scholarly journals Equilibrium states on higher-rank Toeplitz non-commutative solenoids

2019 ◽  
Vol 40 (11) ◽  
pp. 2881-2912 ◽  
Author(s):  
ZAHRA AFSAR ◽  
ASTRID AN HUEF ◽  
IAIN RAEBURN ◽  
AIDAN SIMS
Keyword(s):  
Continuous Functions ◽  
Equilibrium States ◽  
Probability Measures ◽  
Inverse Limits ◽  
Inverse Temperature ◽  
Higher Dimensional ◽  
Direct Limits ◽  
Higher Rank ◽  
Natural Dynamics

We consider a family of higher-dimensional non-commutative tori, which are twisted analogues of the algebras of continuous functions on ordinary tori and their Toeplitz extensions. Just as solenoids are inverse limits of tori, our Toeplitz non-commutative solenoids are direct limits of the Toeplitz extensions of non-commutative tori. We consider natural dynamics on these Toeplitz algebras, and we compute the equilibrium states for these dynamics. We find a large simplex of equilibrium states at each positive inverse temperature, parametrized by the probability measures on an (ordinary) solenoid.

Higher Dimensional Spaces of Functions on the Spectrum of a Uniform Algebra

2007 ◽  
Vol 50 (1) ◽  
pp. 3-10
Author(s):  
Richard F. Basener
Keyword(s):  
Continuous Functions ◽  
Uniform Algebra ◽  
Finite Dimensional ◽  
Higher Dimensional ◽  
Spaces Of Continuous Functions ◽  
The Family ◽  
Continuous Complex ◽  
Finite Dimensional Case ◽  
Complex Valued ◽  
Shed Light

AbstractIn this paper we introduce a nested family of spaces of continuous functions defined on the spectrum of a uniform algebra. The smallest space in the family is the uniform algebra itself. In the “finite dimensional” case, from some point on the spaces will be the space of all continuous complex-valued functions on the spectrum. These spaces are defined in terms of solutions to the nonlinear Cauchy–Riemann equations as introduced by the author in 1976, so they are not generally linear spaces of functions. However, these spaces do shed light on the higher dimensional properties of a uniform algebra. In particular, these spaces are directly related to the generalized Shilov boundary of the uniform algebra (as defined by the author and, independently, by Sibony in the early 1970s).

Activation Functions Defined on Higher-Dimensional Spaces for Approximation on Compact Sets with and without Scaling

2003 ◽  
Vol 15 (9) ◽  
pp. 2199-2226
Author(s):  
Yoshifusa Ito
Keyword(s):  
Fourier Transform ◽  
Activation Function ◽  
Continuous Functions ◽  
Activation Functions ◽  
Compact Sets ◽  
Higher Dimensional ◽  
Hidden Layer ◽  
The Fourier Transform ◽  
Increasing Function ◽  
Sigmoid Functions

Let g be a slowly increasing function of locally bounded variation defined on Rc, 1 ≤c≤d. We investigate when g can be an activation function of the hidden-layer units of three-layer neural networks that approximate continuous functions on compact sets. If the support of the Fourier transform of g includes a converging sequence of points with distinct distances from the origin, it can be an activation function without scaling. If and only if the support of its Fourier transform includes a point other than the origin, it can be an activation function with scaling. We also look for a condition on which an activation function can be used for approximation without rotation. Any nonpolynomial functions can be activation functions with scaling, and many familiar functions, such as sigmoid functions and radial basis functions, can be activation functions without scaling. With or without scaling, some of them defined on Rd can be used without rotation even if they are not spherically symmetric.

scholarly journals Polynomial approximation on spheres - generalizing de la Vallée-Poussin

2011 ◽  
Vol 11 (4) ◽  
pp. 540-552 ◽  
Author(s):  
Ian H. Sloan
Keyword(s):  
Trigonometric Polynomial ◽  
Polynomial Approximation ◽  
Piecewise Linear ◽  
Continuous Functions ◽  
Fast Convergence ◽  
Piecewise Polynomial ◽  
Smooth Functions ◽  
Higher Dimensional ◽  
Trigonometric Polynomial Approximation ◽  
De La Vallée Poussin

AbstractFor trigonometric polynomial approximation on a circle, the century-old de la Vallée-Poussin construction has attractive features: it exhibits uniform convergence for all continuous functions as the degree of the trigonometric polynomial goes to infinity, yet it also has arbitrarily fast convergence for sufficiently smooth functions. This paper presents an explicit generalization of the de la Vallée-Poussin construction to higher dimensional spheres S^d ≤ R^{d+1}. The generalization replaces the C^∞ filter introduced by Rustamov by a piecewise polynomial of minimal degree. For the case of the circle the filter is piecewise linear, and recovers the de la Vallée-Poussin construction, while for the general sphere S^d the filter is a piecewise polynomial of degree d and smoothness C^{d−1}. In all cases the approximation converges uniformly for all continuous functions, and has arbitrarily fast convergence for smooth functions.

