In this chapter we will embark upon the task of systematically identifying important specific phenomena associated with the asymptotic behavior of smooth dynamical systems We will build upon the results of our survey of specific examples in Chapter 1 as well as on the insights gained from the general structural approach outlined and illustrated in Chapter 2 Most of the properties discussed in the present chapter are in fact topological invariants and can be defined for broad classes of topological dynamical systems, including symbolic ones The predominance of topological invariants fits well with the picture that emerges from the considerations of Sections 21, 23, 24, and 26 The considerations of the previous chapter make it very plausible that smooth dynamical systems are virtually never differentiably stable and can only rarely be classified locally up to smooth conjugacy In contrast, structural and the related topological stability seem to be fairly widespread phenomena We will consider three broad classes of asymptotic invariants: (i) growth of the numbers of orbits of various kinds and of the complexity of orbit families, (ii) types of recurrence, and (iii) asymptotic distribution and statistical behavior of orbits The first two classes are of a purely topological nature; they are discussed in the present chapter The last class is naturally related to ergodic theory and hence we will provide an introduction to key aspects of that subject This will require some space so we put that material into a separate chapter The two chapters are intimately connected

Introduction to the Modern Theory of Dynamical Systems: PRINCIPAL CLASSES OF ASYMPTOTIC TOPOLOGICAL INVARIANTS

Introduction to the Modern Theory of Dynamical Systems: INTRODUCTION: WHAT IS LOW-DIMENSIONAL DYNAMICS?

For subshifts of finite type, conformal repellers, and conformal horseshoes, we prove that the set of points where the pointwise dimensions, local entropies, Lyapunov exponents, and Birkhoff averages do not exist simultaneously, carries full topological entropy and full Hausdorff dimension. This follows from a much stronger statement formulated for a class of symbolic dynamical systems which includes subshifts with the specification property. Our proofs strongly rely on the multifractal analysis of dynamical systems and constitute a non-trivial mathematical application of this theory.

Sets of "non-typical" points have full topological entropy and full hausdorff dimension

We consider a large class of partially hyperbolic systems containing, among others, affine maps, frame flows on negatively curved manifolds, and mostly contracting diffeomorphisms If the rate of mixing is sufficiently high, the system satisfies many classical limit theorems of probability theory

/pdf/limit-theorems-for-partially-hyperbolic-systems-1ps42afnrn.pdf

Limit theorems for partially hyperbolic systems

Designed to work as a reference and as a supplement to an advanced course on dynamical systems, this book presents a self-contained and comprehensive account of modern smooth ergodic theory. Among other things, this provides a rigorous mathematical foundation for the phenomenon known as deterministic chaos - the appearance of 'chaotic' motions in pure deterministic dynamical systems. A sufficiently complete description of topological and ergodic properties of systems exhibiting deterministic chaos can be deduced from relatively weak requirements on their local behavior known as nonuniform hyperbolicity conditions. Nonuniform hyperbolicity theory is an important part of the general theory of dynamical systems. Its core is the study of dynamical systems with nonzero Lyapunov exponents both conservative and dissipative, in addition to cocycles and group actions. The results of this theory are widely used in geometry (e.g., geodesic flows and Teichmuller flows), in rigidity theory, in the study of some partial differential equations (e.g., the Schrodinger equation), in the theory of billiards, as well as in applications to physics, biology, engineering, and other fields.

/pdf/nonuniform-hyperbolicity-dynamics-of-systems-with-nonzero-5dks9a0o5q.pdf

Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents

More than 30 years ago R\'enyi [1] introduced the representations of
 real numbers with an arbitrary base as a generalization of the
 -adic representations. One of the most studied problems in this field is
 the link between expansions to base and ergodic properties of the
 corresponding -shift.In this paper we will follow the bibliography of Blanchard [2] and
 give
 an affirmative answer to a question on the size of the set of real numbers
 -shifts.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0143385797079182

Symbolic dynamics for $\beta$-shifts and self-normal numbers

We study the behavior of multifractal spectra on the boundary of their domains of definition. In particular, we show that the dimension of the set of points having minimal (or maximal) pointwise dimension is not necessarily zero. However, this situation may be neglected in the sense of Baire category for Gibbs measures on expanding conformal repellers.

On the completeness of multifractal spectra

We introduce and study a class of endomorphisms which are piecewise smooth and have hyperbolic attractors. We prove the existence of SBR measures and develop the stable manifold theory and ergodic theory of such maps. We show that the Young dimension formula holds if and only if the map is invertible SBR-almost everywhere.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0143385798117996

Jörg Schmeling

Papers

Sets of "non-typical" points have full topological entropy and full hausdorff dimension

Symbolic dynamics for $\beta$-shifts and self-normal numbers

On the completeness of multifractal spectra

Dimension and invertibility of hyperbolic endomorphisms with singularities

Dimension and invertibility of hyperbolic endomorphisms with singularities