arXiv subjects
Publications and source records attributed to V. I. Bakhtin.
It is proved that a parameterized curve in a metric space $X$ is absolutely continuous if and only if its composition with any Lipschitz function on $X$ is absolutely continuous.
In the paper we prove criteria for convexity and concavity of $f$-potentials ($f$-means, Kolmogorov means, weighted quasi-arithmetic means), which particular cases are the arithmetic, geometric, harmonic means, the thermodynamic potential (exponential mean), and the $L^{p}$-norm. Then we compute in quadratures all functions $f$ satisfying these criteria.
An ergodic support $X_0$ of a dynamical system $(X,T)$ with metrizable compact phase space $X$ is the set of all points $x\in X$ such that the corresponding sequence of empirical measures $δ_{x,n} = (δ_x +δ_{Tx}+\dots +δ_{T^{n-1}x})/n$ converges weakly to some ergodic measure. For every invariant probability measure $μ$ on $X$ it is proven that $μ(X_0) =1$ and Choquet distribution $μ^*$ on the set of ergodic measures $\mathop{\mathrm{Erg}} X$ has the natural representation $μ^*(A) =μ(\{ x\in X_0 : \limδ_{x,n} \in A\})$, where $A\subset \mathop{\mathrm{Erg}} X$.
The paper is devoted to the analysis of relationships between principal objects of the spectral theory of dynamical systems (transfer and weighted shift operators) and basic characteristics of information theory and thermodynamic formalism (entropy and topological pressure). We present explicit formulae linking these objects with $t$-entropy and spectral potential. Herewith we uncover the role of inverse rami-rate, forward entropy along with essential set and the property of non-contractibility of a dynamical system.
The article presents new sup-sums principles for integral F-divergence for arbitrary convex function F and arbitrary (not necessarily positive and absolutely continuous) measures. As applications of these results we derive the corresponding sup-sums principle for Kullback--Leibler divergence and work out new `integral' definition for t-entropy explicitly establishing its relation to Kullback--Leibler divergence.
For any transfer operator we establish the equivalence of variational principles for $t$-entropy, the spectral potential and entropy statistic theorem and give new proofs for all these statements.
New direct proofs of variational principles for $t$-entropy, spectral radius of weighted shift operators, and entropy statistic theorem are given. The equivalence of these statements is obtained.
For simplest colored branching processes we prove an analog to the McMillan theorem and calculate Hausdorff dimensions of random fractals defined in terms of the limit behavior of empirical measures generated by finite genetic lines. In this setting the role of Shannon's entropy is played by the Kullback--Leibler divergence and the Hausdorff dimensions are computed by means of the so-called Billingsley--Kullback entropy, defined in the paper.
The article presents a new definition of t-entropy that makes it more explicit and simplifies the process of its calculation.
In this paper we introduce a new functional invariant of discrete time dynamical systems -- the so-called t-entropy. The main result is that this t-entropy is the Legendre dual functional to the logarithm of the spectral radius of the weighted shift operator on $L^1(X,m)$ generated by the dynamical system. This result is called the Variational principle and is similar to the classical variational principle for the topological pressure.
The paper deals with the variational principles for evaluation of the spectral radii of transfer and weighted shift operators associated with a dynamical system. These variational principles have been the matter of numerous investigations and the principal results have been achieved in the situation when the dynamical system is either reversible or it is a topological Markov chain. As the main summands these principles contain the integrals over invariant measures and the Kolmogorov--Sinai entropy. In the article we derive the Variational Principle for an arbitrary dynamical system. It gives the explicit description of the Legendre dual object to the spectral potential. It is shown that in general this principle contains not the Kolmogorov--Sinai entropy but a new invariant of entropy type -- the t-entropy.
In the present paper we introduce positive flows and processes, which generalize the ordinary dynamical systems and stochastic processes. We develop a branch of theory of positive operators based on the concepts of phase and positive algebras, the spectral potential, the dual entropy, equilibrium measures, the action functional, sensitive states, empirical measures and prove within it the law of large numbers with respect to the sensitive states and calculate asymptotics for probabilities of large deviations in terms of the action functional.
The paper presents a criterion for a C*-algebra to be a coefficient algebra associated with a given endomorphism
The paper presents a construction of the crossed product of a C*-algebra by an endomorphism generated by partial isometry