Boris Hasselblatt

Regularity of the Anosov splitting II

We present an optimal result for the regularity of the invariant distributions of an Anosov system in terms of expansion and contraction rates. Compared to earlier results it avoids an ‘infinitesimal loss’ in the Holder exponent.

Chapter 3 Hyperbolic dynamical systems

Publisher Summary This chapter outlines important results in the theory of uniformly hyperbolic dynamical systems on compact spaces as well as its extension to nonuniformly hyperbolic systems and to indicate techniques used in the development of the basic theory. Accordingly, there are comments on possible methods of proof in the earlier parts, whereas later portions give an impressionistic view of several main developments of the subject. The chapter focuses on various aspects of hyperbolicity. There are two intertwined strands of the history of hyperbolic dynamics: Geodesic flows on one hand and hyperbolic phenomena ultimately traceable to some application of dynamical systems. The chapter presents definitions and basic examples as well as the theory of stable and unstable laminations. It describes the two main methods for proving the stable/unstable manifolds theorem, the Hadamard and Perron–Irwin methods.

The Sharkovsky Theorem: A Natural Direct Proof

Abstract We give a natural and direct proof of a famous result by Sharkovsky that gives a complete description of possible sets of periods for interval maps. The new ingredient is the use of Štefan sequences.

Chapter 1 - Partially Hyperbolic Dynamical Systems

This chapter discusses partially hyperbolic dynamical systems. The general guiding standard in this theory is that hyperbolicity in the system provides the mechanism that produces complicated dynamics in both the topological and statistical sense, and that, with respect to ergodic properties, it does so in essence by overcoming the effects of whatever nonhyperbolic dynamics may be present in the system. Partial hyperbolicity is but one possible extension of the notion of classical (complete) hyperbolicity or, in fact, a pair of extensions. Classical hyperbolicity can be described as requiting that the possible uniform rates of exponential relative behavior of orbits come in two collections on either side of one or by requiring that the Mather spectrum of the system consist of parts inside and outside of the unit circle. Partial hyperbolicity merely requires that the Mather spectrum consists of two parts that are separated by some circle centered at the origin (not necessarily the unit circle) and partial require that the Mather spectrum have three annular parts of which the inner one lies inside the unit circle and the outer one lies outside of the unit circle.

Zygmund strong foliations

We show that for a volume-preserving Anosov flow on a 3-manifold the strong stable and unstable foliations are Zygmund-regular. We also exhibit an obstruction to higher regularity, which admits a direct geometric interpretation. Vanishing of this obstruction implies high smoothness of the joint strong subbundle and that the flow is either a suspension or a contact flow.

The Riccati Equation: Pinching of Forcing and Solutions

A problem at the interface of differential geometry and dynamical systems gives rise to the question of what control of solutions of the Riccati equation + x 2 = k(t) with positive right-hand side can be obtained from control of the forcing term k. We show that a known result about “relative” pinching is optimal and refine two known theorems. This gives improved regularity of horospheric foliations and may be of interest in control or filtering theory.

Dimension product structure of hyperbolic sets

We conjecture that the fractal dimension of hyperbolic sets can be computed by adding those of their stable and unstable slices. This would facilitate substantial progress in the calculation or estimation of these dimensions, which are related in deep ways to dynamical properties. We prove the conjecture in a model case of Smale solenoids.

Chapter 1 Principal structures

Publisher Summary Dynamical systems have grown from various roots into a field of great diversity that interacts with many branches of mathematics as well as with the sciences. This chapter presents a survey to describe the general framework for several principal areas of the theory of dynamical systems. A possible use of this survey is as an introduction to mathematicians unfamiliar with dynamics, and it may be interesting to experts as an overview of a diverse field. The chapter focuses on examples, motivations, informal explanations, and discussion of key special cases or simplified versions of general results of dynamical systems. The chapter introduces a collection of important notions in generic terms—that is, without relying on any specific structure of the dynamical system. It introduces basic examples and intersperse further examples, as well as comments on previously introduced ones. The central structural elements are presented in the chapter in the order: the notions of equivalence, principal constructions, recurrence, and orbit growth.

The development of dynamics in the 20th century and the contribution of Jurgen Moser

Jürgen Moser (1928–1999) was one of the most accomplished mathematicians of the second half of the 20th century and his work had a major impact in broad areas of analysis, especially partial differential equations and dynamical systems, and geometry. In this article we discuss only his contributions to dynamics and closely related areas. We feel that the best way to make the reader understand and appreciate the impact of Moser’s work is to discuss it within the framework of some major trends in the development of dynamical systems during the last century. We apologize in advance for omitting many important references, including some key original work. These can be found in secondary sources to which we refer for detailed accounts of various areas. Needless to say, we included in our references all papers by Moser relevant to our discussion. For brief surveys of Moser’s work in all areas see [45, 91]. The leading theme of virtually all of Moser’s work in dynamics is the search for elements of stable behavior in dynamical systems with respect to either initial conditions or perturbations of the system.

A new construction of the Margulis measure for Anosov flows

The Margulis measure for Anosov flows arises from a Hausdorff measure for a natural distance on unstable leaves. This generalizes work of Ursula Hamenstadt for the case of geodesic flows.