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Alexandre I. Danilenko

Publications and source records attributed to Alexandre I. Danilenko.

35 records · Page 2Linked to original sources
arXiv2015

Odometer actions of the Heisenberg group

Let $H_3(\Bbb R)$ denote the 3-dimensional real Heisenberg group. Given a family of lattices $Γ_1\supsetΓ_2\supset\cdots$ in it, let $T$ stand for the associated uniquely ergodic $H_3(\Bbb R)$-{\it odometer}, i.e. the inverse limit of the $H_3(\Bbb R)$-actions by rotations on the homogeneous spaces $H_3(\Bbb R)/Γ_j$, $j\in\Bbb N$. The decomposition of the underlying Koopman unitary representation of $H_3(\Bbb R)$ into a countable direct sum of irreducible components is explicitly described. The ergodic 2-fold self-joinings of $T$ are found. It is shown that in general, the $H_3(\Bbb R)$-odometers are neither isospectral nor spectrally determined.

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arXiv2015

Finite ergodic index and asymmetry for infinite measure preserving actions

Given $k>0$ and an Abelian countable discrete group $G$ with elements of infinite order, we construct $(i)$ rigid funny rank-one infinite measure preserving (i.m.p.) $G$-actions of ergodic index $k$, $(ii)$ 0-type funny rank-one i.m.p. $G$-actions of ergodic index $k$, $(iii)$ funny rank-one i.m.p. $G$-actions $T$ of ergodic index 2 such that the product $T\times T^{-1}$ is not ergodic. It is shown that $T\times T^{-1}$ is conservative for each funny rank-one $G$-action $T$.

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arXiv2014

Actions of finite rank: weak rational ergodicity and partial rigidity

A simple proof of the fact that each rank-one infinite measure preserving (i.m.p.) transformation is subsequence weakly rationally ergodic is found. Some classes of funny rank-one i.m.p. actions of Abelian groups are shown to be subsequence weakly rationally ergodic. A constructive definition of finite funny rank for actions of arbitrary infinite countable groups is given. It is shown that the ergodic i.m.p. transformations of balanced finite funny rank are subsequence weakly rationally ergodic. It is shown that the ergodic probability preserving transformations of exact finite rank, the ergodic Bratteli-Vershik maps corresponding to the "consequtively ordered" Bratteli diagrams of finite rank, some their generalizations and the ergodic IETs are partially rigid.

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arXiv2014

Directional recurrence and directional rigidity for infinite measure preserving actions of nilpotent lattices

Let $Γ$ be a lattice in a simply connected nilpotent Lie group $G$. Given an infinite measure preserving action $T$ of $Γ$ and a "direction" in $G$ (i.e. an element $θ$ of the projective space $P(\goth g)$ of the Lie algebra $\goth g$ of $G$), some notions of recurrence and rigidity for $T$ along $θ$ are introduced. It is shown that the set of recurrent directions $\Cal R(T)$ and the set of rigid directions for $T$ are both $G_δ$. In the case where $G=\Bbb R^d$ and $Γ=\Bbb Z^d$, we prove that (a) for each $G_δ$-subset $Δ$ of $P(\goth g)$ and a countable subset $D\subsetΔ$, there is a rank-one action $T$ such that $D\subset\Cal R(T)\subsetΔ$ and (b) $\Cal R(T)=P(\goth g)$ for a generic infinite measure preserving action $T$ of $Γ$. This answers partly a question from a recent paper by A.~Johnson and A.~{\c S}ahin. Some applications to the directional entropy of Poisson actions are discussed. In the case where $G$ is the Heisenberg group $H_3(\Bbb R)$ and $Γ=H_3(\Bbb Z)$, a rank-one $Γ$-action $T$ is constructed for which $\Cal R(T)$ is not invariant under the natural "adjoint" $G$-action.

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arXiv2012

On self-similarities of ergodic flows

Given an ergodic flow $T=(T_t)_{t\in\Bbb R}$, let $I(T)$ be the set of reals $s\ne 0$ for which the flows $(T_{st})_{t\in\Bbb R}$ and $T$ are isomorphic. It is proved that $I(T)$ is a Borel subset of $\Bbb R^*$. It carries a natural Polish group topology which is stronger than the topology induced from $\Bbb R$. There exists a mixing flow $T$ such that $I(T)$ is an uncountable meager subset of $\Bbb R^*$. For a generic flow $T$, the transformations $T_{t_1}$ and $T_{t_2}$ are spectrally disjoint whenever $|t_1|\ne |t_2|$. A generic transformation (i) embeds into a flow $T$ with $I(T)=\{1\}$ and (ii) does not embed into a flow with $I(T)\ne \{1\}$. For each countable multiplicative subgroup $S\subset\Bbb R^*$, it is constructed a Poisson suspension flow $T$ with simple spectrum such that $I(T)=S$. If $S$ is without rational relations then there is a rank-one weakly mixing rigid flow $T$ with $I(T)=S$.

