Static Theory for Planar Ferromagnets and Antiferromagnets

Here we generalize the “BBH”-asymptotic analysis to a simplified mathematical model for the planar ferromagnets and antiferromagnets. To develop such a static theory is a necessary step for a rigorous mathematical justification of dynamical laws for the magnetic vortices formally derived in [1] and [2]. Received March 15, 2001, Accepted May 16, 2001     

A generalization of Vinograd's theorem for dynamical systems

The theory of dynamical systems has been expanded by the introduction of local dynamical systems [10, 4, 9] and local semidynamical systems [1]. Using integral curves of autonomous ordinary differential equations to illustrate these generalizations, we find that, roughly, the integral curves form a local dynamical system if solutions exist and are unique without requiring existence for all time, and the integral curves form a local semidynamical system if solutions exist and are unique in the positive sense but need not exist for all positive time. In addition to autonomous ordinary differential equations, the enlarged theory of dynamical systems has applications to nonautonomous ordinary differential equations, certain partial differential equations, functional differential equations, and Volterra Integral equations [9, 1, 2, 8], respectively. All of these have metric phase spaces. Since many dynamic considerations are invariant to reparameterizations, it is of interest to known when a local dynamical (or semidynamical) system can be reparameterized to yield a “global” dynamical (or semidynamical) system. For autonomous ordinary differential equations, Vinograd [7] has shown that the local dynamical system on an open subset ofRⁿ formed by integral curves is isomorphic (in the sense of Nemytskii and Stepanov) to a global dynamical system. In an extensive study of isomorphisms, Ura [12] has expanded the Gottschalk-Hedlund notion of an isomorphism and restated Vinograd's result in terms of a reparameterization. In this paper we study the problem of finding a global dynamical (or semidynamical) system which is isomorphic to a given local system. A necessary and sufficient condition is found which is then used to show that the Vinograd result holds on metric spaces.     

Fluctuations of empirical means at low temperature for finite Markov chains with rare transitions in the general case

Summary. The integrated autocovariance and autocorrelation time are essential tools to understand the dynamical behavior of a Markov chain. We study here these two objects for Markov chains with rare transitions with no reversibility assumption. We give upper bounds for the autocovariance and the integrated autocorrelation time, as well as exponential equivalents at low temperature. We also link their slowest modes with the underline energy landscape under mild assumptions. Our proofs will be based on large deviation estimates coming from the theory of Wentzell and Freidlin and others [4, 3, 12], and on coupling arguments (see [6] for a review on the coupling method). Received 5 August 1996 / In revised form: 6 August 1997     

Unifying theory for stability of continuous,discontinuous, and discrete-time dynamical systems

Continuous-time dynamical systems whose motions are continuous with respect to time (called continuous dynamical systems), may be viewed as special cases of continuous-time dynamical systems whose motions are not necessarily continuous with respect to time (called discontinuous dynamical systems, or DDS). We show that the classical Lyapunov stability results for continuous dynamical systems are embedded in the authors’ stability results for DDS (given in [H. Ye, A.N. Michel, L. Hou, Stability theory for hybrid dynamical systems, IEEE Transactions on Automatic Control 43 (4) (1998) 461–474]), in the following sense: if the hypotheses for a given Lyapunov stability result for continuous dynamical systems are satisfied, then the hypotheses of the corresponding stability result for DDS are also satisfied. This shows that the stability results for DDS in [H. Ye, A.N. Michel, L. Hou, Stability theory for hybrid dynamical systems, IEEE Transactions on Automatic Control 43 (4) (1998) 461–474] are much more general than was previously known, and that the quality of the DDS results therein is consistent with that of the classical Lyapunov stability results for continuous dynamical systems.By embedding discrete-time dynamical systems into a class of DDS with equivalent stability properties, we also show that when the hypotheses of the classical Lyapunov stability results for discrete-time dynamical systems are satisfied, then the hypotheses of the corresponding DDS stability results are also satisfied. This shows that the results for DDS in [H. Ye, A.N. Michel, L. Hou Stability theory for hybrid dynamical systems, IEEE Transactions on Automatic Control 43 (4) (1998) 461–474] are much more general than previously known, having connections even with discrete-time dynamical systems!Finally, we demonstrate by the means of a specific example that the stability results for DDS are less conservative than corresponding classical Lyapunov stability results for continuous dynamical systems.     

Periodic solutions of a periodically forced and undamped beam resting on weakly elastic bearings

We study the problem of existence of periodic solutions to a partial differential equation modelling the behavior of an undamped beam subject to an external periodic force. We assume that the ordinary differential equation associated to the first two modes of vibration of the beam has a symmetric homoclinic solution. By using methods borrowed by dynamical systems theory we prove that, if the period is non resonant with the (infinitely many) internal periods of the PDE, the equation has a weak periodic solution of the same period as the external force. In particular we obtain continua of periodic solutions for the undamped beam in absence of external forces. This result may be considered as an infinite dimensional analogue of a result obtained in [16] concerning accumulation of periodic solutions to homoclinic orbits in finite dimensional reversible systems. Matteo Franca: Partially supported by G.N.A.M.P.A. – INdAM (Italy).     

