Flows of stochastic dynamical systems: ergodic theory

We present a version of the Multiplicative Ergodic (Oseledec) Theorem for the flow of a nonlinear stochastic system definedon a smooth compact manifold. This theorem establishes the existence of a Lyapunov spectrum for the flow, which characterises the asymptotic behaviour of the derivative flow. Then we establish the existence of stable manifolds for the flow (on which trajectories cluster) associated with the Lyapunov spectrum. This work is a generalisation of that of Ruelle who deals with ordinary dynamical systems. Finally we give an example of a stochastic system for which the flow is calculated explicitly, and which illustrates the behaviour predicted by the abstract results.     

Products of Random Rectangular Matrices

We study the asymptotic behaviour of points under matrix cocyles generated by rectangular matrices. In particular we prove a random Perron‐Frobenius and a Multiplicative Ergodic Theorem. We also provide an example where such products of random rectangular matrices arise in the theory of random walks in random environments and where the Multiplicative Ergodic Theorem can be used to investigate recurrence problems.     

Partial linearization of differentiable systems

For any given differentiable dynamical system with discrete time on a compact Riemannian manifold of finite dimension d, along its orbits, in this paper, we introduce a type of global linearization under natural moving orthonormal q-frames of the tangent space of the base manifold, 1 ≤ q ≤ d. As an application, we give a new proof of the Oseledec-Million??ikov Multiplicative Ergodic Theorem (MET) for a ergodic smooth systems (M, ν; φ).     

Ergodic theorems for extended real-valued random variables

We first establish a general version of the Birkhoff Ergodic Theorem for quasi-integrable extended real-valued random variables without assuming ergodicity. The key argument involves the Poincaré Recurrence Theorem. Our extension of the Birkhoff Ergodic Theorem is also shown to hold for asymptotic mean stationary sequences. This is formulated in terms of necessary and sufficient conditions. In particular, we examine the case where the probability space is endowed with a metric and we discuss the validity of the Birkhoff Ergodic Theorem for continuous random variables. The interest of our results is illustrated by an application to the convergence of statistical transforms, such as the moment generating function or the characteristic function, to their theoretical counterparts.     

A special case of de Branges' theorem on the inverse monodromy problem

We give a new proof of a special case of de Branges' theorem on the inverse monodromy problem: when an associated Riemann surface is of Widom type with Direct Cauchy Theorem. The proof is based on our previous result (with M.Sodin) on infinite dimensional Jacobi inversion and on Levin's uniqueness theorem for conformal maps onto comb-like domains. Although in this way we can not prove de Branges' Theorem in full generality, our proof is rather constructive and may lead to a multi-dimensional generalization. It could also shed light on the structure of invariant subspaces of Hardy spaces on Riemann surfaces of infinite genus.This work was supported by the Austrian Founds zur Förderung der wissenschaftlichen Forschung, project-number P12985-TEC     

Ergodic theorems for coset spaces

We study in this paper the validity of the Mean Ergodic Theorem along left Følner sequences in a countable amenable group G. Although the Weak Ergodic Theorem always holds along any left Følner sequence in G, we provide examples where the Mean Ergodic Theorem fails in quite dramatic ways. On the other hand, if G does not admit any ICC quotients, e.g., if G is virtually nilpotent, then the Mean Ergodic Theorem holds along any left Følner sequence. In the case when a unitary representation of a countable amenable group is induced from a unitary representation of a “sufficiently thin” subgroup, we show that the Mean Ergodic Theorem holds for the induced representation along any left Følner sequence. Furthermore, we show that every countable (infinite) amenable group L embeds into a countable (not necessarily amenable) group G which admits a unitary representation with the property that for any left Følner sequence (F_n) in L, there exists a sequence (s_n) in G such that the Mean (but not the Weak) Ergodic Theorem fails in a rather strong sense along the (right-translated) sequence (F_ns_n) in G. Finally, we provide examples of countable (not necessarily amenable) groups G with proper, infinite-index subgroups H, so that the Pointwise Ergodic Theorem holds for averages along any strictly increasing and nested sequence of finite subsets of the coset G/H.     

Mean ergodic theorem in symmetric spaces

We investigate the validity of the Mean Ergodic Theorem in symmetric Banach function spaces E. The assertion of that theorem always holds when E is separable, whereas the situation is more delicate when E is non-separable. To describe positive results in the latter setting, we use the connections with the theory of singular traces.     

Rotations of integral matrices (remark on a theorem of YU. V. Linnik)

We provide a proof of a theorem of Yu. V. Linnik in the arithmetic of integral n×n matrices (cf. Yu. V. Linnik, Ergodic Properties of Algebraic Number Fields, Springer-Verlag, Berlin-Heidelberg-New York (1968), p. 154, Theorem VII.2.2), which he stated without proof.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 121, pp. 169–170, 1983.     

Ergodic theorem in variable Lebesgue spaces

We prove an Ergodic Theorem in variable exponent Lebesgue spaces, whenever the exponent is invariant under the transformation. Moreover, a counterexample is provided which shows that the norm convergence fails to hold for an arbitrary exponent.     

The convergence of moments in the martingale central limit theorem

Summary An early extension of the Lindeberg-Feller Theorem was Bernstein's discovery of necessary and sufficient conditions for the convergence of moments in the central limit theorem for sums of independent random variables. In this paper we show that Bernstein's work has a generalisation to martingales. We extend his work in both the independence and the martingale cases by showing that there exists a duality between the behaviour of the moments of the martingale and the behaviour of the sums of squares of the martingale differences. Our proofs are quite unrelated to Bernstein's and are based on Burkholder's inequalities.     