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arXiv2011

Mixing actions of Heisenberg group

Mixing (of all orders) rank-one actions $T$ of Heisenberg group $H_3(\Bbb R)$ are constructed. The restriction of $T$ to the center of $H_3(\Bbb R)$ is simple and commutes only with $T$. Mixing Poisson and mixing Gaussian actions of $H_3(\Bbb R)$ are also constructed. A rigid weakly mixing rank-one action $T$ is constructed such that the restriction of $T$ to the center of $H_3(\Bbb R)$ is not isomorphic to its inverse.

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arXiv2011

Flows with uncountable but meager group of self-similarities

Given an ergodic probability preserving flow $T=(T_t)_{t\in\Bbb R}$, let $I(T):=\{s\in\Bbb R^*\mid T\text{is isomorphic to}(T_{st})_{t\in\Bbb R}\}$. A weakly mixing Gaussian flow $T$ is constructed such that $I(T)$ is uncountable and meager. For a Poisson flow $T$, a subgroup $I_{\text{Po}}(T)\subset I(T)$ of Poissonian self-similarities is introduced. Given a probability measure $κ$ on $\Bbb R^*_+$, a zero-entropy Poisson flow $T$ is constructed such that $I_{\text{Po}}(T)$ is the group of $κ$-quasi-invariance.

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arXiv2011

A survey on spectral multiplicities of ergodic actions

Given a transformation $T$ of a standard measure space $(X,μ)$, let $\Cal M(T)$ denote the set of spectral multiplicities of the Koopman operator $U_T$ defined in $L^2(X,μ)\ominus\Bbb C$ by $U_Tf:=f\circ T$. It is discussed in this survey paper which subsets of $\Bbb N\cup\{\infty\}$ are realizable as $\Cal M(T)$ for various $T$: ergodic, weakly mixing, mixing, Gaussian, Poisson, ergodic infinite measure preserving, etc. The corresponding constructions are considered in detail. Generalizations to actions of Abelian locally compact second countable groups are also discussed.

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arXiv2010

Spectral multiplicities for ergodic flows

Let $E$ be a subset of positive integers such that $E\cap\{1,2\}\ne\emptyset$. A weakly mixing finite measure preserving flow $T=(T_t)_{t\in\Bbb R}$ is constructed such that the set of spectral multiplicities (of the corresponding Koopman unitary representation generated by $T$) is $E$. Moreover, for each non-zero $t\in\Bbb R$, the set of spectral multiplicities of the transformation $T_t$ is also $E$. These results are partly extended to actions of some other locally compact second countable Abelian groups.

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arXiv2010

Mixing constructions with infinite invariant measure and spectral multiplicities

We introduce high staircase infinite measure preserving transformations and prove that they are mixing under a restricted growth condition. This is used to (i) realize each subset $E\subset\Bbb N\cup\{\infty\}$ as the set of essential values of the multiplicity function for the Koopman operator of a mixing ergodic infinite measure preserving transformation, (ii) construct mixing power weakly mixing infinite measure preserving transformations, (iii) construct mixing Poissonian automorphisms with a simple spectrum, etc.

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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 $N$.

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arXiv2009

On new spectral multiplicities for ergodic maps

It is shown that each subset of positive integers that contains 2 is realizable as the set of essential values of the multiplicity function for the Koopman operator of some weakly mixing transformation.

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arXiv2008

Almost continuous orbit equivalence for non-singular homeomorphisms

Let $X$ and $Y$ be Polish spaces with non-atomic Borel measures $μ$ and $ν$ of full support. Suppose that $T$ and $S$ are ergodic non-singular homeomorphisms of $(X,μ)$ and $(Y,ν)$ with continuous Radon-Nikodym derivatives. Suppose that either they are both of type $III_1$ or that they are both of type $III_λ$, $0<λ<1$ and, in the $III_λ$ case, suppose in addition that both `topological asymptotic ranges' (defined in the article) are $\logλ\cdot\Bbb Z$. Then there exist invariant dense $G_δ$-subsets $X'\subset X$ and $Y'\subset Y$ of full measure and a non-singular homeomorphism $ϕ: X' \to Y'$ which is an orbit equivalence between $T|_{X'}$ and $S|_{Y'}$, that is $ϕ\{T^{i}x\} = \{S^{i}x\}$ for all $x \in X'$. Moreover the Radon-Nikodym derivative $dν\circϕ/dμ$ is continuous on $X'$ and, letting $S' = ϕ^{-1}S ϕ$ we have $Tx= {S'}^{n(x)}x$ and $S' = T^{m(x)}x$ where $n$ and $m$ are continuous on $X'$.

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