Mathematics

Following Schweiger's generalization of multidimensional continued fraction algorithms, we consider a very large family of p -adic multidimensional continued fraction algorithms, which include Schneider's algorithm, Ruban's algorithms, and the p -adic Jacobi-Perron algorithm as special cases. The main result is to show that all the transformations in the family are ergodic with respect to the Haar measure.

In 1989 Kusuoka started the study of probability measures on the shift space that are defined with the help of products of matrices. In particular, he derived a sufficient condition for the ergodicity of such measures, which have since been referred to as Kusuoka measures. We observe that repeated measurements of a unitarily evolving quantum system generate a Kusuoka measure on the space of sequences of measurement outcomes. We show that if the measurement consists of scaled projections, then Kusuoka's sufficient ergodicity condition can be significantly simplified. We then prove that this condition is also necessary for ergodicity if the measurement consists of uniformly scaled rank-1 projections (i.e., it is a rank-1 POVM), or of exactly two projections, one of which is rank-1. For the latter class of measurements we also show that the Kusuoka measure is reversible in the sense that every string of outcomes has the same probability of being emitted by the system as its reverse.

We use geometric methods of equivariant dynamical systems to address a long-standing open problem in the theory of nematic liquid crystals, namely a proof of the existence and asymptotic stability of kayaking periodic orbits for which the principal axis of orientation of the molecular field (the director) rotates around the vorticity axis in response to steady shear flow. With a small parameter attached to the symmetric part of the velocity gradient, the problem can be viewed as a symmetry-breaking bifurcation from an orbit of the rotation group~$\SO(3)$ that contains both logrolling (equilibrium) and tumbling (periodic rotation of the director within the plane of shear) regimes as well as a continuum of kayaking orbits. The results turn out to require expansion to second order in the perturbation parameter.

The paper is a continuation of research in the direction of energy function (a smooth Lyapunov function whose set of critical points coincides with the chain recurrent set of a system) construction for discrete dynamical systems. The authors established the existence of an energy function for any A -diffeomorphism of a three-dimensional closed orientable manifold whose non-wandering set consists of chaotic one-dimensional attractor and repeller.

We derive sufficient conditions for a nonholonomic system to preserve a smooth volume form; these conditions become necessary when the density is assumed to only depend on the configuration variables. Moreover, this result can be extended to geodesic flows for arbitrary metric connections and the sufficient condition manifests as integrability of the torsion. As a consequence, volume-preservation of a nonholonomic system is closely related to the torsion of the nonholonomic connection. This result is applied to the Suslov problem for left-invariant systems on Lie groups (where the underlying space is Poisson rather than symplectic).

In the context of the Newtonian N -body problem, we prove the existence of a partially hyperbolic motion with prescribed positive energy and any initial collisionless configuration. Moreover, it is a free time minimizer of the respective supercritical Newtonian action or equivalently a geodesic ray for the respective Jacobi-Maupertuis metric.

We study higher order expansions both in the Berry-Esséen estimate (Edgeworth expansions) and in the local limit theorems for Birkhoff sums of chaotic probability preserving dynamical systems. We establish general results under technical assumptions, discuss the verification of these assumptions and illustrate our results by different examples (subshifts of finite type, Young towers, Sinai billiards, random matrix products), including situations of unbounded observables with integrability order arbitrarily close to the optimal moment condition required in the i.i.d. setting.

A profinite group equipped with an expansive endomorphism is equivalent to a one-sided group shift. We show that these groups have a very restricted structure. More precisely, we show that any such group can be decomposed into a finite sequence of full one-sided group shifts and two finite groups.

In this work we propose a new model of 2D cellular neural networks (CNN) in terms of the lattice FitzHugh-Nagumo equations with boundary feedback and prove a threshold condition for the exponential synchronization of the entire neural network through the \emph{a priori} uniform estimates of solutions and the analysis of dissipative dynamics. The threshold to be satisfied by the gap signals between pairwise boundary cells of the network is expressed by the structural parameters and adjustable. The new result and method of this paper can also be generalized to 3D and higher dimensional FitzHugh-Nagumo type or Hindmarsh-Rose type cellular neural networks.

Let Γ be a geometrically finite discrete subgroup in SO(d+1,1 ) ∘ with parabolic elements. We establish exponential mixing of the geodesic flow on the unit tangent bundle T 1 (Γ∖ H d+1 ) with respect to the Bowen-Margulis-Sullivan measure, which is the unique probability measure on T 1 (Γ∖ H d+1 ) with maximal entropy. As an application, we obtain a resonance free region for the resolvent of the Laplacian on Γ∖ H d+1 . Our approach is to construct a coding for the geodesic flow and then prove a Dolgopyat-type spectral estimate for the corresponding transfer operator.