Elements of Dynamical Systems

Inhaltsverzeichnis

1. Frontmatter

2. Real Dynamics

   V. Kannan

   Abstract

   Real dynamics is the study of those discrete dynamical systems for which the underlying set (called the phase space) is the real line \(\mathbb {R}\) or the unit interval \(I = [0, 1]\), or occasionally some other subset of \(\mathbb {R}\). But the definitions will be given in a more general setting. Most of the examples will be given from real dynamics. Other examples are also provided to see the contrast with real dynamics.

3. Topological Dynamics

   Anima Nagar, C. R. E. Raja

   Abstract

   Given a map \(f : X \longrightarrow X\), we would like to know the asymptotic behaviour of

   $$\begin{aligned} x, f(x), f^{2} (x), \dots , f^{n} (x), \dots \end{aligned}$$

   where \(f^{n} (x)\) is the position of x at time n. Such a sequence is called the trajectory of x.

4. Basic Ergodic Theory

   C. S. Aravinda, Vishesh S. Bhat

   Abstract

   These notes are based on the course of six lectures given by the first named author at the well-run workshop organised at IIT-Delhi in the month of December, 2017. The lectures were intended to be self-contained covering some basic facts in ergodic theory including a discussion of the Birkhoff ergodic theorem which, in a sense, heralded the beginning of ergodic theory.

5. Symbolic Dynamics

   Siddhartha Bhattacharya

   Abstract

   In this chapter, we will study a class of topological dynamical systems known as symbolic dynamical systems. These systems play an important role in coding theory, combinatorial dynamics and theory of cellular automata. In Sect. 2, we introduce the basic concepts associated with such systems. In Sect. 3, we introduce the notion of entropy. In Sect. 4, we compute the measure theoretic entropy of Bernoulli shifts. In Sect. 5, we consider a class of symbolic dynamical systems related to tiling spaces, and prove a result due to M. Szegedy that asserts that any translational tiling of \(\mathbb {Z}^{d}\) by a finite set F is periodic when |F| is prime. The last section is devoted to an algebraic dynamical system known as 3-dot system. Using the concept of directional homoclinic groups we show that \(\mathbb {Z}^{2}\)-actions on symbolic spaces can exhibit strong rigidity property.

6. Complex Dynamics

   S. Sridharan, K. Verma

   Abstract

   These notes are based on a set of lectures given by the second author at the Advanced Instructional School on Ergodic Theory and Dynamical Systems held at IIT Delhi in December 2017. The goal of these lectures was to introduce the audience, that comprised mainly of PhD students, to some basic ideas in complex dynamics in one and several variables. No prior knowledge in dynamics was assumed, nor any originality in the presentation was claimed. The same applies to what follows. In fact, a good fraction of the course was based on the material in Beardon [2] and Steinmetz [9]. The last part on the dynamics of Hénon maps is a summary of some of the work begun in Bedford-Smillie [3]. Other aspects of the dynamics of this class of maps can be found in Fornaess-Sibony [4‐6].

7. Topics in Homogeneous Dynamics and Number Theory

   Anish Ghosh

   Abstract

   This is a survey of some topics at the interface of dynamical systems and number theory, based on lectures delivered at CIRM Luminy, the University of Houston, and IIT Delhi. Specifically, we will be interested in the ergodic theory of group actions on homogeneous spaces and its connections to metric Diophantine approximation. The topics covered in the lectures included the study of the Diophantine approximation of linear forms using dynamics, the study of quadratic forms in particular the famous Oppenheim’s conjecture and its variations, as well as lattice point counting using dynamics. At IIT, non-divergence estimates for unipotent flows and Margulis’ proof of the Borel Harish-Chandra theorem using the non-divergence estimates were also covered.

8. On Certain Unusual Large Subsets Arising as Winning Sets of Some Games

   S. G. Dani

   Abstract

   Consider the space \(\mathbb {R}\) of real numbers. When would we call a subset X of \(\mathbb {R}\) a large set? Of course, the whole of \(\mathbb {R}\) itself or a subset missing only finitely many points would readily qualify to be large. With some understanding of cardinals, we may add to this list the class of subsets whose complements are countable. This includes for instance the sets of all irrational numbers, the set of all transcendental numbers etc. and we recognize these as large sets.