arXiv subjects

Krzysztof Fraczek

Publications and source records attributed to Krzysztof Fraczek.

14 recordsLinked to original sources

arXiv2017

On typicality of translation flows which are disjoint with their inverse

In this paper we prove that translation structures for which the corresponding vertical translation flows is weakly mixing and disjoint with its inverse, form a $G_δ$-dense set in every non-hyperelliptic connected component of the moduli space $\mathcal M$. This is in contrast to hyperelliptic case, where for every translation structure the associated vertical flow is isomorphic to its inverse. To prove the main result, we study limits of the off-diagonal 3-joinings of special representations of vertical translation flows. Moreover, we construct a locally defined continuous embedding of the moduli space into the space of measure-preserving flows to obtain the $G_δ$-condition.

math.DS↗

arXiv2014

Directional localization of light rays in a periodic array of retro-reflector lenses

We show that vertical light rays in almost every periodic array of Eaton lenses do not leave certain strips of bounded width. The light rays are traced by leaves of a non-orientable foliation on a singular plane. We study the flow defined by the induced foliation on the orientation cover of the singular plane. The behavior of that flow and ultimately our claim for the light rays is based on an analysis of the Teichmüller flow and the Kontsevich-Zorich cocycle on the moduli space of two branched, two sheeted torus covers in genus two.

math.DS↗

arXiv2012

Ergodic directions for billiards in a strip with periodically located obstacles

We study the size of the set of ergodic directions for the directional billiard flows on the infinite band $\R\times [0,h]$ with periodically placed linear barriers of length $0<λ<h$. We prove that the set of ergodic directions is always uncountable. Moreover, if $λ/h\in(0,1)$ is rational the Hausdorff dimension of the set of ergodic directions is greater than 1/2. In both cases (rational and irrational) we construct explicitly some sets of ergodic directions.

math.DS↗

arXiv2012

On the self-similarity problem for Gaussian-Kronecker flows

It is shown that a countable symmetric multiplicative subgroup $G=-H\cup H$ with $H\subset\mathbb{R}_+^\ast$ is the group of self-similarities of a Gaussian-Kronecker flow if and only if $H$ is additively $\mathbb{Q}$-independent. In particular, a real number $s\neq\pm1$ is a scale of self-similarity of a Gaussian-Kronecker flow if and only if $s$ is transcendental. We also show that each countable symmetric subgroup of $\mathbb{R}^\ast$ can be realized as the group of self-similarities of a simple spectrum Gaussian flow having the Foias-Stratila property.

math.DS↗

arXiv2011

Ergodic properties of infinite extensions of area-preserving flows

We consider volume-preserving flows $(Φ^f_t)_{t\in\mathbb{R}}$ on $S\times \mathbb{R}$, where $S$ is a closed connected surface of genus $g\geq 2$ and $(Φ^f_t)_{t\in\mathbb{R}}$ has the form $Φ^f_t(x,y)=(ϕ_tx,y+\int_0^t f(ϕ_sx)ds)$, where $(ϕ_t)_{t\in\mathbb{R}}$ is a locally Hamiltonian flow of hyperbolic periodic type on $S$ and $f$ is a smooth real valued function on $S$. We investigate ergodic properties of these infinite measure-preserving flows and prove that if $f$ belongs to a space of finite codimension in $\mathscr{C}^{2+ε}(S)$, then the following dynamical dichotomy holds: if there is a fixed point of $(ϕ_t)_{t\in\mathbb{R}}$ on which $f$ does not vanish, then $(Φ^f_t)_{t\in\mathbb{R}}$ is ergodic, otherwise, if $f$ vanishes on all fixed points, it is reducible, i.e. isomorphic to the trivial extension $(Φ^0_t)_{t\in\mathbb{R}}$. The proof of this result exploits the reduction of $(Φ^f_t)_{t\in\mathbb{R}}$ to a skew product automorphism over an interval exchange transformation of periodic type. If there is a fixed point of $(ϕ_t)_{t\in\mathbb{R}}$ on which $f$ does not vanish, the reduction yields cocycles with symmetric logarithmic singularities, for which we prove ergodicity.

math.DS↗

arXiv2010

On Hausdorff dimension of the set of closed orbits for a cylindrical transformation

We deal with Besicovitch's problem of existence of discrete orbits for transitive cylindrical transformations $T_φ:(x,t)\mapsto(x+α,t+φ(x))$ where $Tx=x+α$ is an irrational rotation on the circle $\T$ and $φ:\T\to\R$ is continuous, i.e.\ we try to estimate how big can be the set $D(α,φ):=\{x\in\T:|φ^{(n)}(x)|\to+\infty\text{as}|n|\to+\infty\}$. We show that for almost every $α$ there exists $φ$ such that the Hausdorff dimension of $D(α,φ)$ is at least $1/2$. We also provide a Diophantine condition on $α$ that guarantees the existence of $φ$ such that the dimension of $D(α,φ)$ is positive. Finally, for some multidimensional rotations $T$ on $\T^d$, $d\geq3$, we construct smooth $φ$ so that the Hausdorff dimension of $D(α,φ)$ is positive.

