution of Toeplitz operators will be described elsewhere.

scholarly journals Smale endomorphisms over graph-directed Markov systems

2020 ◽  
pp. 1-34
Author(s):  
EUGEN MIHAILESCU ◽  
MARIUSZ URBAŃSKI
Keyword(s):  
Equilibrium Measure ◽  
Parabolic Systems ◽  
Skew Product ◽  
Pointwise Dimension ◽  
Markov Systems ◽  
Large Classes ◽  
Equilibrium Measures ◽  
Skew Products ◽  
Natural Extensions ◽  
Induced Maps

We study Smale skew product endomorphisms (introduced in Mihailescu and Urbański [Skew product Smale endomorphisms over countable shifts of finite type. Ergod. Th. & Dynam. Sys. doi: 10.1017/etds.2019.31. Published online June 2019]) now over countable graph-directed Markov systems, and we prove the exact dimensionality of conditional measures in fibers, and then the global exact dimensionality of the equilibrium measure itself. Our results apply to large classes of systems and have many applications. They apply, for instance, to natural extensions of graph-directed Markov systems. Another application is to skew products over parabolic systems. We also give applications in ergodic number theory, for example to the continued fraction expansion, and the backward fraction expansion. In the end we obtain a general formula for the Hausdorff (and pointwise) dimension of equilibrium measures with respect to the induced maps of natural extensions ${\mathcal{T}}_{\unicode[STIX]{x1D6FD}}$ of $\unicode[STIX]{x1D6FD}$ -maps $T_{\unicode[STIX]{x1D6FD}}$ , for arbitrary $\unicode[STIX]{x1D6FD}>1$ .

scholarly journals Equilibrium measures on trees

2021 ◽  
Author(s):  
Nicola Arcozzi ◽  
Matteo Levi
Keyword(s):  
Equilibrium Measure ◽  
Equilibrium Measures ◽  
Local Degree ◽  
Infinite Tree ◽  
Special Case

AbstractWe give a characterization of equilibrium measures for p-capacities on the boundary of an infinite tree of arbitrary (finite) local degree. For $$p=2$$ p = 2 , this provides, in the special case of trees, a converse to a theorem of Benjamini and Schramm, which interpretes the equilibrium measure of a planar graph’s boundary in terms of square tilings of cylinders.

Thermodynamic formalism for countable Markov shifts

1999 ◽  
Vol 19 (6) ◽  
pp. 1565-1593 ◽  
Author(s):  
OMRI M. SARIG
Keyword(s):  
Equilibrium Measure ◽  
Thermodynamic Formalism ◽  
Finite Measure ◽  
Complete Characterization ◽  
Topological Pressure ◽  
Recurrent Functions ◽  
Equilibrium Measures ◽  
Positive Recurrence ◽  
Markov Shifts ◽  
Definition Of

We establish a generalized thermodynamic formalism for topological Markov shifts with a countable number of states. We offer a definition of topological pressure and show that it satisfies a variational principle for the metric entropies. The pressure of $\phi =0$ is the Gurevic entropy. This pressure may be finite even if the topological entropy is infinite. Let $L_\phi$ denote the Ruelle operator for $\phi$. We offer a definition of positive recurrence for $\phi$ and show that it is a necessary and sufficient condition for a Ruelle–Perron–Frobenius theorem to hold: there exist a $\sigma$-finite measure $\nu $, a continuous function $h>0$ and $\lambda >0$ such that $L_\phi ^{*}\nu =\lambda \nu$, $L_\phi h=\lambda h $ and $\lambda ^{-n}L_\phi ^nf\rightarrow h\int f\,d\nu$ for suitable functions $f$. We show that under certain conditions this convergence is uniform and exponential. We prove a decomposition theorem for positive recurrent functions and construct conformal measures and equilibrium measures. We give complete characterization of the situation when the equilibrium measure is a Gibbs measure. We end by giving examples where positive recurrence can be verified. These include functions of the form $$ \phi =\log f\left( \cfrac{1}{x_0+ \cfrac{1}{x_1+\dotsb }}\right), $$ where $f$ is a suitable function on a suitable shift $X$.

