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Adam Kanigowski

Publications and source records attributed to Adam Kanigowski.

At least 37 records · Page 2Linked to original sources
arXiv2020

Rigidity of joinings for some measure preserving systems

We introduce two properties: strong R-property and $C(q)$-property, describing a special way of divergence of nearby trajectories for an abstract measure preserving system. We show that systems satisfying the strong R-property are disjoint (in the sense of Furstenberg) with systems satisfying the $C(q)$-property. Moreover, we show that if $u_t$ is a unipotent flow on $G/Γ$ with $Γ$ irreducible, then $u_t$ satisfies the $C(q)$-property provided that $u_t$ is not of the form $h_t\times\operatorname{id}$, where $h_t$ is the classical horocycle flow. Finally, we show that the strong R-property holds for all (smooth) time changes of horocycle flows and non-trivial time changes of bounded type Heisenberg nilflows.

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arXiv2020

Flexibility of statistical properties for smooth systems satisfying the central limit theorem

In this paper we exhibit new classes of smooth systems which satisfy the Central Limit Theorem (CLT) and have (at least) one of the following properties: (1) zero entropy; (2) weak but not strong mixing; (3) (polynomially) mixing but not $K$; (4) $K$ but not Bernoulli; (5) non Bernoulli and mixing at arbitrary fast polynomial rate. We also give an example of a system satisfying the CLT where the normalizing sequence is regularly varying with index $1$.

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arXiv2020

Prime orbits for some smooth flows on $\mathbb{T}^2$

We consider a class of smooth mixing flows $T^{α,γ}$ on $\mathbb{T}^2$ with one degenerated fixed point $x_0\in \mathbb{T}^2$ of power type $γ\in (-1,0)$. We prove that for a $G_δ$ dense set of $α\in \mathbb{T}$, a prime number theorem for $T^{α,γ}$ holds along a full upper density subsequence. In particular it follows that for every $x\in \mathbb{T}^2\setminus\{x_0\}$, the prime orbit $\mathbb{T}^2$. We also show that there exists a class of smooth weakly mixing flows on $\mathbb{T}^2$ for which a prime number theorem holds. In fact we show that there exists a dense set of smooth functions (in the uniform topology) for which prime number theorem holds quantitatively (with an error term $\log^{-A}N$).

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arXiv2020

Slow entropy of higher rank abelian unipotent actions

We study slow entropy invariants for abelian unipotent actions $U$ on any finite volume homogeneous space $G/Γ$. For every such action we show that the topological slow entropy can be computed directly from the dimension of a special decomposition of $\operatorname{Lie}(G)$ induced by $\operatorname{Lie}(U)$. Moreover, we are able to show that the metric slow entropy of the action coincides with its topological slow entropy. As a corollary, we obtain that the complexity of any abelian horocyclic action is only related to the dimension of $G$. This generalizes the rank one results from [A. Kanigowski, K. Vinhage, D. Wei, Commun. Math. Phys. 370 (2019), no. 2, 449-474.] to higher rank abelian actions.

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arXiv2020

Survey on entropy-type invariants of sub-exponential growth in dynamical systems

Measure-theoretic and topological entropy are classical invariants in the theory of dynamical systems. There are several recently developed entropy type invariants for systems of sub-exponential growth: sequence entropy, slow entropy, Kakutani invariants, scaled entropy, entropy dimensions and entropy convergence rate. They measure the complexity of zero entropy systems by different approaches. These new invariants and corresponding new theories have many applications and interesting properties. This survey paper gives a comprehensive exposition of the slow entropy theory and also discusses some related topics.

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arXiv2020

Prime number theorem for analytic skew products

We establish a prime number theorem for all uniquely ergodic, analytic skew products on the $2$-torus $\mathbb{T}^2$. More precisely, for every irrational $α$ and every $1$-periodic real analytic $g:\mathbb{R}\to\mathbb{R}$ of zero mean, let $T_{α,g} : \mathbb{T}^2 \rightarrow \mathbb{T}^2$ be defined by $(x,y) \mapsto (x+α,y+g(x))$. We prove that if $T_{α, g}$ is uniquely ergodic then, for every $(x,y) \in \mathbb{T}^2$, the sequence $\{T_{α, g}^p(x,y)\}$ is equidistributed on $\mathbb{T}^2$ as $p$ traverses prime numbers. This is the first example of a class of natural, non-algebraic and smooth dynamical systems for which a prime number theorem holds. We also show that such a prime number theorem does not necessarily hold if $g$ is only continuous on $\mathbb{T}^2$.