math.DS↗

arXiv2010

Cocycles over interval exchange transformations and multivalued Hamiltonian flows

We consider interval exchange transformations of periodic type and construct different classes of recurrent ergodic cocycles of dimension $\geq 1$ over this special class of IETs. Then using Poincaré sections we apply this construction to obtain recurrence and ergodicity for some smooth flows on non-compact manifolds which are extensions of multivalued Hamiltonian flows on compact surfaces.

math.DS↗

arXiv2009

Disjointness of interval exchange transformations from systems of probabilistic origin

It is proved that almost every interval exchange transformation given by the symmetric permutation 1->m, 2->m-1,..., m-1->2, m->1, where m>1 is an odd number, is disjoint from ELF systems. The notion of ELF systems was introduced to express the fact that a given system is of probabilistic origin; the following standard classes of systems of probabilistic origin enjoy the ELF property: mixing systems, ergodic Gaussian systems, Poisson suspensions, dynamical systems coming from stationary infinitely divisible processes. Some disjointness properties of special flows built over interval exchange transformations and under piecewise constant roof function are investigated as well.

math.DS↗

arXiv2009

A note on quasi-similarity of Koopman operators

Answering a question of A. Vershik we construct two non-weakly isomorphic ergodic automorphisms for which the associated unitary (Koopman) representations are Markov quasi-similar. We also discuss metric invariants of Markov quasi-similarity in the class of ergodic automorphisms.

math.DS↗

arXiv2009

Density of mild mixing property for vertical flows of Abelian differentials

We prove that if $g\geq 2$ then the set of all Abelian differentials $(M,ω)$ for which the vertical flow is mildly mixing is dense in every stratum of the moduli space $\mathcal{H}_g$. The proof is based on a sufficient condition for special flows over irrational rotations and under piecewise constant roof functions to be mildly mixing.

math.DS↗

arXiv2007

Growth and mixing

Given a bi-Lipschitz measure-preserving homeomorphism of a compact metric measure space of finite dimension, consider the sequence formed by the Lipschitz norms of its iterations. We obtain lower bounds on the growth rate of this sequence assuming that our homeomorphism mixes a Lipschitz function. In particular, we get a universal lower bound which depends on the dimension of the space but not on the rate of mixing. Furthermore, we get a lower bound on the growth rate in the case of rapid mixing. The latter turns out to be sharp: the corresponding example is given by a symbolic dynamical system associated to the Rudin-Shapiro sequence.

math.DS↗

arXiv2006

On mild mixing of special flows over irrational rotations under piecewise smooth functions

It is proved that all special flows over the rotation by an irrational $α$ with bounded partial quotients and under $f$ which is piecewise absolutely continuous with a non-zero sum of jumps are mildly mixing. Such flows are also shown to enjoy a condition which emulates the Ratner condition introduced in \cite{Rat}. As a consequence we construct a smooth vector--field on $\T^2$ with one singularity point such that the corresponding flow $(ϕ_t)_{t\in\R}$ preserves a smooth measure, its set of ergodic components consists of a family of periodic orbits and one component of positive measure on which $(ϕ_t)_{t\in\R}$ is mildly mixing and is spectrally disjoint from all mixing flows.

math.DS↗

arXiv2002

Polynomial growth of the derivative for diffeomorphisms on tori

We consider area--preserving diffeomorphisms on tori with zero entropy. We classify ergodic area--preserving diffeomorphisms of the 3--torus for which the sequence $\{Df^n\}_{n\in{\Bbb N}}$ has polynomial growth. Roughly speaking, the main theorem says that every ergodic area--preserving $C^2$--diffeomorphism with polynomial uniform growth of the derivative is $C^2$--conjugate to a 2--steps skew product of the form \[\tor^3\ni(x_1,x_2,x_3)\mapsto (x_1+α,\ep x_2+β(x_1),x_3+γ(x_1,x_2))\in\tor^3,\] where $\ep=\pm 1$. We also indicate why there is no 4--dimensional analogue of the above result. Random diffeomorphisms on the 2--torus are studied as well.

math.DS↗

arXiv2000

On cocycles with values in the group SU(2)

In this paper we introduce the notion of degree for $C^1$-cocycles over irrational rotations on the circle with values in the group SU(2). It is shown that if a $C^1$-cocycle $ϕ:S^1\to SU(2)$ over an irrational rotation by $α$ has nonzero degree, then the skew product $S^1\times SU(2)\ni(x,g)\mapsto (x+α,gϕ(x))\in S^1\times SU(2)$ is not ergodic and the group of essential values of $ϕ$ is equal to the maximal Abelian subgroup of SU(2). Moreover, if $ϕ$ is of class $C^2$ (with some additional assumptions) the Lebesgue component in the spectrum of the skew product has countable multiplicity. Possible values of degree are discussed, too.

math.DS↗