Hausdorff Measures versus Equilibrium States of Conformal Infinite Iterated Function Systems

Dealing with infinite iterated function systems we introduce and develop the ergodic theory of Hölder systems of functions similarly as in [HU] and [HMU]. In the context of conformal infinite iterated function systems we prove the volume lemma for the Hausdorff dimension of the projection onto the limit set of a shift invariant measure. This can be considered as a Billingsley type result. Our cenral goal is to demonstrate the appearance of the "singularity-absolute continuity" dichotomy for equilibrium states of Hölder systems of functions which has been observed in [PUZ,I] and [PUZ,II] (see also [DU1] and [DU2]) in the setting of rational functions of the Riemann sphere.     

Universal minimal topological dynamical systems

Rokhlin (1963) showed that any aperiodic dynamical system with finite entropy admits a countable generating partition. Krieger (1970) showed that aperiodic ergodic systems with entropy < log a, admit a generating partition with no more than a sets. In Symbolic Dynamics terminology, these results can be phrased— ℕ^ℤ is a universal system in the category of aperiodic systems, and [a]^ℤ is a universal system for aperiodic ergodic systems with entropy < log a. Weiss ([We89], 1989) presented a Minimal system, on a Compact space (a subshift of ) which is universal for aperiodic systems. In this work we present a joint generalization of both results: given ɛ, there exists a minimal subshift of [a]^ℤ, universal for aperiodic ergodic systems with entropy < log a − ɛ.     

Weighted mean topological dimension

This paper is devoted to the investigation of weighted mean topological dimension in dynamical systems. We show that the weighted mean dimension is not larger than the weighted metric mean dimension, which generalizes the classical result of Lindenstrauss and Weiss [16]. We also establish the relationship between the weighted mean dimension and the weighted topological entropy of dynamical systems, which indicates that each system with finite weighted topological entropy or small boundary property has zero weighted mean dimension.     

Nonmonotone systems decomposable into monotone systems with negative feedback

Motivated by the work of Angeli and Sontag [Monotone control systems, IEEE Trans. Automat. Control 48 (2003) 1684-1698] and Enciso and Sontag [On the global attractivity of abstract dynamical systems satisfying a small gain hypothesis, with applications to biological delay systems, Discrete Continuous Dynamical Systems, to appear] in control theory, we show that certain finite and infinite dimensional semi-dynamical systems with “negative feedback” can be decomposed into a monotone “open-loop” system with “inputs” and a decreasing “output” function. The original system is reconstituted by “plugging the output into the input”. Employing a technique of Gouzé [A criterion of global convergence to equilibrium for differential systems with an application to Lotka-Volterra systems, Rapport de Recherche 894, INRIA] and Cosner [Comparison principles for systems that embed in cooperative systems, with applications to diffusive Lotka-Volterra models, Dynam. Cont., Discrete Impulsive Systems 3 (1997) 283-303] of imbedding the system into a larger symmetric monotone system, we are able to obtain information on the asymptotic behavior of solutions, including existence of positively invariant sets and global convergence.     

Some properties of the exit measure for super Brownian motion

 We consider the exit measure of super Brownian motion with a stable branching mechanism of a smooth domain D of ℝ ^d . We derive lower bounds for the hitting probability of small balls for the exit measure and upper bounds in the critical dimension. This completes results given by Sheu [22] and generalizes the results of Abraham and Le Gall [2]. Because of the links between exits measure and partial differential equations, those results imply bounds on solutions of elliptic semi-linear PDE. We also give the Hausdorff dimension of the support of the exit measure and show it is totally disconnected in high dimension. Eventually we prove the exit measure is singular with respect to the surface measure on ∂D in the critical dimension. Our main tool is the subordinated Brownian snake introduced by Bertoin, Le Gall and Le Jan [4]. Received: 6 December 1999 / Revised version: 9 June 2000 / Published online: 11 December 2001     

Iterated forcing in quadratic form theory

In [Sp1] and [B/Sp] it has been shown that the existence of quadratic spaces of uncountable dimension over finite or countable fields sharing the property that every infinite dimensional subspace has its orthogonal complement of at most countable dimension is independent of the axioms of ZFC set theory. Such a space will be called astrong Gross space in the sequel. Cardinal invariants of the continuum decide whether strong Gross spaces exist or not. Namely, when b=ω₁ a strong Gross space of dimension ℵ₁ exists. When p>ω₁ such spaces do not exist. Here we answer the question what happens with strong Gross spaces in case b>ω₁ or p=ω₁. This work forms part of the author’s Habilitationsschrift at the ETH Zürich. The author is supported by the Basic Research Foundation of the Israel Academy of Sciences.     

Large dynamics of Yang–Mills theory: mean dimension formula

We study the Yang–Mills anti-self-dual (ASD) equation over the cylinder as a non-linear evolution equation. We consider a dynamical system consisting of bounded orbits of this evolution equation. This system contains many chaotic orbits, and moreover becomes an infinite dimensional and infinite entropy system. We study the mean dimension of this huge dynamical system. Mean dimension is a topological invariant of dynamical systems introduced by Gromov. We prove the exact formula of the mean dimension by developing a new technique based on the metric mean dimension theory of Lindenstrauss–Weiss.     