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arXiv2020

Polynomial 3-mixing for smooth time-changes of horocycle flows

Let $(h_t)_{t\in \mathbb{R}}$ be the horocycle flow acting on $(M,μ)=(Γ\backslash \text{SL}(2,\mathbb{R}),μ)$, where $Γ$ is a co-compact lattice in $\text{SL}(2,\mathbb{R})$ and $μ$ is the homogeneous probability measure locally given by the Haar measure on $\text{SL}(2,\mathbb{R})$. Let $τ\in W^6(M)$ be a strictly positive function and let $μ^τ$ be the measure equivalent to $μ$ with density $τ$. We consider the time changed flow $(h_t^τ)_{t\in \mathbb{R}}$ and we show that there exists $γ=γ(M,τ)>0$ and a constant $C>0$ such that for any $ f_0, f_1, f_2\in W^6(M)$ and for all $0=t_0<t_1<t_2$, we have $$\ \left|\int_M \prod_{i=0}^{2} f_i\circ h^τ_{t_i} d μ^τ-\prod_{i=0}^{2}\int_M f_i d μ^τ\right|\leq C \left(\prod_{i=0}^{2} \|f_i\|_6\right) \left(\min_{0\leq i<j\leq 2} |t_i-t_j|\right)^{-γ}.$$ With the same techniques, we establish polynomial mixing of all orders under the additional assumption of $τ$ being fully supported on the discrete series.

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arXiv2019

Bernoulli property for certain skew products over hyperbolic systems

We study the Bernoulli property for a class of partially hyperbolic systems arising from skew products. More precisely, we consider a hyperbolic map $(T,M,μ)$, where $μ$ is a Gibbs measure, an aperiodic Hölder continuous cocycle $ϕ:M\to \mathbb R$ with zero mean and a zero-entropy flow $(K_t,N,ν)$. We then study the skew product $$ T_ϕ(x,y)=(Tx,K_{ϕ(x)}y), $$ acting on $(M\times N,μ\times ν)$. We show that if $(K_t)$ is of slow growth and has good equidistribution properties, then $T_ϕ$ remains Bernoulli. In particular, our main result applies to $(K_t)$ being a typical translation flow on a surface of genus $\geq 1$ or a smooth reparametrization of isometric flows on $\mathbb T^2$. This provides examples of non-algebraic, partially hyperbolic systems which are Bernoulli and for which the center is non-isometric (in fact might be weakly mixing).

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arXiv2019

Spectral disjointness of rescalings of some surface flows

We study self-similarity problem for two classes of flows: (1) special flows over circle rotations and under roof functions with symmetric logarithmic singularities (2) special flows over interval exchange transformations and under roof functions which are of two types * piecewise constant with one additional discontinuity which is not a discontinuity of the IET; * piecewise linear over exchanged intervals with non-zero slope. We show that if $\{T^f_t\}_{t\in\mathbb R}$ is as in (1) then for a full measure set of rotations, and for every two distinct natural numbers $K$ and $L$, we have that $\{T^f_{Kt}\}_{t\in\mathbb R}$ and $\{T^f_{Lt}\}_{t\in\mathbb R}$ are spectrally disjoint. Similarly, if $\{T^f_t\}_{t\in\mathbb R}$ is as in (2), then for a full measure set of IET's, a.e. position of the additional discontinuity (of $f$, in piecewise constant case) and every two distinct natural numbers $K$ and $L$, the flows $\{T^f_{Kt}\}_{t\in\mathbb R}$ and $\{T^f_{Lt}\}_{t\in\mathbb R}$ are spectrally disjoint.

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arXiv2019

Lebesgue spectrum of countable multiplicity for conservative flows on the torus

We study the spectral measures of conservative mixing flows on the 2-torus having one degenerate singularity. We show that, for a sufficiently strong singularity, the spectrum of these flows is typically Lebesgue with infinite multiplicity. For this, we use two main ingredients: 1) a proof of absolute continuity of the maximal spectral type for this class of non-uniformly stretching flows that have an irregular decay of correlations, 2) a geometric criterion that yields infinite Lebesgue multiplicity of the spectrum and that is well adapted to rapidly mixing flows.

