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Anton V. Solomko

Publications and source records attributed to Anton V. Solomko.

5 recordsLinked to original sources
arXiv2016

On isomorphism problem for von Neumann flows with one discontinuity

A von Neumann flow is a special flow over an irrational rotation of the circle and under a piecewise $C^1$ roof function with a non-zero sum of jumps. We prove that the absolute value of the slope is a (measure theoretic) invariant in the class of von Neumann special flows with one discontinuity, i.e. two ergodic von Neumann flows with one discontinuity are not isomorphic if the slopes of the roof functions have different absolute values, regardless of the irrational rotation in the base.

math.DS↗
arXiv2016

On rank of von Neumann special flows

We prove that special flows over an ergodic rotation of the circle under a $C^1$ roof function with one discontinuity do not have local rank one. In particular, any such flow has infinite rank.

math.DS↗
arXiv2011

New spectral multiplicities for ergodic actions

Let G be a locally compact second countable Abelian group. Given a measure preserving action T of G on a standard probability space, let M(T) denote the set of essential values of the spectral multiplicity function of the Koopman unitary representation of G associated with T. In the case when G is either a discrete countable Abelian group or R^n, n>0, it is shown that the sets of the form {p,q,pq}, {p,q,r,pq,pr,qr,pqr} etc. or any multiplicative (and additive) subsemigroup of N are realizable as M(T) for a weakly mixing G-action T.

math.DS↗
arXiv2010

Ergodic Abelian actions with homogeneous spectrum

It is shown that for each $N>0$ and for a wide class of Abelian non-compact locally compact second countable groups $G$ including all infinite countable discrete ones and $\Bbb R^{d_1}\times\Bbb Z^{d_2}$ with $d_1,d_2\ge 0$, there exists a weakly mixing probability preserving $G$-action with a homogeneous spectrum of multiplicity $N$.

math.DS↗