THERMODYNAMIC FORMALISM FOR RANDOM TRANSFORMATIONS REVISITED

2008 ◽  
Vol 08 (01) ◽  
pp. 77-102 ◽  
Author(s):  
YURI KIFER
Keyword(s):  
Finite Type ◽  
Type Theorem ◽  
Thermodynamic Formalism ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Topological Mixing ◽  
Subshifts Of Finite Type

We return to the thermodynamic formalism constructions for random expanding in average transformations and for random subshifts of finite type with random rates of topological mixing, as well as to the Perron–Frobenius type theorem for certain random positive linear operators. Our previous expositions in [14, 19] and [21] were based on constructions which left some gaps and inaccuracies related to the measurability and uniqueness issues. Our approach here is based on Hilbert projective norms which were already applied in [5] for the thermodynamic formalism constructions for random subshifts of finite type but our method is somewhat different and more general so that it enables us to treat simultaneously both expanding and subshift cases.

scholarly journals The Approximation of Bivariate Blending Variant Szász Operators Based Brenke Type Polynomials

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Behar Baxhaku ◽  
Ramadan Zejnullahu ◽  
Artan Berisha
Keyword(s):  
Modulus Of Continuity ◽  
Weighted Space ◽  
Type Theorem ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Weighted Modulus Of Continuity ◽  
Rate Of Approximation ◽  
Szász Operators ◽  
Weighted Modulus

We have constructed a new sequence of positive linear operators with two variables by using Szasz-Kantorovich-Chlodowsky operators and Brenke polynomials. We give some inequalities for the operators by means of partial and full modulus of continuity and obtain a Lipschitz type theorem. Furthermore, we study the convergence of Szasz-Kantorovich-Chlodowsky-Brenke operators in weighted space of function with two variables and estimate the rate of approximation in terms of the weighted modulus of continuity.

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.

Discrete Lyapunov exponents and Hausdorff dimension

2000 ◽  
Vol 20 (1) ◽  
pp. 145-172 ◽  
Author(s):  
SHMUEL FRIEDLAND
Keyword(s):  
Hausdorff Dimension ◽  
Finite Type ◽  
Lyapunov Exponents ◽  
Thermodynamic Formalism ◽  
Sufficient Conditions ◽  
Kleinian Groups ◽  
Limit Sets ◽  
Geometrically Finite ◽  
Subshifts Of Finite Type ◽  
Discrete Analogs

We study certain metrics on subshifts of finite type for which we define the discrete analogs of Lyapunov exponents. We prove Young's formula for $\mu$-Hausdorff dimension. We give sufficient conditions on the above metrics for which the Hausdorff dimension is given by thermodynamic formalism. We apply these results to the Hausdorff dimension of the limit sets of geometrically finite, purely loxodromic, Kleinian groups.

scholarly journals A Voronovskaya-type theorem for a positive linear operator

2006 ◽  
Vol 2006 ◽  
pp. 1-7 ◽  
Author(s):  
Alexandra Ciupa
Keyword(s):  
Linear Operator ◽  
Exponential Growth ◽  
Positive Linear Operator ◽  
Type Theorem ◽  
Continuous Functions ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Voronovskaya Type Theorem

We consider a sequence of positive linear operators which approximates continuous functions having exponential growth at infinity. For these operators, we give a Voronovskaya-type theorem

Approximation by mixed positive linear operators based on second-kind beta transform

2021 ◽  
Author(s):  
Naokant Deo ◽  
Ram Pratap
Keyword(s):  
Bounded Variation ◽  
Type Theorem ◽  
Weighted Approximation ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Functions Of Bounded Variation ◽  
Approximation Operators ◽  
Voronovskaya Type Theorem ◽  
Mixed Approximation ◽  
Direct Results

In this paper, we consider mixed approximation operators based on second-kind beta transform using Szász–Mirakjan operators. For the proposed operators, we establish some direct results, Voronovskaya-type theorem, quantitative Voronovskaya-type theorem, Grüss–Voronovskaya-type theorem, weighted approximation and functions of bounded variation.

scholarly journals Certain approximation properties of Brenke polynomials using Jakimovski–Leviatan operators

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Shahid Ahmad Wani ◽  
M. Mursaleen ◽  
Kottakkaran Sooppy Nisar
Keyword(s):  
Order Of Convergence ◽  
Type Theorem ◽  
Weighted Approximation ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Approximation Properties ◽  
Voronovskaya Type Theorem

AbstractIn this article, we establish the approximation by Durrmeyer type Jakimovski–Leviatan operators involving the Brenke type polynomials. The positive linear operators are constructed for the Brenke polynomials, and thus approximation properties for these polynomials are obtained. The order of convergence and the weighted approximation are also considered. Finally, the Voronovskaya type theorem is demonstrated for some particular case of these polynomials.

Approximation in statistical sense to B -continuous functions by positive linear operators

2010 ◽  
Vol 47 (3) ◽  
pp. 289-298 ◽  
Author(s):  
Fadime Dirik ◽  
Oktay Duman ◽  
Kamil Demirci
Keyword(s):  
Compact Subset ◽  
Real Line ◽  
Statistical Convergence ◽  
Approximation Theorem ◽  
Continuous Functions ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Statistical Approximation ◽  
Classical Version ◽  
Statistical Sense

In the present work, using the concept of A -statistical convergence for double real sequences, we obtain a statistical approximation theorem for sequences of positive linear operators defined on the space of all real valued B -continuous functions on a compact subset of the real line. Furthermore, we display an application which shows that our new result is stronger than its classical version.

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