arXiv subjects
Publications and source records attributed to Klaudiusz Czudek.
We show that every $C^2$ minimal action of Thompson group $T$ on the circle is ergodic with respect to the Lebesgue measure. If such action is not minimal then the Lebesgue measure of the exceptional minimal set is zero.
We construct an e-chain on a locally compact space with the unique stationary distribution such that the strong law of large numbers does not hold. This answers negatively the question asked by Ö. Stenflo.
In this short paper, we show a characterization of Birkhoff Billiards inside discs which is related to the expansion of the formal Lazutkin conjugacy at the boundary.
We consider a finite number of orientation preserving $C^2$ interval diffeomorphisms and apply them randomly in such a way that the expected Lyapunov exponents at the boundary points are positive. We prove the exponential decay of correlations for Lipschitz observables with respect to the unique stationary measure supported on the interior of the interval. The key step is to show the exponential synchronization in average.
In De Simoi J., Kaloshin V., Wei Q. "Dynamical spectral rigidity among $\mathbb{Z}_2$-symmetric strictly convex domains close to a circle" (Appendix B coauthored with H. Hezari) Ann. of Math. 186.1 (2017), pp. 277-314 deformational spectral rigidity of $\mathbb{Z}_2$ symmetric domains close to the circle has been shown. One of the steps of the proof was to express the maximal symmetric periodic orbits in the Lazutkin parametrization. Here using the action-angle variables we find the second order approximation of Lazutkin coordinates of the maximal symmetric periodic orbits on the ellipses.
It has been shown that in one dimension the environment viewed by the particle process (EVP process) in quasi periodic random environment is uniquely ergodic and mixing under mild additional assumptions. Here we construct an analytic quasi periodic environment on higher dimensional torus such that the EVP process is not uniquely ergodic. The stationary measures in this counterexample are necessarily atomless. We show also that the EVP process is mixing with respect to any ergodic stationary measure under some smoothness assumption.
We prove the Central Limit Theorem and superpolynomial mixing for environment viewed for the particle process in quasi periodic Diophantine random environment. The main ingredients are smoothness estimates for the solution of the Poisson equation and local limit asymptotics for certain accelerated walks.
Fix an irrational number $α$ and a smooth, positive, real function $\mathfrak{p}$ on the circle. If current position is $x\in \mathbb R/\mathbb Z$ then in the next step jump to $x+α$ with probability $\mathfrak{p}(x)$ or to $x-α$ with probability $1-\mathfrak{p}(x)$. In 1999 Sinai has proven that if $\mathfrak{p}$ is asymmetric (in certain sense) or $α$ is Diophantine then the Markov process possesses a unique stationary distribution. Next year Conze and Guivarc'h showed the uniqueness of stationary distribution for an arbitrary irrational angle $α$. In this note we present a new proof of latter result.
Fix an irrational number $α$ and a real function $\mathfrak{p}$ on the circle with $0<\mathfrak{p}<1$. If a particle is placed at a point $x\in \mathbb R/\mathbb Z$, then in the next step it jumps to $x+α$ with probability $\mathfrak{p}(x)$ and to $x-α$ with probability $1-\mathfrak{p}(x)$. Sinai and Kaloshin proved that if $\mathfrak{p}$ is smooth then the random walk is uniquely ergodic and mixing, unless $α$ is Liouville and $\mathfrak{p}$ is symmetric. Unique ergodicity in the general case has been obtained by Conze and Guivarc'h. Here we give an alternative proof of the latter as well as some generic result about mixing, which partially solves a recent open problem.
Fix an irrational number $α$, and consider a random walk on the circle in which at each step one moves to $x+α$ or $x-α$ with probabilities $1/2, 1/2$ provided the current position is $x$. If an observable is given we can study a process called an additive functional of this random walk. One can formulate certain relations between the regularity of the observable and the Diophantine properties of $α$ implying the central limit theorem. It is proven here that for every Liouville angle there exists a smooth observable such that the central limit theorem fails. We construct also a Liouville angle such that the central limit theorem fails with some analytic observable. For Diophantine angles some counterexample is given as well. An interesting question remained open.
The central limit theorem for Markov chains generated by iterated function systems consisting of orientation preserving homeomorphisms of the interval is proved. We study also ergodicity of such systems.
We consider systems of two specific piecewise linear homeomorphisms of the unit interval, so called the Alsedà-Misiurewicz systems, and investigate the basic properties of Markov chains which arise when these two transformations are applied randomly with probabilities depending on the point of the interval. Though this iterated function system is not contracting in average and known methods do not apply, stability and the strong law of large numbers are proven.
We prove the $Λ$-variation diminishing property of the Bernstein and Kantorovich polynomials. Next we apply this result to characterize the space $CΛBV_c$ as the closure of the space of polynomials in the $ΛBV$ norm. A new proof of the separability of $CΛBV_c$ is given.
We show that not every family of generalized microscopic sets forms an ideal. Moreover, we prove that some of these families have some weaker additivity properties and some of them do not have even that.