Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userUnder- and over-independence in measure preserving systems
111
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We introduce the notions of over- and under-independence for weakly mixing and (free) ergodic measure preserving actions and establish new results which complement and extend the theorems obtained in [BoFW] and [A]. Here is a sample of results obtained in this paper: $cdot$ (Existence of density-1 UI and OI set) Let $(X,mathcal{B},mu,T)$ be an invertible probability measure preserving weakly mixing system. Then for any $dinmathbb{N}$, any non-constant integer-valued polynomials $p_{1},p_{2},dots,p_{d}$ such that $p_{i}-p_{j}$ are also non-constant for all $i eq j$, (i) there is $Ainmathcal{B}$ such that the set $${ninmathbb{N}colonmu(Acap T^{p_{1}(n)}Acapdotscap T^{p_{d}(n)}A)<mu(A)^{d+1}}$$ is of density 1. (ii) there is $Ainmathcal{B}$ such that the set $${ninmathbb{N}colonmu(Acap T^{p_{1}(n)}Acapdotscap T^{p_{d}(n)}A)>mu(A)^{d+1}}$$ is of density 1. $cdot$ (Existence of Ces`aro OI set) Let $(X,mathcal{B},mu,T)$ be a free, invertible, ergodic probability measure preserving system and $Minmathbb{N}$. %Suppose that $X$ contains an ergodic component which is aperiodic. Then there is $Ainmathcal{B}$ such that $$frac{1}{N}sum_{n=M}^{N+M-1}mu(Acap T^{n}A)>mu(A)^{2}$$ for all $Ninmathbb{N}$. $cdot$ (Nonexistence of Ces`aro UI set) Let $(X,mathcal{B},mu,T)$ be an invertible probability measure preserving system. For any measurable set $A$ satisfying $mu(A) in (0,1)$, there exist infinitely many $N in mathbb{N}$ such that $$frac{1}{N} sum_{n=0}^{N-1} mu ( A cap T^{n}A) > mu(A)^2.$$
rate research
Read More
Let $(X,mathcal{B},mu)$ be a standard probability space. We give new fundamental results determining solutions to the coboundary equation: begin{eqnarray*} f = g - g circ T end{eqnarray*} where $f in L^p$ and $T$ is ergodic invertible measure preserving on $(X, mathcal{B}, mu )$. We extend previous results by showing for any measurable $f$ that is non-zero on a set of positive measure, the class of measure preserving $T$ with a measurable solution $g$ is meager (including the case where $int_X f dmu = 0$). From this fact, a natural question arises: given $f$, does there always exist a solution pair $T$ and $g$? In regards to this question, our main results are: (i) Given measurable $f$, there exists an ergodic invertible measure preserving transformation $T$ and measurable function $g$ such that $f(x) = g(x) - g(Tx)$ for a.e. $xin X$, if and only if $int_{f > 0} f dmu = - int_{f < 0} f dmu$ (whether finite or $infty$). (ii) Given mean-zero $f in L^p$ for $p geq 1$, there exists an ergodic invertible measure preserving $T$ and $g in L^{p-1}$ such that $f(x) = g(x) - g( Tx )$ for a.e. $x in X$. (iii) In some sense, the previous existence result is the best possible. For $p geq 1$, there exist mean-zero $f in L^p$ such that for any ergodic invertible measure preserving $T$ and any measurable $g$ such that $f(x) = g(x) - g(Tx)$ a.e., then $g otin L^q$ for $q > p - 1$. Also, we show this situation is generic for mean-zero $f in L^p$. Finally, it is shown that we cannot expect finite moments for solutions $g$, when $f in L^1$. In particular, given any $phi : mathbb{R} to mathbb{R}$ such that $lim_{xto infty} phi (x) = infty$, there exist mean-zero $f in L^1$ such that for any solutions $T$ and $g$, the transfer function $g$ satisfies: begin{eqnarray*} int_{X} phi big( | g(x) | big) dmu = infty. end{eqnarray*}
We consider both geometric and measure-theoretic shrinking targets for ergodic maps, investigating when they are visible or invisible. Some Baire category theorems are proved, and particular constructions are given when the underlying map is fixed. Open questions about shrinking targets are also described.
A group $G$ is said to be periodic if for any $gin G$ there exists a positive integer $n$ with $g^n=id$. We prove that a finitely generated periodic group of homeomorphisms on the 2-torus that preserves a measure $mu$ is finite. Moreover if the group consists in homeomorphisms isotopic to the identity, then it is abelian and acts freely on $mathbb{T}^2$. In the Appendix, we show that every finitely generated 2-group of toral homeomorphisms is finite.
An important problem in the theory of finite dynamical systems is to link the structure of a system with its dynamics. This paper contains such a link for a family of nonlinear systems over an arbitrary finite field. For systems that can be described by monomials, one can obtain information about the limit cycle structure from the structure of the monomials. In particular, the paper contains a sufficient condition for a monomial system to have only fixed points as limit cycles. The condition is derived by reducing the problem to the study of a Boolean monomial system and a linear system over a finite ring.
We investigate Weierstrass functions with roughness parameter $gamma$ that are Holder continuous with coefficient $H={loggamma}/{log frac12}.$ Analytical access is provided by an embedding into a dynamical system related to the baker transform where the graphs of the functions are identified as their global attractors. They possess stable manifolds hosting Sinai-Bowen-Ruelle (SBR) measures. We systematically exploit a telescoping property of associated measures to give an alternative proof of the absolute continuity of the SBR measure for large enough $gamma$ with square-integrable density. Telescoping allows a macroscopic argument using the transversality of the flow related to the mapping describing the stable manifold. The smoothness of the SBR measure can be used to compute the Hausdorff dimension of the graphs of the original Weierstrass functions and investigate their local times.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
