School Program

1. Aspects of non-singular ergodic theory (Jon Aaronson)
The course will give an introduction to the following four major aspects of non-singular ergodic theory:
(i) Generalization of the classical ergodic theory of probability preserving transformations, culminating in the ratio ergodic theorem, the analogue of the\individual" ergodic theorem of G.D. Birkhoff in the general case.
(ii) The study of the properties that are valid \in the absence of invariant probabilities", e.g. existence of weakly wandering sets.
(iii) \Infinite ergodic theory" the study of the properties of transformations preserving infinite measures; this will be done via a few "strategic examples".
(iv) Study of systems not admitting invariant measures (Type III systems).

2. Iteration of entire functions (Walter Bergweiler)
The lecture series will give an introduction to complex dynamics, with emphasis on the iteration theory of transcendental entire functions. We will first review some background from function theory such as normal families, the Arzela-Ascoli Theorem, Marty's Theorem, Montel's Theorem, Picard's Theorem and Zalcman's Lemma. Then Fatou and Julia sets will be introduced and their basic properties will be discussed. This is followed by a discussion of periodic points, both the local theory (with a discussion of the Schroder, Bottcher and Abel functional equations) and the global theory, the classification of periodic Fatou components, and the relation of periodic Fatou components to singularities of the inverse function.
Special attention will be paid to aspects where the theory for transcendental entire functions differs from those for rational functions. Thus there will be a thorough discussion of Baker and wandering domains, as well as the escaping set of entire functions.

3. Dynamical systems of algebraic origin (S. G Dani)
This course will aim at giving an introduction to ergodic and dynamical properties of systems arising algebraically. We begin with simple systems such as the rotation of the circle, translations and affine transformations of tori, describe their properties and their significance in the study of dynamical systems. We shall then introduce homogeneous spaces of the modular group SL(2;R) with finite measure and relate them to hyperbolic geometry of surfaces of negative curvature. The ergodic and dynamical properties of the geodesic and horocycle ows and the connections with Diophantine approximation will be described. The general results in the theory relating to the Oppenheim conjecture and the Littlewood conjecture will be indicated.

4. Introduction to Ergodic Optimization(Renaud Leplaideur)
Dynamical systems usually have uncountably many invariant measures. The Thermodynamic formalism is a way to particularize one of them  via some functional called pressure associated to a given function called potential. In that case the associated measure is called an equilibrium state
Ergodic optimization consists in studying what happens when we introduce the temperature as a parameter and decrease it to the absolute zero. In that case, the equilibrium reaches a ground-state. The main problem is to identify this ground-state.
The course will introduce the basic tools of thermodynamic formalisms and study on some example what happens at zero temperature.

5. Introduction to Differentiable Ergodic Theory(Stefano Luzzatto)
This course will give an introduction to Differentiable Ergodic Theory, that is, the ergodic theory of differentiable transformations on manifolds. While abstract ergodic theory generally assumes the existence of an invariant measure, in this setting the given reference measure is not generally invariant and a key problem is to study the existence of invariant measures as well as their properties. In the course we will focus on one-dimensional maps and give several examples as well as general results of the theory. In particular we will discuss the question of the existence of measures which are absolutely continuous with respect to Lebesgue measure.

6. Dynamics of distal actions on locally compact groups(Riddhi Shah)
The action of group Γ on a topological space X is said to be distal if fao any two distict points x, y Є X and a Є X, the closure of {(r(x), r(y) : rЄ Γ)} (in the cartesian product space) does not contain (a; a). Distality, introduced by Hilbert, has been studied in a variety of different contexts.
We shall begin by introducing various generalities about distality and its relation with minimality. We then specialise to the case when X = G, a locally compact group. The group G is said to be distal if the conjugation action of G on itself is distal. We discuss in detail the relation of distality of groups with the behaviour of powers of probability measures on the group and concentration functions, as also with growth properties of the groups.

7. Some Properties of components of Fatou sets(Ajaya Singh)
Let F be a family of analytic functions in a domain D of extended plane is said to be normal or mormal family if every sequence of functions {fn} of F contains a subsequnce which converginces to a limit function f ≠ ∞ on each compact subset of D or contains a subsequence which converges uniformly to ∞ every compact subset of D.
Let f rational or an entire, where function f 0 (z) = z and define inductively,
f n(z) = f (f n-1 (z)), n = 1, 2, 3, . . .
Here f n called the nth of f. Let F = {z : {f n} is well defined and is normal in some neighborhood of z}
then F is called Fatou set or set of normality and J =C - F is called the Julia set. Fatou and Julia sets are completely invariant under map f.
In this talk, I plan to discuss some basic principles of Complex Dynamics, different types of the components of Fatou sets and give some properties of periodic and wandering components of certain functions.

8. Applications of Approximation Theory to Complex Dynamics(A. P. Singh)
Let f be a transcendental entire function. For positive integer n, let fn denote the nth iterate of the function. The set F(f)={z Є ℂ: {f n} is defined and normal some neighbourhood of z} is called Fatou Set. Its complement in denoted by J is called Julia Set.
Let U be a component of F(f), then by completely invariant of Fatou Set, f(U) lies in a component V of the Fatou Set. In fact, V \ F(U) is at most a single point.
If Un∩Um= ф for different integers m and n, where Un denotes the component of F(f) which contains f n(U) then U is called wandering domain, else it is called pre-periodic and if Un= U, then U is called periodic domain.
In these lectures we plan to develop the theory of Complex dynamics and give applications of approximation theory of entire functions to construct entire functions having wandering domains with various properties.

9. Introduction to Dynamical systems(Kedar Nath Uprety)
In this talk I will give elementary course on Dynamical System such as : Planar Linear Systems, Phase Portraits for Planar Systems, Flows on the Line, Fixed Point and Stability, Population Growth, Linear Stability Analysis, Bifurcations.
PhasePlane: Phase Portraits, Fixed Points and Linearization. Related Examples and Problems.

Tentative Schedule
 09:00-10:30 10:40-12:10 12:10-13:30 13:30-15:00 15:30-16:30 Mon Inauguration Dani Lunch Uprety Prob. Solving Tue Dani Luzzatto AP Singh Prob. Solving Wed Dani Luzzatto Thu Dani Luzzatto AP Singh Prob. Solving Fri Luzzatto AP Singh AP Singh Prob. Solving Sat Sun Leplaideur Bergweiler Mon Leplaideur Shah Bergweiler Prob. Solving Tue Aaronson Leplaideur Bergweiler Prob. Solving Wed Aaronson Shah Thu Leplaideur Shah Bergweiler Prob. Solving Fri Aaronson Leplaideur Bergweiler Prob. Solving Sat Shah Aaronson