The Experts below are selected from a list of 1887 Experts worldwide ranked by ideXlab platform
L D Faddeev - One of the best experts on this subject based on the ideXlab platform.
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discrete heisenberg weyl group and modular group
Letters in Mathematical Physics, 1995Co-Authors: L D FaddeevAbstract:It is shown that the generators of two discrete Heisenberg-Weyl groups with Irrational Rotation numbers θ and −1/θ generate the whole algebraB of operators onL2(R). The natural action of the modular group inB is implied. Applications to dynamical algebras appearing in lattice regularization and some duality principles are discussed.
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discrete heisenberg weyl group and modular group
arXiv: High Energy Physics - Theory, 1995Co-Authors: L D FaddeevAbstract:It is shown that the generators of two discrete Heisenberg-Weyl groups with Irrational Rotation numbers $\theta$ and $-1/ \theta$ generate the whole algebra $\cal B$ of bounded operators on $L_2(\bf R)$. The natural action of the modular group in $\cal B$ is implied. Applications to dynamical algebras appearing in lattice regularization and some duality principles are discussed.
P. J. Stacey - One of the best experts on this subject based on the ideXlab platform.
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an inductive limit model for the k theory of the generator interchanging antiautomorphism of an Irrational Rotation algebra
Canadian Mathematical Bulletin, 2003Co-Authors: P. J. StaceyAbstract:Let Abe the universal C � -algebra generated by two unitaries U, V satisfying VU = e 2�iUV and letbe the antiautomorphism of Ainterchanging U and V. The K-theory of R� = {a ∈ A� : �(a) = a � } is computed. Whenis Irrational, an inductive limit of algebras of the form Mq(C(T)) ⊕ Mq'(R) ⊕ Mq(R) is constructed which has complexification Aand the same K-theory as R�.
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inductive limit decompositions of real structures in Irrational Rotation algebras
Indiana University Mathematics Journal, 2002Co-Authors: P. J. StaceyAbstract:An Irrational Rotation C*-algebra can be written as a direct limit of algebras, each of which is the direct sum of two matrix algebras over the algebra of continuous functions on the circle. The invariance of such decompositions under involutory antiautomorphisms is investigated, and a complete answer obtained for toral antiautomorphisms.
Konstantin Khanin - One of the best experts on this subject based on the ideXlab platform.
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hausdorff dimension of invariant measure of circle diffeomorphisms with a break point
Ergodic Theory and Dynamical Systems, 2019Co-Authors: Konstantin Khanin, Sasa KocicAbstract:We prove that, for almost all Irrational , is zero. This result cannot be extended to all Irrational Rotation numbers.
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renormalization conjecture and rigidity theory for circle diffeomorphisms with breaks
Geometric and Functional Analysis, 2014Co-Authors: Konstantin Khanin, Sasa KocicAbstract:We prove the renormalization conjecture for circle diffeomorphisms with breaks, i.e., that the renormalizations of any two C 2+α -smooth (α ∈ (0, 1)) circle diffeomorphisms with a break point, with the same Irrational Rotation number and the same size of the break, approach each other exponentially fast in the C 2-topology. As was shown in [KKM], this result implies the following strong rigidity statement: for almost all Irrational numbers ρ, any two circle diffeomorphisms with a break, with the same Rotation number ρ and the same size of the break, are C 1-smoothly conjugate to each other. As we proved in [KK13], the latter claim cannot be extended to all Irrational Rotation numbers. These results can be considered an extension of Herman’s theory on the linearization of circle diffeomorphisms.
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robust rigidity for circle diffeomorphisms with singularities
Inventiones Mathematicae, 2007Co-Authors: Konstantin Khanin, A TeplinskyAbstract:We prove that under certain regularity conditions imposed on the renormalizations of two circle diffeomorphisms with singularities, their C 1-smooth equivalence follows from exponential convergence of those renormalizations. As an easy corollary, any two analytical critical circle maps with the same order of critical points and the same Irrational Rotation number are C 1-smoothly conjugate.
