Infinite-Dimensional Dynamical Systems and Projections

Diploma

ABSTRACT

We address three problems arising in the theory of infinite-dimensional dynamical systems. First, we study the extent to which the Hausdorff dimension and the dimension spectrum of a fractal measure supported on a compact subset of a Banach space are affected by a typical mapping into a finite-dimensional Euclidean space. We prove that a typical mapping preserves these quantities up to a factor involving the thickness of the support of the measure. Second, we prove a weighted Sobolev-Lieb-Thirring inequality and we use this inequality to derive a physically relevant upper bound on the dimension of the global attractor associated with the viscous lake equations. Finally, we show that in a general setting one may deduce the accuracy of the projection of a dynamical system solely from observation of the projected system.

INTRODUCTION

The connection between the theory of dynamical systems and the long-time behavior of solutions of a priori infinite-dimensional continuum systems described by partial differential equations is of great importance. Indeed, the application of dynamical ideas to areas of mathematical physics such as turbulence theory and fluid dynamics depends on this relationship. One views the equation of interest as the generator of a semiflow or a flow on a suitable function space. Ergodic theory and dimension theory may then be brought to bear on the analysis of asymptotic behavior in both the deterministic and stochastic contexts.
My research is based on two general lines of inquiry. Intrinsic questions concern the nature of the flow and its asymptotic properties. Examples of such problems include global attractor existence, attractor dimension estimates, inertial manifold existence, and the ergodic properties of invariant measures. The second line of inquiry deals with measurement and reconstruction from experimental data or a finite-dimensional truncation of the flow. One effectively projects the phase space onto a finite-dimensional space in order to reconstruct dynamical objects of interest and compute dynamical invariants. How accurately does the projection of the dynamical system reflect the dynamical system itself? Can the accuracy of the projection be deduced solely from observation of the projected system? We address problems with origins in both lines of inquiry.
In Chapter 2, we study the extent to which the Hausdorff dimension and the dimension spectrum of a fractal measure supported on a compact subset of a Banach space are affected by a typical mapping into a finite-dimensional Euclidean space. Let X be a compact subset of a Banach space B with thickness exponent τ(X) and Hausdorff dimension dimH(X). Let M be any subspace of the Borel measurable functions from B to Rm that contains the space of linear functions and is contained in the space of locally Lipschitz functions. We prove that for almost every (in the sense of prevalence) function f ∈ M, one has dimH(f(X)) > min{m,dimH(X)/(1 + τ(X))}. We also prove an analogous result for a certain part of the dimension spectra of Borel probability measures supported on X. The factor 1/(1 + τ(X)) can be improved to 1/(1 + τ(X)/2) if B is a Hilbert space. Since dimension cannot increase under a locally Lipschitz function, these theorems become dimension preservation results when τ(X) = 0. We conjecture that many of the attractors associated with the evolution equations of mathematical physics have zero thickness. The sharpness of our results in the case τ(X) =6 0 is discussed.
In Chapter 3, we consider the motion of an incompressible fluid confined to a shallow basin with varying bottom topography. A two-dimensional shallow water model has been derived from a three-dimensional anisotropic eddy viscosity model and has been shown to be globally well posed in [40]. The dynamical system associated with the shallow water model is studied. We show that this system possesses a global attractor and that the Hausdorff and box-counting dimensions of this attractor are bounded above by a value proportional to the weighted L2norm of the wind forcing function. A weighted Sobolev-Lieb-Thirring inequality plays the key role in obtaining the dimension estimate.
In Chapter 4, we study the extent to which the accuracy of a projection may be deduced solely from observation of the projected system. Let A be a compact
invariant set for a map f on Rn and let φ : Rn → Rm where n > m be a “typical” smooth map. When can we say that A and φ(A) are similar, based only on knowledge of the images in Rm of trajectories in A? For example, under what conditions on φ(A) (and the induced dynamics thereon) are A and φ(A) homeomorphic? Are their Lyapunov exponents the same? Or, more precisely, which of their Lyapunov exponents are the same? We address these questions with respect to both the general class of smooth mappings φ and the subclass of delay coordinate mappings.
In answering these questions, a fundamental problem arises about an arbitrary compact set A in Rn. For x ∈ A, what is the smallest integer d such that there is a C1 manifold of dimension d that contains all points of A that lie in some neighborhood of x? We define a tangent space TxA in a natural way and show that the answer is d = dim(TxA). As a consequence we obtain a Platonic version of the Whitney embedding theorem.