arXiv subjects

Przemysław Berk

Publications and source records attributed to Przemysław Berk.

14 recordsLinked to original sources

arXiv2026

On disjointness of 3-IETs from mixing and lightly mixing systems

In this note we show that all interval exchange transformations of three intervals (3-IETs) are spectrally disjoint from all mixing and disjoint from lightly mixing systems. Until now, it was known that almost every 3-IET is rigid, which implies this property. However, not all 3-IETs are rigid, hence our result is a strict improvement of current knowledge. We then use this result, among many others, to prove that although some 3-IETs are mildly mixing, they are always disjoint in the sense of Furstenberg from lightly mixing systems.

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arXiv2026

Coboundaries of 3-IETs

In this note, we investigate coboundaries of interval exchange transformations of three intervals, or 3-IETs. More precisely, we study differentiable functions whose derivative is absolutely continuous, and whose integral is zero and whose derivative also has integral zero. We show that these functions are coboundaries for a typical 3-IET if and only if their values at the endpoints of the domain are zero. We also show the existence of exceptional counterexamples for both possible endpoint behaviors. Our results are obtained by studying the properties of associated skew products.

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arXiv2026

Disjointness of rescalings of smooth area preserving flows on surfaces

We consider the problem of \emph{disjointness of rescalings} $(φ_{κt})_{t\in \mathbb{R}}$, $κ\in\mathbb{R}$ of a flow $(φ_{t})_{t\in \mathbb{R}}$ in the context of smooth flows preserving a smooth invariant measure, or, equivalently, locally Hamiltonian flows on compact orientable surfaces. We show that, when the genus of the surface is $g\ge 2$, almost every locally Hamiltonian flow with 2g-2 non-degenerate simple saddles is such that any distinct two rational rescalings $(φ_{κt})_{t\in \mathbb{R}}$ and $(φ_{κ' t})_{t\in \mathbb{R}}$ with $κ=p/q$ and $κ'=p'/q'$ of different absolute values, are disjoint. Previous results on disjointness of rescalings were available only for rescalings for locally Hamiltonian flows and their special flow representations in genus one. The result is proved using a criterion for disjointness based on the study of the distribution of Birkhoff sums of a special representation and in particular estimates on their exponential tails decay. A key novel geometric ingredient in the proof is the existence of a sequence of rigidity times which display what we call bounded-type rigidity, so that a large set of points comes back in time $q$ with distance $O(1/q)$. To produce such bounded-type rigidity times we exploit a particular way of degeneration of a translation surface to a flat torus for which the vertical flow has bounded-type rotation number.

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arXiv2026

Coexistence of two contrasting recurrence properties of certain non-integrable cocycles

We study the recurrence properties of certain skew products over symmetric interval exchange transformations, including rotations, with cocycles of the form $f(x)=-\frac{1}{x^a}+\frac{1}{(1-x)^a}$, where $a>1$. We prove that typically, such systems are dissipative. However, at the same time they are \emph{topologically recurrent}, i.e. for every open rectangle $A\subset[0,1)\times \R$, there exists an infinite sequence $(q_n)_{n=1}^{\infty}$ such that $T^{q_n}_f(A)\cap A\neq\emptyset$.

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arXiv2026

On the ergodicity of anti-symmetric skew products with singularities and its applications

We introduce a novel method for proving ergodicity for skew products of interval exchange transformations (IETs) with piecewise smooth cocycles having singularities at the ends of exchanged intervals. This approach is inspired by Borel-Cantelli-type arguments from Fayad and Lemańczyk (2006). The key innovation of our method lies in its applicability to singularities beyond the logarithmic type, whereas previous techniques were restricted to logarithmic singularities. Our approach is particularly effective for proving the ergodicity of skew products for symmetric IETs and antisymmetric cocycles. Moreover, its most significant advantage is its ability to study the equidistribution of error terms in the spectral decomposition of Birkhoff integrals for locally Hamiltonian flows on compact surfaces, applicable not only when all saddles are perfect (harmonic) but also in the case of some non-perfect saddles.

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arXiv2024

Ergodic properties of infinite extension of symmetric interval exchange transformations

We prove that skew products with the cocycle given by the function $f(x)=a(x-1/2)$ with $a\neq 0$ are ergodic for every ergodic symmetric IET in the base, thus giving the full characterization of ergodic extensions in this family. Moreover, we prove that under an additional natural assumption of unique ergodicity on the IET, we can replace $f$ with any differentiable function with a non-zero sum of jumps. Finally, by considering weakly mixing IETs instead of just ergodic, we show that the skew products with cocycle given by $f$ have infinite ergodic index.