Poisson Statistics of Eigenvalues in the Hierarchical Anderson Model

We study the eigenvalue statistics for the hieracharchial Anderson model of Molchanov [21–23,27,28]. We prove Poisson fluctuations at arbitrary disorder, when the the model has a spectral dimension d < 1. The proof is based on Minami’s technique [25] and we give an elementary exposition of the probabilistic arguments. Submitted: October 8, 2007. Accepted: December 17, 2007.     

Mel’nikov-Samoilenko adiabatic stability problem

We develop a symplectic method for the investigation of invariant submanifolds of nonautonomous Hamiltonian systems and ergodic measures on them. The so-called Mel’nikov-Samoilenko problem for the case of adiabatically perturbed completely integrable oscillator-type Hamiltonian systems is studied on the basis of a new construction of “ virtual” canonical transformations. Dedicated to the memory of Viktor Koz’mich Mel’nikov, colleague and teacher, a talented Moscow mathematician, without whom the theory of dynamical systems would not be so attractive. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 6, pp. 787–803, June, 2006.     

Three-page approach to knot theory. Encoding and local moves

In the present paper, we suggest a new combinatorial approach to knot theory based on embeddings of knots and links into a union of three half-planes with the same boundary. The idea to embed knots into a “book” is quite natural and was considered already in [1]. Among recent papers on embeddings of knots into a book with infinitely many pages, we mention [2] and [3] (see also references therein). The restriction of the number of pages to three (or any other number ≥3) provides a convenient way toencode links by words in a finite alphabet. For those words, we give a finite set of local changes that realizes the equivalence of links by analogy with the Reidemeister moves for planar link diagrams. This work is partially supported by Russian Foundation for Basic Research grant No. 99-01-00090. Moscow State University. Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 33, No. 4, pp. 25–37, October–December, 1999. Translated by I. A. Dynnikov     

Normal Currents: Structure, Duality Pairings and div–curl Lemmas

The paper gives a decomposition of a general normal r-dimensional current [5] into the sum of three measures of which the first is an r-dimensional rectifiable measure, the second is the Cantor part of the current, and the third is Lebesgue absolutely continuous. This is analogous to the well-known decomposition of the derivative of a function of bounded variation into the jump, Cantor, and absolutely continuous parts; in fact the last is a special case of the result for (n–1)-dimensional normal currents. Further, Whitney’s cap product [15] is recast in the language of the approach to flat chains by Federer [5] and a special case (viz., currents of dimension n – 1) is shown to be closely related to the measure-valued duality pairings between vector measures with curl a measure and L^∞ vectorfields with L^∞ divergence as established by Anzellotti [2] and Kohn & Témam [6]. Finally, the cap product is shown to be jointly weak* continuous in the two factors of the product in a way similar to the compensated compactness theory; in the cases of (n – 1)-dimensional objects this reduces to results closely related to the div–curl lemmas of the standard compensated compactness theory. Received: June 2007     

Tracing of pseudotrajectories of dynamical systems and stability of prolongations of orbits

We investigate properties of dynamical systems associated with the approximation of pseudotrajectories of a dynamical system by its trajectories. According to modern terminology, a property of this sort is called the “property of tracing pseudotrajectories” (also known in the English literature as the “shadowing property”). We prove that dynamical systems given by mappings of a compact set into itself and possessing this property are systems with stable prolongation of orbits. We construct examples of mappings of an interval into itself that prove that the inverse statement is not true, i.e., that dynamical systems with stable prolongation of orbits may not possess the property of tracing pseudotrajectories. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 8, pp. 1016–1024, August, 1997.     

Conley index for random dynamical systems

Conley index theory is a very powerful tool in the study of dynamical systems. In this paper, we generalize Conley index theory to discrete random dynamical systems. Our constructions are basically the random version of Franks and Richeson in [J. Franks, D. Richeson, Shift equivalence and the Conley index, Trans. Amer. Math. Soc. 352 (2000) 3305-3322] for maps, and the relations of isolated invariant sets between time-continuous random dynamical systems and corresponding time-h maps are discussed. Two examples are presented to illustrate results in this paper.     

Global stability of a generalized epidemic model

The competitive exclusion principle is one of the most interesting and important phenomena in both theoretical epidemiology and biology. We show that the equilibrium in which only the strain with the maximum basic reproductive number exists is globally asymptotically stable by using an average Lyapunov function theorem and some dynamical system theory. This result is anticipated by H.J. Bremermann and H.R. Thieme (1989) [6] where they showed that the equilibrium is locally stable — the global result has not been established previously.     

Distribution of frequencies of digits via multifractal analysis

We study the Hausdorff dimension of a large class of sets in the real line defined in terms of the distribution of frequencies of digits for the representation in some integer base. In particular, our results unify and extend classical work of Borel, Besicovitch, Eggleston, and Billingsley in several directions. Our methods are based on recent results concerning the multifractal analysis of dynamical systems and often allow us to obtain explicit expressions for the Hausdorff dimension. This work is still another illustration of the role that the theory of dynamical systems can play in number theory.