Rakesh K. Bansal - One of the best experts on this subject based on the ideXlab platform.
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On Match Lengths, Zero Entropy, and Large Deviations—With Application to Sliding Window Lempel–Ziv Algorithm
2016Co-Authors: Siddharth Jain, Rakesh K. BansalAbstract:Abstract — The sliding window Lempel–Ziv (SWLZ) algorithm that makes use of recurrence times and match lengths has been studied from various perspectives in information theory literature. In this paper, we undertake a finer study of these quantities under two different scenarios: 1) zero entropy sources that are characterized by strong long-term memory and 2) the processes with weak memory as described through various mixing conditions. For zero entropy sources, a general statement on match length is obtained. It is used in the proof of almost sure optimality of fixed shift variant of Lempel–Ziv (FSLZ) and SWLZ algorithms given in literature. Through an example of stationary and ergodic processes generated by an Irrational Rotation, we establish that for a window of size nw, a compression ratio given by O(log nw/nwa), where a depends on nw and approaches 1 as nw → ∞, is obtained under the application of FSLZ and SWLZ algorithms. In addition, we give a general expression for the compression ratio for a class of stationary and ergodic processes with zero entropy. Next, we extend the study of Ornstein and Weiss on the asymptotic behavior of the normalized version of recurrence times and establish the large deviation property for a class of mixing processes. In addition, an estimator of entropy based on recurrence times is proposed for which large deviation principle is proved for sources satisfying similar mixing conditions. Index Terms — Irrational Rotation, zero entropy, entropy dimension, recurrence times, match lengths, sliding window Lempel-Ziv, large deviation property, mixing conditions, entropy estimator, exponential convergence
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On Match Lengths and the Asymptotic Behavior of Sliding Window Lempel-Ziv Algorithm for Zero Entropy Sequences
2016Co-Authors: Siddharth Jain, Rakesh K. BansalAbstract:Abstract—The Sliding Window Lempel-Ziv (SWLZ) algorithm has been studied from various perspectives in information theory literature. In this paper, we provide a general law which defines the asymptotics of match length for stationary and ergodic zero entropy processes. Moreover, we use this law to choose the match length Lo in the almost sure optimality proof of Fixed Shift Variant of Lempel-Ziv (FSLZ) and SWLZ algorithms given in literature. First, through an example of stationary and ergodic processes generated by Irrational Rotation we establish that for a window size of nw a compression ratio given by O ( lognwnwa) where a is arbitrarily close to 1 and 0 < a < 1, is obtained under the application of FSLZ and SWLZ algorithms. Further, we give a general expression for the compression ratio for a class of stationary and totally ergodic processes with zero entropy. I
Michele Triestino - One of the best experts on this subject based on the ideXlab platform.
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on the invariant distributions of c 2 circle diffeomorphisms of Irrational Rotation number
Mathematische Zeitschrift, 2013Co-Authors: Andres Navas, Michele TriestinoAbstract:Although invariant measures are a fundamental tool in Dynamical Systems, very little is known about distributions (i.e. linear functionals defined on some space of smooth functions on the underlying space) that remain invariant under a dynamics. Perhaps the most general definite result in this direction is the remarkable theorem of Avila and Kocsard [1] according to which no C∞ circle diffeomorphism of Irrational Rotation number has an invariant distribution different from (a scalar multiple of integration with respect to) the (unique) invariant (probability) measure. The main result of this Note is an analogous result in low regularity. Unlike [1] which involves very hard computations, our approach is more conceptual. It relies on the work of Douady and Yoccoz [3] concerning automorphic measures for circle diffeomorphisms.
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on the invariant distributions of c 2 circle diffeomorphisms of Irrational Rotation number
arXiv: Dynamical Systems, 2012Co-Authors: Andres Navas, Michele TriestinoAbstract:We show that no C^2 circle diffeomorphism of Irrational Rotation number has invariant 1-distributions other than (scalar multiples of) the invariant measure. We also show that this is false in the C^1 context by giving both minimal and non-minimal examples.
