Search arXiv⌕ Search

arXiv subjects

Adam Kanigowski

Publications and source records attributed to Adam Kanigowski.

At least 19 recordsLinked to original sources
arXiv2026

Visible Measures along $\Omega(n)$ and Distribution of Horocycle Orbits

Let $\Omega(n)$ denote the number of prime factors of $n$, counted with multiplicities. We study the set $Acc^\Omega(x)$ of weak-$^*$ limits of the sequence $\frac{1}{N}\sum_{n\leq N}\delta_{T^{\Omega(n)}x}$ in $\sigma$-compact dynamical systems $ (X,T)$, demonstrating that if $x \in X$ is quasi-generic for an ergodic measure $\mu$, then $\mu \in Acc^\Omega(x)$. This extends a result of Bergelson and Richter, who studied the problem in the setting of uniquely ergodic systems. We give a more precise description of the set $Acc^\Omega(x)$ in the case of the horocycle flow on non-compact quotients of $SL(2,\mathbb{R})$. We show that for every non-periodic $x\in X$, in addition to Haar measure, there exists sequences $(s_n), (c_n) \subseteq \mathbb{R}$ such that $$ \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-\frac{r^2}{2}}\nu^{i}_{s_n-2\log|1+c_nr|} dr\in Acc^{\Omega}(x), $$ where $\{ \nu^{i}_{s} \}_{i \leq k}$ denotes the one parameter family of periodic measures in each of the $k$ inequivalent cusps. Depending on Diophantine properties of the non-periodic point $x$, we show that $Acc^\Omega(x)$ contains a full two parameter family of such periodic measures, as well as the Dirac measure at each cusp. In particular, these results yield almost-everywhere divergence of pointwise averages along $\Omega(n)$ for the non-compact horocycle flow.

math.DS↗
arXiv2026

Weak mixing for area preserving flows on surfaces

Let $(\phi_t)$ be an area-preserving smooth flow on a compact, connected, orientable surface $\mathcal M$ with at least one but finitely many fixed points. Assume that $(\phi_t)$ is analytic (up to a canonical change of coordinates) in the neighborhood of each saddle fixed point. We show that the flow $(\phi_t)$ is weakly mixing on each of its (finitely many) quasi-minimal components.

math.DS↗
arXiv2025

Density of orbits of horocycle flows at sub-quadratic polynomial times

Let $\Gamma\subset PSL(2,\mathbb{R})$ be such that the space $X=\Gamma\backslash PSL(2,\mathbb{R})$ is not compact. Let $(h_t)$ be the horocycle flow acting on $X$. We show that for every $x\in X$ that is not periodic for $(h_t)$ and for every $\delta\in (0,1)$ the orbit $\{h_{n^{2-\delta}}x\}_{n\in \mathbb{N}}$ is dense in $X$. Assuming additionally the Hardy-Littlewood conjecture we show that for every non-periodic $x\in X$, $\{h_{p}x\}_{p- \text{prime}}$ is dense in $X$. Finally we show that for $\Gamma=PSL(2,\mathbb{Z})$, $\{h_{n^2}y_q\}_{n<q}$ equidistribute, as $q\to \infty$ along primes congruent to $1 \pmod{4}$, towards Haar measure, where $\{y_q\}$ is a sequence of periodic points of period $q$.

math.DS↗
arXiv2025

The influence of the maximal summand on ergodic sums of non-integrable observables over rotations

