From one-dimensional to infinite-dimensional dynamical systems: Ideal turbulence

There is a very short chain that joins dynamical systems with the simplest phase space (real line) and dynamical systems with the “most complicated” phase space containing random functions, as well. This statement is justified in this paper. By using “simple” examples of dynamical systems (one-dimensional and two-dimensional boundary-value problems), we consider notions that generally characterize the phenomenon of turbulence—first of all, the emergence of structures (including the cascade process of emergence of coherent structures of decreasing scales) and self-stochasticity.     

Transition to turbulence in coupled maps on hierarchical lattices

General hierarchical lattices of coupled maps are considered as dynamical systems. These models may describe many processes occurring in heterogeneous media with tree-like structures. The transition to turbulence via spatiotemporal intermittency is investigated for these geometries. Critical exponents associated to the onset of turbulence are calculated as functions of the parameters of the systems. No evidence of non-trivial collective behavior is observed in the global quantity used to characterize the spatiotemporal dynamics.     

Orthonormal expansions of invariant densities for expanding maps

We give a novel way of constructing the density function for the absolutely continuous invariant measure of piecewise expanding C^ω Markov maps. This is a classical problem, with one of the standard approaches being Ulam's method [Problems in Modern Mathematics, Interscience, New York, 1960] of phase space discretisation.Our method hinges instead on the expansion of the density function with respect to an L² orthonormal basis, and the computation of the expansion coefficients in terms of the periodic orbits of the expanding map. The efficiency of the method, and its extension to C^k expanding maps, are also discussed.     

From greedy to lazy expansions and their driving dynamics

In this paper we study the ergodic properties of non-greedy series expansions to non-integer bases β > 1. It is shown that the so-called ‘lazy’ expansion is isomorphic to the ‘greedy’ expansion. Furthermore, a class of expansions to base β > 1, β , ‘in between’ the lazy and the greedy expansions are introduced and studies. It is shown that these expansions are isomorphic to expansions of the form Tx = βx + (mod 1).     

From graphs to ortholattices and equivariant maps

A functor is constructed from the category of graphs and graph homomorphisms to the category of spaces with involutions and equivariant homotopy classes of maps. This can sometimes be used to prove lower bounds on chromatic numbers, and was inspired by Lovász's proof of Kneser's conjecture. Ortholattices occur as an intermediate step between graphs and spaces, and the correspondence between graphs and ortholattices is analyzed.     

Randomness relative to Cantor expansions

Imagine a sequence in which the first letter comes from a binary alphabet, the second letter can be chosen on an alphabet with 10 elements, the third letter can be chosen on an alphabet with 3 elements and so on. Such sequences occur in various physical contexts, in which the coding of experimental outcome varies with scale. When can such a sequence be called random? In this paper we offer a solution to the above question using the approach to randomness proposed by Algorithmic Information Theory.     

Lifting factor maps to resolving maps

We consider Smale spaces, that is, homeomorphisms of a compact metric spaces possessing canonical coordinates of contracting (stable) and expanding (unstable) directions. Examples of such dynamical systems include the basic sets for Smale's Axiom A systems. We also assume that each point of the space is non-wandering and that there is a dense orbit. We show that any almost one-to-one factor map between two such systems may be lifted in a certain sense to a factor map which is injective on the local stable sets (i.e., s-resolving). We derive several corollaries. One is a refinement of Bowen's result that every irreducible Smale space is a factor of an irreducible shift of finite type by an almost one-to-one factor map. We are able to show that there exists such a factor which is the composition of an s-resolving map and a u-resolving map. Supported in part by a grant from NSERC, Canada.     

Topological duality and lattice expansions,II: Lattice expansions with quasioperators

The main objective of this paper (the second of two parts) is to show that quasioperators can be dealt with smoothly in the topological duality established in Part I. A quasioperator is an operation on a lattice that either is join preserving and meet reversing in each argument or is meet preserving and join reversing in each argument. The paper discusses several common examples, including orthocomplementation on the closed subspaces of a fixed Hilbert space (sending meets to joins), modal operators ? and □ on a bounded modal lattice (preserving joins, resp. meets), residuation on a bounded residuated lattice (sending joins to meets in the first argument and meets to meets in the second). This paper introduces a refinement of the topological duality of Part I that makes explicit the topological distinction between the duals of meet homomorphisms and of join homomorphisms. As a result, quasioperators can be represented by certain continuous maps on the topological duals.     

Study of the higher-order asymptotic behavior of quantum field expansions in the theory of two-dimensional fully developed turbulence

We use a simplified (0+1)-dimensional theory to develop approaches for studying the higher-order asymptotic behavior of quantum field expansions in the two-dimensional theory of fully developed turbulence. We consider the asymptotic behavior of the correlation function in the small-time limit in the theory of fully developed turbulence and derive and investigate the stationarity equation. We show that the perturbation series in this limit has a finite convergence radius.     

Lifting homotopy T-algebra maps to strict maps

The settings for homotopical algebra—categories such as simplicial groups, simplicial rings, _A∞$A_{\infty }$ spaces, _E∞$E_{\infty }$ ring spectra, etc.—are often equivalent to categories of algebras over some monad or triple T. In such cases, T is acting on a nice simplicial model category in such a way that T descends to a monad on the homotopy category and defines a category of homotopy T-algebras. In this setting there is a forgetful functor from the homotopy category of T-algebras to the category of homotopy T-algebras.     

Moments of distributions related to digital expansions

The paper studies certain (probability) measures of binomial type defined in a recursive way on the unit interval. These measures are related to the sum-of-digit function and similar quantities. In particular, we undertake an asymptotic analysis of the moments of the corresponding distributions. This is done by a combination of a method based on the Mellin transform and the depoissonisation technique.     

Duality with expanding maps and shrinking maps, and its applications to Gauss maps

We study expanding maps and shrinking maps of subvarieties of Grassmann varieties in arbitrary characteristic. The shrinking map was studied independently by Landsberg and Piontkowski in order to characterize Gauss images. To develop their method, we introduce the expanding map, which is a dual notion of the shrinking map and is a generalization of the Gauss map. Then we give a characterization of separable Gauss maps and their images, which yields results for the following topics: (1) Linearity of general fibers of separable Gauss maps; (2) Generalization of the characterization of Gauss images; (3) Duality on one-dimensional parameter spaces of linear subvarieties lying in developable varieties.     

Asymptotic expansions of solutions to singularly perturbed systems

Under the condition that a degenerate system has an exponentially stable integral manifold, an asymptotic expansion of the Cauchy problem that generalizes the well known Vasil'eva expansion is constructed for a perturbed system.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 4, pp. 552–560, April, 1993.     

Harmonic analysis operators related to symmetrized Jacobi expansions

Following a symmetrization procedure proposed recently by Nowak and Stempak, we consider the setting of symmetrized Jacobi expansions. In this framework we investigate mapping properties of several fundamental harmonic analysis operators, including Riesz transforms, Poisson semigroup maximal operator, Littlewood–Paley–Stein square functions and multipliers of Laplace and Laplace–Stieltjes transform type. Our paper delivers also some new results in the original setting of classical Jacobi expansions.