Speciality:
01.01.05 (Probability theory and mathematical statistics)
Birth date:
02.03.1968
E-mail: ,
Website: https://www.itp.ac.ru Keywords: dynamical systems; symbolic dynamics; topological Markov chains; Ruelle-Perron-Frobenius operators; spectral theory of nonnegative matrices; directed graphs.
Subject:
Abramov's formula for the entropy of the special flow constructed from a dynamical system $(T,X,\mu)$ and a nonnegative function $f$ on the space $X$ is generalized to the case when the measure $\mu$ is infinite ($\mu(X)=\infty$) and the integral of $f$ with respect to $\mu$ is finite $(\limits\int_{X}fd\mu < \infty)$. In this generalized Abramov formula the Kringel entropy is taken as the metric entropy of $\mu$ with respect to $T.$ For finite topological Markov chains, the Parry conjecture is proved. It states that any Holder function all of whose integrals with respect to invariant probability measures are nonnegative is cohomologous to a nonnegative Holder function. It is shown that under adding a new row and a new column to a matrix the typical change of the spectral properties for each of its eigenvalues is the following: one largest Jordan block disappears and all the others remain the same. It is proved that this typical change of the spectral properties takes place for the Perron eigenvalue of any principal submatrix of co-order one of an irreducible nonnegative matrix. The same result holds under a typical rank one perturbation. For a typical perturbation of rank $r,$ the spectral properties of a fixed eigenvalue are changed in the following way: $r$ largest Jordan blocks disappear and the others are reserved in the Jordan form of the perturbed matrix. A criterion for a strongly connected subdigraph $S$ to be maximal in the original strongly connected digraph $D$ is obtained (by definition, $S$ is maximal if and only if any strongly connected subdigraph that contains $S$ is either $S$ or $D$). It is shown that any two maximal strongly connected subdigraphs have no common vertices if and only if the diameter of $D$ is one less that its order $n$, the digraph $D$ has a (unique) Hamiltonian circuit and there are at least two pairs of vertices such that the distance between them is equal to $n-1.$ These results are used for studying the connectivity properties and the spectral properties of vertex-deleted subdigraphs with the biggest Perron eigenvalue.
Main publications:
Savchenko S. V. Periodicheskie tochki schetnykh topologicheskikh markovskikh tsepei // Matem. sbornik, 1995, 186 (10), 103–140.
Savchenko S. V. Spetsialnye potoki, postroennye po schetnym TMTs // Funk. anal. i ego pril., 1998, 32 (1), 40–53.
Gurevich B. M., Savchenko S. V. Termodinamicheskii formalizm dlya simvolicheskikh tsepei Markova so schetnym chislom sostoyanii // UMN, 1998, 53 (2), 3–106.
Savchenko S. V. O spektralnykh svoistvakh nerazlozhimoi neotritsatelnoi matritsy i ee glavnykh podmatrits koporyadka odin // UMN, 2000, 55 (1), 191–192.