scholarly journals Real Functions and the Extension of Generalized Probability Measures

2013 ◽  
Vol 55 (1) ◽  
pp. 85-94
Author(s):  
Jana Havlíčková
Keyword(s):  
Category Theory ◽  
Continuous Functions ◽  
Probability Measures ◽  
Fuzzy Probability ◽  
Classical Probability ◽  
Random Events ◽  
Elementary Methods ◽  
Generalized Probability ◽  
Fuzzy Probability Theory

Abstract In the classical probability, as well as in the fuzzy probability theory, random events and probability measures are modelled by functions into the closed unit interval [0,1]. Using elementary methods of category theory, we present a classification of the extensions of generalized probability measures (probability measures and integrals with respect to probability measures) from a suitable class of generalized random events to a larger class having some additional (algebraic and/or topological) properties. The classification puts into a perspective the classical and some recent constructions related to the extension of sequentially continuous functions.

scholarly journals Cocycles on groupoids arising from -actions

2021 ◽  
pp. 1-32
Author(s):  
CARLA FARSI ◽  
LEONARD HUANG ◽  
ALEX KUMJIAN ◽  
JUDITH PACKER
Keyword(s):  
Continuous Functions ◽  
Locally Compact Abelian Group ◽  
Compact Metric Space ◽  
Kms States ◽  
Existence And Uniqueness Results ◽  
Commuting Operators ◽  
Uniqueness Results ◽  
Unit Space ◽  
Higher Rank

Abstract We consider groupoids constructed from a finite number of commuting local homeomorphisms acting on a compact metric space and study generalized Ruelle operators and $ C^{\ast } $ -algebras associated to these groupoids. We provide a new characterization of $ 1 $ -cocycles on these groupoids taking values in a locally compact abelian group, given in terms of $ k $ -tuples of continuous functions on the unit space satisfying certain canonical identities. Using this, we develop an extended Ruelle–Perron–Frobenius theory for dynamical systems of several commuting operators ( $ k $ -Ruelle triples and commuting Ruelle operators). Results on KMS states on $ C^{\ast } $ -algebras constructed from these groupoids are derived. When the groupoids being studied come from higher-rank graphs, our results recover existence and uniqueness results for KMS states associated to the graphs.

scholarly journals Equidistribution results for geodesic flows

2013 ◽  
Vol 34 (3) ◽  
pp. 742-764
Author(s):  
ABDELHAMID AMROUN
Keyword(s):  
Geodesic Flow ◽  
Large Deviation ◽  
Riemannian Metric ◽  
Upper Bounds ◽  
Equilibrium States ◽  
Topological Pressure ◽  
Probability Measures ◽  
Geodesic Flows ◽  
Lower And Upper Bounds ◽  
Unit Tangent Bundle

AbstractUsing the works of Mañé [On the topological entropy of the geodesic flows.J. Differential Geom.45(1989), 74–93] and Paternain [Topological pressure for geodesic flows.Ann. Sci. Éc. Norm. Supér.(4)33(2000), 121–138] we study the distribution of geodesic arcs with respect to equilibrium states of the geodesic flow on a closed manifold, equipped with a$\mathcal {C}^{\infty }$Riemannian metric. We prove large-deviation lower and upper bounds and a contraction principle for the geodesic flow in the space of probability measures of the unit tangent bundle. We deduce a way of approximating equilibrium states for continuous potentials.

Dynamical properties of shift maps on inverse limits with a set valued function

2016 ◽  
Vol 38 (4) ◽  
pp. 1499-1524 ◽  
Author(s):  
JUDY KENNEDY ◽  
VAN NALL
Keyword(s):  
Topological Structure ◽  
Inverse Limit ◽  
Natural Generalization ◽  
Continuous Functions ◽  
Invariant Sets ◽  
Inverse Limits ◽  
Limit Space ◽  
Shift Map ◽  
Single Well ◽  
Inverse Limit Space

Set-valued functions from an interval into the closed subsets of an interval arise in various areas of science and mathematical modeling. Research has shown that the dynamics of a single-valued function on a compact space are closely linked to the dynamics of the shift map on the inverse limit with the function as the sole bonding map. For example, it has been shown that with Devaney’s definition of chaos the bonding function is chaotic if and only if the shift map is chaotic. One reason for caring about this connection is that the shift map is a homeomorphism on the inverse limit, and therefore the topological structure of the inverse-limit space must reflect in its richness the dynamics of the shift map. In the set-valued case there may not be a natural definition for chaos since there is not a single well-defined orbit for each point. However, the shift map is a continuous single-valued function so it together with the inverse-limit space form a dynamical system which can be chaotic in any of the usual senses. For the set-valued case we demonstrate with theorems and examples rich topological structure in the inverse limit when the shift map is chaotic (on certain invariant sets). We then connect that chaos to a property of the set-valued function that is a natural generalization of an important chaos producing property of continuous functions.

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