For $R_{\alpha}$ being an irrational rotation of angle $\alpha$ on the one torus $\mathbb{T}$ and $\phi(x)=\frac{1}{x}-\frac{1}{1-x}$, we compare the behavior of the Birkhoff sum $S_N(\phi)=\sum_{k=0}^{N-1}(\phi\circ R_{\alpha}^k)(x)$ with the successive entry $(\phi\circ R_{\alpha}^N)(x)$. In particular, we are interested in the almost sure limsup behavior of $\frac{(\phi\circ R_{\alpha}^N)(x)}{S_N(\phi)(x)}$. We show that depending on the Diophantine properties of $\alpha$ we have that the limsup either equals $0$ or $\infty$. Moreover, we show that those $\alpha$ for which the limsup equals $0$ form an atypical set in the sense that its Hausdorff dimension equals $\frac{1}{2}$. These results have consequences in studying a reparametrization $(T_t)$ of the linear flow $(L_t)$ with direction $(1,\alpha)$ on the two torus $\mathbb{T}^2$ with function $\varphi$, where $\varphi$ is a smooth non-negative function that has exactly two (non-degenerate) zeros at $\bf p$ and $\bf q$. We prove that for a full measure set $(\alpha, {\bf p}, {\bf q})\in \mathbb{T}\times \mathbb{T}^2\times \mathbb{T}^2$ the special flow $(T_t)$ exhibits extreme historic behavior proving a conjecture given by Andersson and Guih\'eneuf.

math.DS↗
arXiv2025

Singularity of the spectrum of typical minimal smooth area-preserving flows in any genus

We consider smooth flows preserving a smooth invariant measure, or, equivalently, locally Hamiltonian flows on compact orientable surfaces, and show that almost every such locally Hamiltonian flow with only simple saddles has singular spectrum. Furthermore, we prove that for almost every pair of such flows, the elements of the pair are spectrally disjoint. More generally, the results from which these statements are deduced are singularity of the spectrum and pairwise spectral disjointness for special flows over full measure sets of interval exchange transformations under a roof with symmetric logarithmic singularities. Spectral singularity is proved using a criterion based on tightness of Birkhoff sums with exponential tail decay. The assumptions of the criterion are verified exploiting the cancellations proved by the last author to prove the absence of mixing in this class of flows, by showing that the latter can be combined with rigidity by exploiting the local product structure of Rauzy-Veech induction. Pairwise spectral disjointness then follows by producing mixing times (for the second flow), using a new mechanism for shearing based on what we call resonant rigidity times.

math.DS↗
arXiv2024

Multiple mixing for parabolic systems

The famous Rokhlin Problem asks whether mixing implies higher order mixing. So far, all the known examples of zero entropy, mixing dynamical systems enjoy a variant of the mixing via shearing mechanism. In this paper we introduce the notion of locally uniformly shearing systems (LUS) which is a rigorous way of describing the mixing via shearing mechanism. We prove that all LUS flows are mixing of all orders. We then show that mixing smooth flows on surfaces and smooth time-changes of unipotent flow are LUS. We also introduce the notion of quantitative LUS. We show that polynomially mixing systems that are polynomially LUS are in fact polynomially mixing of all orders. As a consequence we show that Kochergin flows on $\mathbb{T}^2$ (for a.e. irrational frequency) as well as smooth time-changes of unipotent flows are polynomially mixing of all orders.

math.DS↗
arXiv2024

Horocycle flow at product of two primes

We show that if $\Gamma$ is a co-compact arithmetic lattice in $SL(2,\mathbb{R})$ or $\Gamma=SL(2,\mathbb{Z})$ then the horocycle orbit of every non-periodic point $x\in SL(2,\mathbb{R})/\Gamma$ equidistributes (with respect to Haar measure) when sampled at integers having exactly two prime factors.

math.DS↗
arXiv2023

On the local Fourier uniformity problem for small sets

We consider vanishing properties of exponential sums of the Liouville function $\lambda$ of the form $$ \lim_{H\to\infty}\limsup_{X\to\infty}\frac{1}{\log X}\sum_{m\leq X}\frac{1}{m}\sup_{\alpha\in C}\bigg|\frac{1}{H}\sum_{h\leq H}\lambda(m+h)e^{2\pi ih\alpha}\bigg|=0, $$ where $C\subset\mathbb{T}$. The case $C=\mathbb{T}$ corresponds to the local $1$-Fourier uniformity conjecture of Tao, a central open problem in the study of multiplicative functions with far-reaching number-theoretic applications. We show that the above holds for any closed set $C\subset\mathbb{T}$ of zero Lebesgue measure. Moreover, we prove that extending this to any set $C$ with non-empty interior is equivalent to the $C=\mathbb{T}$ case, which shows that our results are essentially optimal without resolving the full conjecture. We also consider higher-order variants. We prove that if the linear phase $e^{2\pi ih\alpha}$ is replaced by a polynomial phase $e^{2\pi ih^t\alpha}$ for $t\geq 2$ then the statement remains true for any set $C$ of upper box-counting dimension $<1/t$. The statement also remains true if the supremum over linear phases is replaced with a supremum over all nilsequences coming form a compact countable ergodic subsets of any $t$-step nilpotent Lie group. Furthermore, we discuss the unweighted version of the local $1$-Fourier uniformity problem, showing its validity for a class of ``rigid'' sets (of full Hausdorff dimension) and proving a density result for all closed subsets of zero Lebesgue measure.

