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.
Biography
Graduated from Faculty of Mathematics and Mechanics of M.V. Lomonosov Moscow State University (MSU) in 1992 (department of mathematical statistics and random processes). Ph.D. thesis was defended in 1996. A list of my works contains more than 15 titles.
Member of Moscow Mathematical Society.
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.
S. V. Savchenko, “On the number of noncritical vertices in strongly connected digraphs”, Mat. Zametki, 79:5 (2006), 743–755; Math. Notes, 79:5 (2006), 687–696
S. V. Savchenko, “Laurent expansion for the determinant of the matrix of scalar resolvents”, Mat. Sb., 196:5 (2005), 121–144; Sb. Math., 196:5 (2005), 743–764
S. V. Savchenko, “On the Change in the Spectral Properties of a Matrix under Perturbations of Sufficiently Low Rank”, Funktsional. Anal. i Prilozhen., 38:1 (2004), 85–88; Funct. Anal. Appl., 38:1 (2004), 69–71
S. V. Savchenko, “Typical Changes in Spectral Properties under Perturbations by a Rank-One Operator”, Mat. Zametki, 74:4 (2003), 590–602; Math. Notes, 74:4 (2003), 557–568
S. V. Savchenko, “Maximal suborgraphs with the biggest Perron number”, Uspekhi Mat. Nauk, 56:6(342) (2001), 165–166; Russian Math. Surveys, 56:6 (2001), 1181–1182
2000
6.
S. V. Savchenko, “Spectral properties of an indecomposable non-negative matrix and its principal submatrices of co-order one”, Uspekhi Mat. Nauk, 55:1(331) (2000), 191–192; Russian Math. Surveys, 55:1 (2000), 184–185
S. V. Savchenko, “Stability of the upper bound of a spectrum under perturbations by diagonal matrices for a class of self-adjoint operators”, Uspekhi Mat. Nauk, 53:2(320) (1998), 163–164; Russian Math. Surveys, 53:2 (1998), 406–407
11.
B. M. Gurevich, S. V. Savchenko, “Thermodynamic formalism for countable symbolic Markov chains”, Uspekhi Mat. Nauk, 53:2(320) (1998), 3–106; Russian Math. Surveys, 53:2 (1998), 245–344
S. V. Savchenko, “On the ground state of free and random discrete Hamiltonians perturbed by an operator of rank one for a critical value of the coupling constant”, TMF, 114:1 (1998), 94–103; Theoret. and Math. Phys., 114:1 (1998), 73–80
S. V. Savchenko, “On the probability of the existenceof a localized basic state for a discrete Schrödinger equation with random potential, perturbed by a compact operator”, Teor. Veroyatnost. i Primenen., 43:1 (1998), 166–171; Theory Probab. Appl., 43:1 (1999), 158–162
S. V. Savchenko, “On the spectra of finite submatrices of infinite irreducible matrices”, Uspekhi Mat. Nauk, 52:3(315) (1997), 175–176; Russian Math. Surveys, 52:3 (1997), 619–620
S. V. Savchenko, “Equilibrium states with incomplete supports and periodic trajectories”, Mat. Zametki, 59:2 (1996), 230–253; Math. Notes, 59:2 (1996), 163–179
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