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arXiv2019

Kakutani Equivalence of Unipotent Flows

We study Kakutani equivalence in the class of unipotent flows acting on finite volume quotients of semisimple Lie groups. For every such flow we compute the Kakutani invariant of M. Ratner, the value of which being explicitly given by the Jordan block structure of the unipotent element generating the flow. This in particular answers a question of M. Ratner. Moreover it follows that the only standard unipotent flows are given by $\begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix} \times \operatorname{id}$ acting on $(SL(2,\mathbb{R}) \times G')/Γ'$, where $Γ'$ is an irreducible lattice in $SL(2,\mathbb{R}) \times G'$ (with the possibility that $G' = \{e\}$).

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arXiv2018

Bernoulli property for homogeneous systems

Let $G$ be a semisimple Lie group with Haar measure $μ$ and let $Γ$ be an irreducible lattice in $G$. For $g\in G$, we consider left translation $L_g$ acting on $(G\backslashΓ,μ)$. We show that if $L_g$ is $K$ (which is equivalent to positive entropy of $L_g$) then $L_g$ is a Bernoulli automorphism. As a corollary, we also obtain analogous results for homogeneous flows.

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arXiv2018

Rigidity of a class of smooth singular flows on $\mathbb T^2$

We study joining rigidity in the class of von Neumann flows with one singularity. They are given by a smooth vector field $\mathcal{X}$ on $\mathbb T^2\setminus \{a\}$, where $\mathcal{X}$ is not defined at $a\in \mathbb T^2$. It follows that the phase space can be decomposed into a (topological disc) $D_\mathcal{X}$ and an ergodic component $E_\mathcal{X}=\mathbb T^2\setminus D_\mathcal{X}$. Let $ω_\mathcal{X}$ be the 1-form associated to $\mathcal{X}$. We show that if $|\int_{E_{\mathcal{X}_1}}dω_{\mathcal{X}_1}|\neq |\int_{E_{\mathcal{X}_2}}dω_{\mathcal{X}_2}|$, then the corresponding flows $(v_t^{\mathcal{X}_1})$ and $(v_t^{\mathcal{X}_2})$ are disjoint. It also follows that for every $\mathcal{X}$ there is a uniquely associated frequency $α=α_{\mathcal{X}}\in \mathbb T$. We show that for a full measure set of $α\in \mathbb T$ the class of smooth time changes of $(v_t^\mathcal{X_α})$ is joining rigid, i.e. every two smooth time changes are either cohomologous or disjoint. This gives a natural class of flows for which the answer to a problem of Ratner (Problem 3 in \cite{Rat4}) is positive.

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arXiv2018

Mutliple mixing and disjointness for time changes of bounded-type Heisenberg nilflows

We study time changes of bounded type Heisenberg nilflows $(ϕ_t)$ acting on the Heisenberg nilmanifold $M$. We show that for every positive $τ\in W^s(M)$, $s~>~7/2$, every non-trivial time change $(ϕ_t^τ)$ enjoys the Ratner property. As a consequence every mixing time change is mixing of all orders. Moreover we show that for every $τ\in W^s(M)$, $s>9/2$ and every $p,q\in \mathbb{N}$, $p\neq q$, $(ϕ_{pt}^τ)$ and $(ϕ_{qt}^τ)$ are disjoint. As a consequence Sarnak's Conjecture on Möbius disjointness holds for all such time changes.

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arXiv2018

On disjointness properties of some parabolic flows

The Ratner property, a quantitative form of divergence of nearby trajectories, is a central feature in the study of parabolic homogeneous flows. Discovered by Marina Ratner and used in her 1980th seminal works on horocycle flows, it pushed forward the disjointness theory of such systems. In this paper, exploiting a recent variation of the Ratner property, we prove new disjointness phenomena for smooth parabolic flows beyond the homogeneous world. In particular, we establish a general disjointness criterion based on the switchable Ratner property. We then apply this new criterion to study disjointness properties of smooth time changes of horocycle flows and smooth Arnol'd flows on the torus, focusing in particular on disjointness of distinct flow rescalings. As a consequence, we answer a question by Marina Ratner on the Moebius orthogonality of time-changes of horocycle flows. In fact, we prove Moebius orthogonality for all smooth time-changes of horocycle flows and uniquely ergodic realizations of Arnol'd flows considered.

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arXiv2018

Horocycle flow on negative variable curvature surface is standard

We provide a new proof that the horocycle flow preserving the Margulis measure on a variable negative curvature surface is standard. This was first proved by Ratner. The main purpose of this note is to provide a simplified case of the arguments for Kakutani equivalence of unipotent flows on homogeneous spaces, which have similar but more complicated structures, as well as illustrate the versatility of the method by applying it to a non-homogeneous flow.

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