math.DS↗
arXiv2023

A non-mixing Arnold flow on a surface

We construct a smooth area preserving flow on a genus 2 surface with exactly one open uniquely ergodic component, that is asymmetrically bounded by separatrices of non-degenerate saddles and that is nevertheless not mixing.

math.DS↗
arXiv2023

Limit theorems for low dimensional generalized $(T,T^{-1})$ transformations

We consider generalized $(T, T^{-1})$ transformations such that the base map satisfies a multiple mixing local limit theorem and anticoncentration large deviation bounds and in the fiber we have $\mathbb{R}^d$ actions with $d=1$ or $2$ which are exponentially mixing of all orders. If the skewing cocycle has zero drift, we show that the ergodic sums satisfy the same limit theorems as the random walks in random scenery studied by Kesten and Spitzer (1979) and Bolthausen (1989). The proofs rely on the quenched CLT for the fiber action and the control of the quenched variance. This paper complements our previous work where the classical central limit theorem is obtained for a large class of generalized $(T, T^{-1})$ transformations.

math.DS↗
arXiv2021

Counterexamples to a rigidity conjecture

We discuss several counterexamples to a rigidity conjecture of K. Khanin, which states that under some quantitative condition on non-existence of periodic orbits, $C^0$ conjugacy implies $C^1$ (even $C^\infty$) conjugacy. We construct examples of non-rigid diffeomorphisms on the $2$-torus, which satisfy the assumptions of Khanin's (but not of Krikorian's) conjecture. We also construct examples of flows which are topologically conjugate, but not $C^1$ conjugate, in contradiction to a natural generalization of the conjecture to flows. These latter examples are based on results on solutions of the cohomological equation and suggest that the structure of the space of invariant distributions has to be taken into account in rigidity questions.

math.DS↗
arXiv2021

Exponential mixing implies Bernoulli

Let $f$ be a $C^{1+\alpha}$ diffeomorphism of a compact manifold $M$ preserving a smooth measure $\mu$. We show that if $f:(M,\mu)\to (M,\mu)$ is exponentially mixing then it is Bernoulli.

math.DS↗
arXiv2021

On arithmetic functions orthogonal to deterministic sequences

We prove Veech's conjecture on the equivalence of Sarnak's conjecture on M\"obius orthogonality with a Kolmogorov type property of Furstenberg systems of the M\''obius function. This yields a combinatorial condition on the M\"obius function itself which is equivalent to Sarnak's conjecture. As a matter of fact, our arguments remain valid in a larger context: we characterize all bounded arithmetic functions orthogonal to all topological systems whose all ergodic measures yield systems from a fixed characteristic class (zero entropy class is an example of such a characteristic class) with the characterization persisting in the logarithmic setup. As a corollary, we obtain that the logarithmic Sarnak's conjecture holds if and only if the logarithmic M\''obius orthogonality is satisfied for all dynamical systems whose ergodic measures yield nilsystems.

math.DS↗
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 $1$.

math.DS↗
arXiv2020

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

We consider a class of smooth mixing flows $T^{\alpha,\gamma}$ on $\mathbb{T}^2$ with one degenerated fixed point $x_0\in \mathbb{T}^2$ of power type $\gamma\in (-1,0)$. We prove that for a $G_\delta$ dense set of $\alpha\in \mathbb{T}$, a prime number theorem for $T^{\alpha,\gamma}$ 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 $\log^{-A}N$).

math.DS↗
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 $G/\Gamma$. 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.

math.DS↗