Entropy and Ergodicity of Boole-Type Transformations

Blackmore, Denis; Balinsky, Alexander; Kycia, R. A.; Prykarpatski, Anatolij K.

doi:10.3390/e23111405

Cited by 3 publications

(5 citation statements)

References 60 publications

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“…for any 0 . z z ± ≠ ∈ Then, the following theorem, based on Proposition 3.1 and the ergodicity results of [15,29,30] that the Lebesgue measure λ on  is also ergodic with respect to the mappings ( 25) and ( 26), we obtain that any function 2 ( ) g L ∈  allows the direct sum orthogonal decomposition (22).…”

Section: Main Invariant Decomposition Results and Examplesmentioning

confidence: 91%

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Hilbert Space Decomposition Properties of Complex Functions and Their Applications

Vovk,

Pukach,

Dilnyi

et al. 2023

Contemp. Math.

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We analyzed the classical problem of decomposing the Hilbert space of holomorphic functions, especially their splitting into the product or sum of domain-separated components. For the Bergman space of analytical functions, we obtained a special decomposition satisfying the assigned growth degree properties. Concerning a general Hilbert space of analytical functions on a connected domain, we studied its α-invariant decomposition and related ergodic consequences. As an interesting consequence, we obtained the decomposition theorem for an ergodic α-mapping on the Bergman space of holomorphic functions.

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Section: Main Invariant Decomposition Results and Examplesmentioning

confidence: 91%

“…Proof. Proof easily follows from the ergodicity property of the Lebesgue measure λ on  stated in [29] and the α-invariance property of Remark 3.2. Consider now a so-called [31] internal function ( , ) : ω ω ± ± ∈  Taking into account now that the fixed points , ( , )…”

Section: Main Invariant Decomposition Results and Examplesmentioning

confidence: 99%

Hilbert Space Decomposition Properties of Complex Functions and Their Applications

Vovk,

Pukach,

Dilnyi

et al. 2023

Contemp. Math.

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“…2 ((R m ) ⊗N ; C) is the corresponding symmetric ground-state wave function of the related quantum Hamiltonian system, satisfying the conditions ( 49) and (49), reduced on the invariant subspace Φ N . Moreover, the following general expressions hold: Ω(η) = 1 and for any ω ∈ L (µ)…”

Section: Non-relativistic Quantum Current Algebra and Its Cyclic Repr...mentioning

confidence: 99%

“…In particular, it was observed [25][26][27][28][29][30][31][32][33] that these structures are canonically related to each other. Mathematical properties, lying in a background of their analytical description, make it possible to study additional important parameters [34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50] of different hydrodynamic and magnetohydrodynamic systems, amongst which we will mention integral invariants, describing such internal fluid motion peculiarities as vortices, topological singularities [51] and other different instability states, strongly depending [52,53] on imposed isentropic fluid motion constraints. Being interested in their general properties and mathematical structures, responsible for their existence and behavior, we present a detailed enough differential geometrical approach to investigating thermodynamically quasi-stationary isentropic fluid motions, paying more attention to analytical argumentation of tricks and techniques used during the presentation.…”

Section: Diffeomorphism Group Structure and Functional Phase Space De...mentioning

confidence: 99%

“…It was observed [25][26][27][28][29][30][31][32][33] that the corresponding mathematical structures used for describing the analytical properties of hydrodynamical systems are canonically related to each other. Simultaneously, mathematical properties, against a background of their analytical description, make it possible to study additional important parameters [34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50] of different hydrodynamic and magnetohydrodynamic systems. Amongst these, we will mention integral invariants, describing such internal fluid motion peculiarities as vortices, topological singularities [51] and other different instability states, strongly depending [52,53] on imposed isentropic fluid motion constraints.…”

Section: Introductionmentioning

confidence: 99%

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Quantum Current Algebra in Action: Linearization, Integrability of Classical and Factorization of Quantum Nonlinear Dynamical Systems

Prykarpatski

2022

Universe

Self Cite

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This review is devoted to the universal algebraic and geometric properties of the non-relativistic quantum current algebra symmetry and to their representations subject to applications in describing geometrical and analytical properties of quantum and classical integrable Hamiltonian systems of theoretical and mathematical physics. The Fock space, the non-relativistic quantum current algebra symmetry and its cyclic representations on separable Hilbert spaces are reviewed and described in detail. The unitary current algebra family of operators and generating functional equations are described. A generating functional method to constructing irreducible current algebra representations is reviewed, and the ergodicity of the corresponding representation Hilbert space measure is mentioned. The algebraic properties of the so called coherent states are also reviewed, generated by cyclic representations of the Heisenberg algebra on Hilbert spaces. Unbelievable and impressive applications of coherent states to the theory of nonlinear dynamical systems on Hilbert spaces are described, along with their linearization and integrability. Moreover, we present a further development of these results within the modern Lie-algebraic approach to nonlinear dynamical systems on Poissonian functional manifolds, which proved to be both unexpected and important for the classification of integrable Hamiltonian flows on Hilbert spaces. The quantum current Lie algebra symmetry properties and their functional representations, interpreted as a universal algebraic structure of symmetries of completely integrable nonlinear dynamical systems of theoretical and mathematical physics on functional manifolds, are analyzed in detail. Based on the current algebra symmetry structure and their functional representations, an effective integrability criterion is formulated for a wide class of completely integrable Hamiltonian systems on functional manifolds. The related algebraic structure of the Poissonian operators and an effective algorithm of their analytical construction are described. The current algebra representations in separable Hilbert spaces and the factorized structure of quantum integrable many-particle Hamiltonian systems are reviewed. The related current algebra-based Hamiltonian reconstruction of the many-particle oscillatory and Calogero–Moser–Sutherland quantum models are reviewed and discussed in detail. The related quasi-classical quantum current algebra density representations and the collective variable approach in equilibrium statistical physics are reviewed. In addition, the classical Wigner type current algebra representation and its application to non-equilibrium classical statistical mechanics are described, and the construction of the Lie–Poisson structure on the phase space of the infinite hierarchy of distribution functions is presented. The related Boltzmann–Bogolubov type kinetic equation for the generating functional of many-particle distribution functions is constructed, and the invariant reduction scheme, compatible with imposed correlation functions constraints, is suggested and analyzed in detail. We also review current algebra functional representations and their geometric structure subject to the analytical description of quasi-stationary hydrodynamic flows and their magneto-hydrodynamic generalizations. A unified geometric description of the ideal idiabatic liquid dynamics is presented, and its Hamiltonian structure is analyzed. A special chapter of the review is devoted to recent results on the description of modified current Lie algebra symmetries on torus and their Lie-algebraic structures, related to integrable so-called heavenly type spatially many-dimensional dynamical systems on functional manifolds.

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On calculation of the exact Lyapunov exponent of affine Boole transformations

Barczy

2023

Dynamical Systems

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Entropy and Ergodicity of Boole-Type Transformations

Cited by 3 publications

References 60 publications

Hilbert Space Decomposition Properties of Complex Functions and Their Applications

Hilbert Space Decomposition Properties of Complex Functions and Their Applications

Quantum Current Algebra in Action: Linearization, Integrability of Classical and Factorization of Quantum Nonlinear Dynamical Systems

On calculation of the exact Lyapunov exponent of affine Boole transformations

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