These notes originated from a graduate course given at Cornell University during the fall of 1998. One of the aims of the course was to present Sobolev inequalities and some of their applications in the context of analysis on manifolds—including Harnack inequalities and heat kernel estimates—to an audience not necessarily very familiar with analysis in general and Sobolev inequalities in particular. The first part (Chapters 1–2) introduces the reader to Sobolev inequalities in ℝn. An important application, Moser's proof of the elliptic Harnack inequality for uniformly elliptic divergence form second order differential operators, is treated in detail. In the second part (Chapters 3–4), Sobolev inequalities on complete non-compact Riemannian manifolds are discussed: What is their meaning and when do they hold true? How does one prove them? This discussion is illustrated by the treatment of some explicit examples. In the third and last part, Chapter 5, families of local Sobolev and Poincaré inequalities are introduced. These turn out to be crucial for taking full advantage of Sobolev inequality techniques on Riemannian manifolds. For instance, complete Riemannian manifolds satisfying a scale-invariant parabolic Harnack inequality are characterized in terms of Poincaré inequalities and volume growth. These notes give the first detailed exposition of this fundamental result.

We warn the reader that no effort has been made to include a comprehensive bibliography. Many important papers related to the topics presented in these notes are not mentioned. Actually, the literature on Sobolev inequalities is so vast that it would certainly be difficult to list it all.

Introduction

Notation concerning Riemannian manifolds

We want now to replace the Euclidean space ℝn by a Riemannian manifold M and consider the possibility of having some kind of Sobolev inequalities. This brings in a whole new point of view. On Euclidean space, we could only discuss whether inequalities were true or not. In the more general setting of Riemannian manifolds, we can investigate the relations between various functional inequalities and the relations between these functional inequalities and the geometry of the manifold. We can search for necessary and/or sufficient conditions for a given Sobolev-type inequality to hold true. This leads to a better understanding of what information about M is encoded in various Sobolev-type inequalities.

Sobolev inequalities are useful when developing analysis on Riemannian manifolds, even more so than on Euclidean space, because other tools such as Fourier analysis are not available any more. This is particularly true when one studies large scale behavior of solutions of partial differential equations such as the Laplace and heat equations.

In the sequel, we will focus on complete, non-compact Riemannian manifolds. For compact manifolds, local Euclidean-type Sobolev inequalities are always satisfied and the interesting questions have to do with controlling the constants arising in these inequalities in geometric terms. We refer the interested reader to where this is discussed at length.

In previous chapters we discussed dynamical systems mainly from a geometrical or topological point of view. The geometrical approach is intuitively appealing and lends itself to suggestive graphical representations. Therefore, it has been tremendously successful in the study of low-dimensional systems: continuous-time systems with one, two or three variables; discrete-time systems with one or two variables. For higher-dimensional systems, however, the approach has encountered rather formidable obstacles and rigorous results and classifications are few. Thus, it is sometimes convenient to change perspective and adopt a different approach, based on the concept of measure, and aimed at the investigation of the statistical properties of ensembles of orbits. This requires the use and understanding of some basic notions and results, to which we devote this chapter. The ergodic theory of dynamical systems often parallels its geometric counterpart and many concepts discussed in chapters 3–8, such as invariant, indecomposable and attracting sets, attractors, and Lyapunov characteristic exponents will be reconsidered in a different light, thereby enhancing our understanding of them. We shall see that the ergodic approach is very powerful and effective for dealing with basic issues such as chaotic behaviour and predictability, and investigating the relationship between deterministic and stochastic systems.

From the point of view of ergodic theory, there is no essential difference between discrete- and continuous-time dynamical systems. Therefore, in what follows, we develop the discussion mostly in terms of maps, mentioning from time to time special problems occurring for flows.

In chapter 3 we discussed the behaviour of a dynamical system when it is displaced from its state of rest, or equilibrium, and, in particular, we studied the conditions under which the displaced system does not wander too far from equilibrium or even converges back to it as time goes by. For such cases, we call the equilibrium stable or asymptotically stable. But what happens if we perturb an unstable equilibrium?

For an autonomous linear system, if we exclude unlikely borderline cases such as centres, the answer to this question is straightforward: orbits will diverge without bound.

The situation is much more complicated and interesting for nonlinear systems. First of all, in this case we cannot speak of the equilibrium, unless we have established its uniqueness. Secondly, for nonlinear systems, stability is not necessarily global and if perturbations take the system outside the basin of attraction of a locally stable equilibrium, it will not converge back to it. Thirdly, besides convergence to a point and divergence to infinity, the asymptotic behaviour of nonlinear systems includes a wealth of possibilities of various degrees of complexity.

As we mentioned in chapter 1, closed-form solutions of nonlinear dynamical systems are generally not available, and consequently, exact analytical results are, and will presumably remain, severely limited. If we want to study interesting dynamical problems described by nonlinear differential or difference equations, we must change our orientation and adapt our goals to the available means.

In chapter 5 we studied bifurcations, that is, qualitative changes in the orbit structures of dynamical systems, which take place when parameters are varied. In chapter 6 we discussed chaos and provided a precise characterisation of chaotic dynamics. In this chapter we take up again the question of transition in a system's behaviour, with a view to understanding how complex dynamics and chaos appear as parameters change. This problem is often discussed under the label ‘routes to chaos’. The present state of the art does not permit us to define the prerequisites of chaotic behaviour with sufficient precision and generality, and we do not have a complete and exhaustive list of all such possible routes. In what follows we limit our investigation to a small number of ‘canonical’ transitions to chaos which are most commonly encountered in applications. In our discussion we omit many technical details and refer the reader to the relevant references.

Period-doubling route to chaos

Although the period-doubling route to chaos could be discussed in a rather general framework (cf. Eckmann 1981, pp. 648–9), here we shall treat it in the context of noninvertible one-dimensional maps, because they provide an interesting topic per se and are by far the most common type of dynamical system encountered in applications. Before describing the period-doubling scenario in detail, we discuss some general results covering a broad class of one-dimensional maps.

Dynamics is the study of the movement through time of variables such as heartbeat, temperature, species population, voltage, production, employment, prices and so forth.

This is often achieved by means of equations linking the values of variables at different, uniformly spaced instants of time, i.e., difference equations, or by systems relating the values of variables to their time derivatives, i.e., ordinary differential equations. Dynamical phenomena can also be investigated by other types of mathematical representations, such as partial differential equations, lattice maps or cellular automata. In this book, however, we shall concentrate on the study of systems of difference and differential equations and their dynamical behaviour.

In the following chapters we shall occasionally use models drawn from economics to illustrate the main concepts and methods. However, in general, the mathematical properties of equations will be discussed independently of their applications.

A static problem

To provide a first, broad idea of the problems posed by dynamic vis-à-vis static analysis, we shall now introduce an elementary model that could be labelled as ‘supply-demand-price interaction in a single market’. Our model considers the quantities supplied and demanded of a single good, defined as functions of a single variable, its price, p. In economic parlance, this would be called partial analysis since the effect of prices and quantities determined in the markets of all other goods is neglected. It is assumed that the demand function D(p) is decreasing in p (the lower the price, the greater the amount that people wish to buy), while the supply function S(p) is increasing in p (the higher the price, the greater the amount that people wish to supply).