On special flows over IETs that are not isomorphic to their inverses
Przemysław Berk Krzysztof Frączek
Discrete & Continuous Dynamical Systems - A 2015, 35(3): 829-855 doi: 10.3934/dcds.2015.35.829
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 [6]. 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.
keywords: joinings Non-reversibility of flows special flows over IETs.
Disjointness of interval exchange transformations from systems of probabilistic origin
Jacek Brzykcy Krzysztof Frączek
Discrete & Continuous Dynamical Systems - A 2010, 27(1): 53-73 doi: 10.3934/dcds.2010.27.53
We prove the disjointness of almost all interval exchange transformations from ELF systems (systems of probabilistic origin) for a countable subset of permutations including the symmetric permutations

$ 1\ 2\ \ldots \ m-1 \ m $
$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ m\ m-1 \ldots \ 2\ 1 $ for m=3,5,7.

Some disjointness properties of special flows built over interval exchange transformations and under piecewise constant roof function are investigated as well.

keywords: disjointness of dynamical systems. Interval exchange transformations
Ratner's property and mild mixing for special flows over two-dimensional rotations
Krzysztof Frączek Mariusz Lemańczyk
Journal of Modern Dynamics 2010, 4(4): 609-635 doi: 10.3934/jmd.2010.4.609
We consider special flows over two-dimensional rotations by $(\alpha,\beta)$ on $\T^2$ and under piecewise $C^2$ roof functions $f$ satisfying von Neumann's condition $\int_{\T^2}f_x(x,y)dxdy\ne 0$ or $\int_{\T^2}f_y(x,y)dxdy\ne 0 $. Such flows are shown to be always weakly mixing and never partially rigid. It is proved that while specifying to a subclass of roof functions and to ergodic rotations for which $\alpha$ and $\beta$ are of bounded partial quotients the corresponding special flows enjoy the so-called weak Ratner property. As a consequence, such flows turn out to be mildly mixing.
keywords: Ratner's property. Measure-preserving flows mild mixing special flows
Growth and mixing
Krzysztof Frączek Leonid Polterovich
Journal of Modern Dynamics 2008, 2(2): 315-338 doi: 10.3934/jmd.2008.2.315
Given a bi-Lipschitz measure-preserving homeomorphism of a finite dimensional compact metric measure space, consider the sequence of 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
keywords: Growth rate of homeomorphism the rate of mixing.
A class of mixing special flows over two--dimensional rotations
Krzysztof Frączek Mariusz Lemańczyk
Discrete & Continuous Dynamical Systems - A 2015, 35(10): 4823-4829 doi: 10.3934/dcds.2015.35.4823
We consider special flows over two-dimensional rotations by $(\alpha,\beta)$ on $\mathbb{T}^2$ and under piecewise $C^2$ roof functions $f$ satisfying von Neumann's condition \[\int_{\mathbb{T}^2}f_x(x,y)\,dx\,dy\neq 0\quad\text{ and }\quad \int_{\mathbb{T}^2}f_y(x,y)\,dx \,dy\neq 0.\] For an uncountable set of $(\alpha,\beta)$ with both $\alpha$ and $\beta$ of unbounded partial quotients the mixing property is proved to hold.
keywords: special flows Measure-preserving flows stretching. von Neumann's condition mixing
Polynomial growth of the derivative for diffeomorphisms on tori
Krzysztof Frączek
Discrete & Continuous Dynamical Systems - A 2004, 11(2&3): 489-516 doi: 10.3934/dcds.2004.11.489
We consider area--preserving zero entropy ergodic diffeomorphisms on tori. We classify such diffeomorphisms for which the sequence {$Df^n$} has a polynomial growth on the $3$-torus: they are necessary of the form

$\mathbb T^3\quad (x_1,x_2,x_3)\mapsto (x_1+\alpha,\varepsilon x_2+\beta(x_1),x_3+\gamma(x_1,x_2))\in\mathbb T^3,

where $\varepsilon =\pm 1$. We also indicate why there is no $4$-dimensional analogue of the above result. Random diffeomorphisms on the $2$-torus are studied as well.

keywords: polynomial growth of the derivative random diffeomorphisms. Area-preserving diffeomorphisms
Mild mixing property for special flows under piecewise constant functions
Krzysztof Frączek M. Lemańczyk E. Lesigne
Discrete & Continuous Dynamical Systems - A 2007, 19(4): 691-710 doi: 10.3934/dcds.2007.19.691
We give a condition on a piecewise constant roof function and an irrational rotation by $\alpha$ on the circle to give rise to a special flow having the mild mixing property. Such flows will also satisfy Ratner's property. As a consequence we obtain a class of mildly mixing singular flows on the two-torus that arise from quasi-periodic Hamiltonians flows by velocity changes.
keywords: Mild mixing property measure-preserving flows special flows.
Genericity on curves and applications: pseudo-integrable billiards, Eaton lenses and gap distributions
Krzysztof Frączek Ronggang Shi Corinna Ulcigrai
Journal of Modern Dynamics 2018, 12(1): 55-122 doi: 10.3934/jmd.2018004

In this paper we prove results on Birkhoff and Oseledets genericity along certain curves in the space of affine lattices and in moduli spaces of translation surfaces. In the space of affine lattices $ASL_2( \mathbb{R})/ASL_2( \mathbb{Z})$, we prove that almost every point on a curve with some non-degeneracy assumptions is Birkhoff generic for the geodesic flow. This implies almost everywhere genericity for some curves in the locus of branched covers of the torus inside the stratum $\mathscr{H}(1,1)$ of translation surfaces. For these curves we also prove that almost every point is Oseledets generic for the Kontsevitch-Zorich cocycle, generalizing a recent result by Chaika and Eskin. As applications, we first consider a class of pseudo-integrable billiards, billiards in ellipses with barriers, and prove that for almost every parameter, the billiard flow is uniquely ergodic within the region of phase space in which it is trapped. We then consider any periodic array of Eaton retroreflector lenses, placed on vertices of a lattice, and prove that in almost every direction light rays are each confined to a band of finite width. Finally, a result on the gap distribution of fractional parts of the sequence of square roots of positive integers is also obtained.

keywords: Homogeneous dynamics translation surfaces Kontsevich-Zorich cocycle equidistribution ergodic theorem Oseledets theorem Eaton lenses gap distribution pseudo-integrable billiards

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