scholarly journals Gibbs and equilibrium measures for some families of subshifts

2012 ◽  
Vol 33 (3) ◽  
pp. 934-953 ◽  
Author(s):  
TOM MEYEROVITCH
Keyword(s):  
Finite Type ◽  
Gibbs Measure ◽  
Equilibrium Measure ◽  
Gibbs Measures ◽  
Type Theorem ◽  
Equilibrium Measures ◽  
Strongly Irreducible ◽  
Subshifts Of Finite Type ◽  
Summable Variation

AbstractFor subshifts of finite type (SFTs), any equilibrium measure is Gibbs, as long as $f$ has $d$-summable variation. This is a theorem of Lanford and Ruelle. Conversely, a theorem of Dobrušin states that for strongly irreducible subshifts, shift-invariant Gibbs measures are equilibrium measures. Here we prove a generalization of the Lanford–Ruelle theorem: for all subshifts, any equilibrium measure for a function with $d$-summable variation is ‘topologically Gibbs’. This is a relaxed notion which coincides with the usual notion of a Gibbs measure for SFTs. In the second part of the paper, we study Gibbs and equilibrium measures for some interesting families of subshifts: $\beta $-shifts, Dyck shifts and Kalikow-type shifts (defined below). In all of these cases, a Lanford–Ruelle-type theorem holds. For each of these families, we provide a specific proof of the result.

Variation of the equilibrium measure and theS-property of a stationary compact set

2011 ◽  
Vol 66 (1) ◽  
pp. 176-178 ◽  
Author(s):  
Andrei Martínez-Finkelshtein ◽  
Evguenii A Rakhmanov ◽  
Sergey P Suetin
Keyword(s):  
Equilibrium Measure ◽  
Compact Set

scholarly journals Equilibrium measures for certain isometric extensions of Anosov systems

2016 ◽  
Vol 38 (3) ◽  
pp. 1154-1167 ◽  
Author(s):  
RALF SPATZIER ◽  
DANIEL VISSCHER
Keyword(s):  
Equilibrium Measure ◽  
Closed Manifold ◽  
Unique Equilibrium ◽  
Hölder Continuous ◽  
Equilibrium Measures ◽  
Negatively Curved ◽  
Continuous Potential ◽  
Anosov Systems ◽  
Frame Flow

We prove that for the frame flow on a negatively curved, closed manifold of odd dimension other than 7, and a Hölder continuous potential that is constant on fibers, there is a unique equilibrium measure. Brin and Gromov’s theorem on the ergodicity of frame flows follows as a corollary. Our methods also give a corresponding result for automorphisms of the Heisenberg manifold fibering over the torus.

scholarly journals Deformations of asymptotically cylindricalG2-manifolds

2008 ◽  
Vol 145 (2) ◽  
pp. 311-348 ◽  
Author(s):  
JOHANNES NORDSTRÖM
Keyword(s):  
Fundamental Group ◽  
Moduli Space ◽  
Smooth Manifold ◽  
Double Cover ◽  
Dimensional Manifold ◽  
Local Properties ◽  
Torsion Free

AbstractWe prove that for a 7-dimensional manifoldMwith cylindrical ends the moduli space of exponentially asymptotically cylindrical torsion-freeG2-structures is a smooth manifold (if non-empty), and study some of its local properties. We also show that the holonomy of the induced metric of an exponentially asymptotically cylindricalG2-manifold is exactlyG2if and only if the fundamental group π1(M) is finite and neitherMnor any double cover ofMis homeomorphic to a cylinder.

Equilibrium measures for rational maps

1986 ◽  
Vol 6 (3) ◽  
pp. 393-399 ◽  
Author(s):  
Artur Oscar Lopes
Keyword(s):  
Equilibrium Measure ◽  
Julia Set ◽  
Maximal Entropy ◽  
Rational Maps ◽  
Rational Map ◽  
Equilibrium Measures ◽  
Polynomial Map ◽  
Two Measures ◽  
Measure Of Maximal Entropy

AbstractFor a polynomial map the measure of maximal entropy is the equilibrium measure for the logarithm potential in the Julia set [1], [4].Here we will show that in the case where f is a rational map such that f(∞) = ∞ and the Julia set is bounded, then the two measures mentioned above are equal if and only if f is a polynomial.

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