Minimal models for noninvertible and not uniquely ergodic systems

Let (Y, S) be a (not necessarilly invertible) topological dynamical system on a zero-dimensional metric spaceY without periodic points. Then there exists a minimal system (X, T) with the same simplex of invariant measures as (Y, S). More precisely, there exists a Borel isomorphism between full sets inY andX such that the adjoint map on measures is a homeomorphism between the corresponding sets of invariant measures in the weak topology. As an application we construct a minimal system carrying isomorphic copies of all nonatomic invariant measures.     

An application of ergodic theory to a problem in geometric ramsey theory

LetE be a measurable subset of ℝ ^k ,k>2, with XXX(E)>0. LetV = {0,υ ₁, …,υ _k+1} ε ℝ ^k , whereυ ₁, …,υ _k+1 are affinely independent. We show that forr large enough, we can find an isometric copy ofrV arbitrarily close toE. This is a generalization of a theorem of Furstenberg, Katznelson and Weiss [FKW] showing a similar property for ℝ²,V = {0,υ ₁,υ ₂}.     

On the multiplicative ergodic theorem for uniquely ergodic systems

We consider the question of uniform convergence in the multiplicative ergodic theorem

相似文献   

From infinite ergodic theory to number theory (and possibly back)

Some basic facts of infinite ergodic theory are reviewed in a form suitable to be applied to interval maps with number theoretic significance such as the Farey map. This is an enlarged version of the lecture notes accompanying a short course on Infinite Ergodic Theory at the First meeting of the (mostly) young italian hyperbolicians (Corinaldo, Italy, June 8-12, 2009).     

Topological mixing and uniquely ergodic systems

Every ergodic transformation (X, T, ℬ,μ) has an isomorphic system (Y, U, ν) which is uniquely ergodic and topologically mixing. This work is a part of an M.Sc. thesis written at The Hebrew University of Jerusalem under the supervision of Professor B. Weiss to whom the author is greatly indebted.     

Comparison of interacting diffusions and an application to their ergodic theory

Summary A general comparison argument for expectations of certain multitime functionals of infinite systems of linearly interacting diffusions differing in the diffusion coefficient is derived. As an application we prove clustering occurs in the case when the symmetrized interaction kernel is recurrent, and the components take values in an interval bounded on one side. The technique also gives an alternative proof of clustering in the case of compact intervals.     

Applications of the Wulff construction to the number theory

We apply the geometric construction of solutions of some variational problems of combinatorics in order to estimate the number of partitions and of plane partitions of an integer. Bibliography: 11 titles.Published in Zapiski Nauchnykh Seminarov POMI, Vol. 292, 2002, pp. 153–160.     

An application of gomory cuts in number theory

Frobenius has stated the following problem. Suppose thata ₁, a₂, ?, a_n are given positive integers and g.c.d. (a ₁, ? , a_n) = 1. The problem is to determine the greatest positive integerg so that the equation $$\sum\limits_{i = 1}^n {a_i x_i = g} $$ has no nonnegative integer solution. Showing the interrelation of the original problem and discrete optimization we give lower bounds for this number using Gomory cuts which are tools for solving discrete programming problems. In the first section an important theorem is cited after some remarks. In Section 2 we state a parametric knapsack problem. The Frobenius problem is equivalent with finding the value of the parameter where the optimal objective function value is maximal. The basis of this reformulation is the above mentioned theorem. Gomory's cutting plane method is applied for the knapsack problem in Section 3. Only one cut is generated and we make one dual simplex step after cutting the linear programming optimum of the knapsack problem. Applying this result we gain lower bounds for the Frobenius problem in Section 4. In the last section we show that the bounds are sharp in the sense that there are examples with arbitrary many coefficients where the lower bounds and the exact solution of the Frobenius problem coincide.     

An analytic approach to the ergodic theory of a stochastic variational inequality

In an earlier work made by the first author with J. Turi (Degenerate Dirichlet Problems Related to the Invariant Measure of Elasto-Plastic Oscillators, AMO, 2008), the solution of a stochastic variational inequality modeling an elasto-perfectly-plastic oscillator has been studied. The existence and uniqueness of an invariant measure have been proven. Nonlocal problems have been introduced in this context. In this work, we present a new characterization of the invariant measure. The key finding is the connection between nonlocal PDEs and local PDEs which can be interpreted with short cycles of the Markov process solution of the stochastic variational inequality.     

Discrete Radon transforms and applications to ergodic theory

We prove L ^p boundedness of certain non-translation-invariant discrete maximal Radon transforms and discrete singular Radon transforms. We also prove maximal, pointwise, and L ^p ergodic theorems for certain families of non-commuting operators.     

Realisation of measured dynamics as uniquely ergodic minimal homeomorphisms on manifolds

We prove that the family of measured dynamical systems which can be realised as uniquely ergodic minimal homeomorphisms on a given manifold (of dimension at least two) is stable under measured extension. As a corollary, any ergodic system with an irrational eigenvalue is isomorphic to a uniquely ergodic minimal homeomorphism on the two-torus. The proof uses the following improvement of Weiss relative version of Jewett–Krieger theorem: any extension between two ergodic systems is isomorphic to a skew-product on Cantor sets.     

Equipartition of interval partitions and an application to number theory

We give wider application and simpler proofs of results describing the rate at which the digits of one number-theoretic expansion determine those of another. The proofs are based on general measure-theoretic covering arguments and not on the dynamics of specific maps.

     

$$C_0$$-semigroups associated with uniquely ergodic Kantorovich modifications of operators

We extend to the context of \(L^p\) spaces and \(C_0\)-semigroups of operators our previous results from Heilmann and Ra?a (Positivity 21:897–910, 2017.  https://doi.org/10.1007/s11117-016-0441-1), concerning the eigenstructure and iterates of uniquely ergodic Kantorovich modifications of linking operators.     

An application of an optimal behaviour of the greedy solution in number theory

Leta ₁,a ₂, ...,a _n be relative prime positive integers. The Frobenius problem is to determine the greatest integer not belonging to the set { _j=1 ⁿ a _j x _j :xZ ₊ ⁿ }. The Frobenius problem belongs to the combinatorial number theory, which is very rich in methods. In this paper the Frobenius problem is handled by integer programming which is a new tool in this field. Some new upper bounds and exact solutions of subproblems are provided. A lot of earlier results obtained with very different methods can be discussed in a unified way.     

Construction of curious minimal uniquely ergodic homeomorphisms on manifolds: the Denjoy-Rees technique

In [Rees, M., A minimal positive entropy homeomorphism of the 2-torus, J. London Math. Soc. 23 (1981) 537-550], Mary Rees has constructed a minimal homeomorphism of the n-torus with positive topological entropy. This homeomorphism f is obtained by enriching the dynamics of an irrational rotation R. We improve Rees construction, allowing to start with any homeomorphism R instead of an irrational rotation and to control precisely the measurable dynamics of f. This yields in particular the following result: Any compact manifold of dimensiond?2which carries a minimal uniquely ergodic homeomorphism also carries a minimal uniquely ergodic homeomorphism with positive topological entropy.More generally, given some homeomorphism R of a compact manifold and some homeomorphism hC of a Cantor set, we construct a homeomorphism f which “looks like” R from the topological viewpoint and “looks like” R×hC from the measurable viewpoint. This construction can be seen as a partial answer to the following realisability question: which measurable dynamical systems are represented by homeomorphisms on manifolds?