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arXiv2024

Ergodicity of skew-products over typical IETs

We prove ergodicity of a class of infinite measure preserving systems, called skew-products. More precisely, we consider systems of the form \[ {T_f}:{[0, 1) \times \mathbb{R}}\to{[0, 1) \times \mathbb{R}},\quad {T_f(x, t)}:={(T(x), t+f(x))}, \] where $T$ is an interval exchange transformation and $f$ is a piece-wise constant function with a finite number of discontinuities. We show that such system is ergodic with respect to ${Leb}_{[0,1)\times \mathbb{R}}$ for a typical choice of parameters of $T$ and $f$.

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arXiv2024

On the ergodicity of infinite antisymmetric extensions of symmetric IETs

In this article, we consider skew product extensions over symmetric interval exchange transformations with respect to the cocycle $f(x)=χ_{(0,1/2)}-χ_{(1/2,1)}$. More precisely, we prove that for almost every interval exchange transformation $T$ with symmetric combinatorial data, the skew product $T_f: [0, 1) \times \mathbb Z \to [0, 1) \times \mathbb Z$ given by $T_f(x,r)=(T(x),r+f(x))$ is ergodic with respect to the product of the Lebesgue and counting measure.

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arXiv2024

Rigidity for piece-wise smooth circle maps and certain GIETs

The goal of this article is to show a rigidity property of conjugacies of generalized interval exchange transformations (GIETs). More precisely, we show that if two piecewise $C^3$ GIETs $f$ and $g$ of generic rotation number with mean-non-linearity 0 are homeomorphic, boundary-equivalent and their renormalizations approach in an appropriate way the set of affine interval exchange transformations, then their respective renormalizations converge to each other and the conjugating map is $C^1$. Moreover, if $f$ and $g$ are GIETs with rotation type combinatorial data, generic rotation number and they are break-equivalent as piecewise circle diffeomorphisms, they are actually $C^1$-conjugated as circle diffeomorphisms. These results generalize the work of K. Cunha and D. Smania \cite{cunha_rigidity_2014} in the case of piecewise $C^3$ circle maps, where the authors prove an analogous result for GIETs with rotation type combinatorial data and bounded rotation number.

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arXiv2023

Joining properties of automorphisms disjoint with all ergodic systems

We study the class $Erg^\perp$ of automorphisms which are disjoint with all ergodic systems. We prove that the identities are the only multipliers of $Erg^\perp,$ that is, each automorphism whose every joining with an element of $Erg^{\perp}$ yields a system which is again an element of $Erg^{\perp}$, must be an identity. Despite this fact, we show that $Erg^\perp$ is closed by taking Cartesian products. Finally, we prove that there are non-identity elements in $Erg^\perp$ whose self-joinings always yield elements in $Erg^\perp$. This shows that there are non-trivial characteristic classes included in $Erg^\perp$.

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arXiv2023

Ergodicity of explicit logarithmic cocycles over IETs

We prove ergodicity in a class of skew-product extensions of interval exchange transformations given by cocycles with logarithmic singularities. This, in particular, gives explicit examples of ergodic $\mathbb{R}$-extensions of minimal locally Hamiltonian flows with non-degenerate saddles in genus two. More generally, given any symmetric irreducible permutation, we show that for almost every choice of lengths vector, the skew-product built over the IET with the given permutation and lengths vector given by a cocycle, with symmetric, logarithmic singularities, which is \emph{odd} when restricted to each continuity subinterval is ergodic.

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arXiv2022

Backward Rauzy-Veech algorithm and horizontal saddle connections

The goal of this note is to study the action of the backward Rauzy-Veech algorithm on the translation surfaces with horizontal saddle connections. In particular, we prove that the orbit of a translation surface via the aforementioned algorithm is $\infty$-complete if and only if the surface does not posses horizontal saddle connections. Moreover, we show that appearance of all saddle connections as sides of the polygonal representations of the translations surface along the backward Rauzy-Veech induction orbit is equivalent to the minimality of the horizontal translation flow.

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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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arXiv2014

On special flows over IETs that are not isomorphic to their inverses

In this paper we give a criterion for a special flow to be not isomorphic to its inverse which is a refine of a result in \cite{Fr-Ku-Le}. We apply this criterion to special flows $T^f$ built over ergodic interval exchange transformations $T:[0,1)\to[0,1)$ (IETs) and under piecewise absolutely continuous roof functions $f:[0,1)\to\mathbb{R}_+$. We show that for almost every IET $T$ if $f$ is absolutely continuous over exchanged intervals and has non-zero sum of jumps then the special flow $T^f$ is not isomorphic to its inverse. The same conclusion is valid for a typical piecewise constant roof function.

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