We consider a transformation of a normalized measure space such that the image of any point is a finite set. We call such transformation $m$-transformation. In this case the orbit of any point looks like a tree. In the study of $m$-transformations we are interested in the properties of the trees.
An $m$-transformation generates a stochastic kernel and a new measure. Using these objects, we introduce analogies of some main concept of ergodic theory: ergodicity, Koopman and Frobenius-Perron operators etc. We prove ergodic theorems and consider examples. We also indicate possible applications to fractal geometry and give a generalization of our construction. Some results which have analogies in the classical ergodic theory we are proved using standard methods. Other results have no analogies.
J. - P. P. P. P. Venot和作者用不同的方法证明了混合秩1的无穷保保测变换的中心化子是平凡的。在这篇笔记中，作者提出了他的加入证明。我们还考虑建设与代数间隔以及一类西顿建设产生新的混合秩一变换的例子。关于Gordin提出的零熵系统存在同宿遍历作用的问题，我们还讨论了Sidon秩1构造的某些修改的Poisson悬置。
As a natural counterpart to Nakada's $α$-continued fraction maps, we study a one-parameter family of continued fraction transformations with an indifferent fixed point. We prove that matching holds for Lebesgue almost every parameter in this family and that the exceptional set has Hausdorff dimension 1. Due to this matching property, we can construct a planar version of the natural extension for a large part of the parameter space. We use this to obtain an explicit expression for the density of the unique infinite $σ$-finite absolutely continuous invariant measure and to compute the Krengel en…
For any infinite zero-density integer set M, we found a rigid measure-preserving transformation mixing along M by answering Bergelson's question. Gaussian and Poisson suspensions over infinite constructions are suggested as suitable examples. We also discuss rigid rank one transformations of finite measure that mix along a given set, and some analogues for group actions.
An ergodic self-joining of an infinite rank-one transformation is a part of the weak limit of off-diagonal measures. A class of uncountaible cardinality of nonisomorphic transformations with polynomial weak closure is presented. Such actions have minimal self-joinings and some unusual spectral properties. For any set $M$ of positive integers there exists an infinite transformation $T$ such that the products $T\otimes T^m$ have simple singular spectrum as $1<m\in M$, and Lebesgue spectrum as $1<m\notin M$. There are similar effects for the corresponding Gaussian and Poisson suspensions.
Quantization for a probability distribution refers to the idea of estimating a given probability by a discrete probability supported by a finite set. In this article, we consider a probability distribution generated by an infinite system of affine transformations $\{S_{ij}\}$ on $\mathbb R^2$ with associated probabilities $\{p_{ij}\}$ such that $p_{ij}>0$ for all $i, j\in \mathbb N$ and $\sum_{i, j=1}^\infty p_{ij}=1$. For such a probability measure $P$, the optimal sets of $n$-means and the $n$th quantization error are calculated for every natural number $n$. It is shown that the distribution…
In this paper we investigate translated cone exchange transformations, a new family of piecewise isometries and renormalize its first return map to a subset of its partition. As a consequence we show that the existence of an embedding of an interval exchange transformation into a map of this family implies the existence of infinitely many bounded invariant sets. We also prove the existence of infinitely many periodic islands, accumulating on the real line, as well as non-ergodicity of our family of maps close to the origin.
We analyze finite and infinite words coming from the symbolic version of Chacon's transformation, focusing on distances between such words. Our main result is that if W = 0010 0010 1 0010 ... is the infinite word usually associated with Chacon's transformation, then the Hamming distance between W and any positive shift of W is strictly greater than 2/9; moreover, this bound is sharp. This yields an alternate proof that Chacon's transformation is non-rigid and (using King's weak closure theorem) has trivial centralizer.
Infinite-dimensional Bäcklund transformations between isotropic and anisotropic plasma equilibria. Infinite symmetries of anisotropic plasma equilibria
We prove that the joint distribution of the occupation time ratios for ergodic transformations preserving an infinite measure converges to a multidimensional version of Lamperti's generalized arcsine distribution, in the sense of strong distributional convergence. Our results can be applied to interval maps and Markov chains. We adopt the double Laplace transform method, which has been utilized in the study of occupation times of diffusions on multiray. We also discuss the inverse problem.
For a class of irrational numbers, depending on their Diophantine properties, we construct explicit rank-one transformations that are totally ergodic and not weakly mixing. We classify when the measure is finite or infinite. In the finite case they are isomorphic to irrational rotations. We also obtain rank-one nonrigid infinite invariant measures for irrational rotations, and, for each Krieger type, nonsingular measures on irrational rotations. In the third version, in the infinite case we use the constructions to provide examples of non-weakly mixing infinite measure-preserving ergodic trans…
We study the group of interval exchange transformations. Let $T$ be an $m$-interval exchange transformation. By the rank of $T$ we mean the dimension of the $\mathbb{Q}$-vector space spanned by the lengths of the exchanged subintervals. We prove that if $T$ satisfies Keane's infinite distinct orbit condition and $\text{rank}(T)>1+\lfloor m/2 \rfloor$ then the only interval exchange transformations which commute with $T$ are its powers. In the case that $T$ is a minimal 3-interval exchange transformation, we prove a more precise result: $T$ has a trivial centralizer in the group of interval exc…
We extend the results of our 2020 paper in the Annali della Scuola Normale Superiore di Pisa, Classe di Scienze. There, we associated to each of an infinite family of triangle Fuchsian groups a one-parameter family of continued fraction maps and showed that the matching (or, synchronization) intervals are of full measure. Here, we find planar extensions of each of the maps, and prove the continuity of the entropy function associated to each one-parameter family. We also introduce a notion of "first pointwise expansive power" of an eventually expansive interval map. We prove that for every map …
In this paper, we propose a dual set membership filter for nonlinear dynamic systems with unknown but bounded noises, and it has three distinctive properties. Firstly, the nonlinear system is translated into the linear system by leveraging a semi-infinite programming, rather than linearizing the nonlinear function. In fact, the semi-infinite programming is to find an ellipsoid bounding the nonlinear transformation of an ellipsoid, which aims to compute a tight ellipsoid to cover the state. Secondly, the duality result of the semi-infinite programming is derived by a rigorous analysis, then a f…