A Cartan’s Second Main Theorem Approach in Nevanlinna Theory

In 2002, in the paper entitled “A subspace theorem approach to integral points on curves”, Corvaja and Zannier started the program of studying integral points on algebraic varieties by using Schmidt’s subspace theorem in Diophantine approximation. Since then, the program has led a great progress in the study of Diophantine approximation. It is known that the counterpart of Schmidt’s subspace in Nevanlinna theory is H. Cartan’s Second Main Theorem. In recent years, the method of Corvaja and Zannier has been adapted by a number of authors and a big progress has been made in extending the Second Main Theorem to holomorphic mappings from C into arbitrary projective variety intersecting general divisors by using H. Cartan’s original theorem. We call such method “a Cartan’s Second Main Theorem approach”. In this survey paper, we give a systematic study of such approach, as well as survey some recent important results in this direction including the recent work of the author with Paul Voja.     

A Quantum Version of Spectral Decomposition Theorem of dynamical systems,quantum chaos hierarchy: Ergodic,mixing and exact

In this paper we study Spectral Decomposition Theorem (Lasota and Mackey, 1985) and translate it to quantum language by means of the Wigner transform. We obtain a Quantum Version of Spectral Decomposition Theorem (QSDT) which enables us to achieve three distinct goals: First, to rank Quantum Ergodic Hierarchy levels (Castagnino and Lombardi, 2009, Gomez and Castagnino, 2014). Second, to analyze the classical limit in quantum ergodic systems and quantum mixing systems. And third, and maybe most important feature, to find a relevant and simple connection between the first three levels of Quantum Ergodic Hierarchy (ergodic, exact and mixing) and quantum spectrum. Finally, we illustrate the physical relevance of QSDT applying it to two examples: Microwave billiards (Stockmann, 1999, Stoffregen et al. 1995) and a phenomenological Gamow model type (Laura and Castagnino, 1998, Omnès, 1994).     

Notes on Embedding Theorems for Local Hardy spaces

In this note we establish an embedding theorem (Theorem 2.4) for local Hardy spaces in the sense of GOLDBERG [G]. This result is a non-homogeneous version of the theorem of BAERNSTEIN and SAWYER (Theorem BS). Also applying this theorem we establish embedding theorem and Fourier embedding theorem (Theorem 4.2, Theorem 4.3 and Corollary 4.4) for local Hardy spaces.     

关于单叶函数系数之—基本引理及其应用

<正> 设函数 f(z)=z+a_2Z~2+…在单位圆|z|<1上是正则的单叶的.这种函数的全体形成一族 S.S 中满足条件|f(z)|1上是单叶的,除开极点ζ=∞是正则的.这种函数的全体形成一族∑.∑中满足条件|F(ζ)|>R的函     

Growth properties of the q-Dunkl transform in the space $$L^{p}_{q,\alpha }({\mathbb {R}}_{q},|x|^{2\alpha +1}d_{q}x)$$ L q , α p ( R q , | x | 2 α + 1 d q x )

The Ramanujan Journal - Our aim in this work is to prove an analogue of Titchmarsh’s theorem [19, Theorem 84] and Younis’s theorem [20, Theorem 3.3] on the image under the q-Dunkl...     

基于脉冲微分方程研究具有竞争关系的捕食系统的持久性

基于脉冲微分方程理论,考虑到在现实生活中,种群内部和种群之间都存在相互竞争,故本文在捕食与被捕食系统中引入竞争关系,建立了具有Hassell-Varley功能性反应的一类食饵与一类捕食者系统.利用比较定理得到此系统的有界性和生物学家比较关注的系统持久性的充分条件,即定理3.1和定理3.2.最后本文对得到的结论进行了阐释,并给出了相应的生物学意义.     

A convexity theorem for noncommutative gradient flows

In this survey we shall prove a convexity theorem for gradient actions of reductive Lie groups on Riemannian symmetric spaces. After studying general properties of gradient maps, this proof is established by (1) an explicit calculation on the hyperbolic plane followed by a transfer of the results to general reductive Lie groups, (2) a reduction to a problem on abelian spaces using Kostant's Convexity Theorem, (3) an application of Fenchel's Convexity Theorem. In the final section the theorem is applied to gradient actions on other homogeneous spaces and we show, that Hilgert's Convexity Theorem for moment maps can be derived from the results.     

A common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying a contractive condition of integral type

In this paper, we proved a common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying a contractive condition of integral type and a property (E.A) introduced in [M. Aamri, D. El Moutawakil, Some new common fixed point theorems under strict contractive conditions, J. Math. Anal. Appl. 270 (2002) 181-188]. Our theorem generalizes Theorem 2.2 of [M. Aamri, D. El Moutawakil, Common fixed points under contractive conditions in symmetric spaces, Appl. Math. E-Notes 3 (2003) 156-162] and Theorem 2 of [M. Aamri, D. El Moutawakil, Some new common fixed point theorems under strict contractive conditions, J. Math. Anal. Appl. 270 (2002) 181-188].     

Ergodic decomposition

Ergodic systems, being indecomposable are important part of the study of dynamical systems but if a system is not ergodic, it is natural to ask the following question:

Is it possible to split it into ergodic systems in such a way that the study of the former reduces to the study of latter ones?

Also, it will be interesting to see if the latter ones inherit some properties of the former one. This document answers this question for measurable maps defined on complete separable metric spaces with Borel probability measure, using the Rokhlin Disintegration Theorem.     

Ergodic potential

Potential Theory for ergodic Markov chains (with a discrete state spare and a continuous parameter) is developed in terms of the fundamental matrix of a chain.A notion of an ergodic potential for a chain is introduced and a form of Riesz decomposition theorem for measures is proved. Ergodic potentials of charges (with total charge zero) are shown to play the role of Green potentials for transient chains.