# ADVANCED GEOMETRY 2 cod. 23013

1° year of course - First semester
Professor
- Anna BENINI
Geometria (MAT/03)
Field
Attività formative affini o integrative
Type of training activity
Related/supplementary
48 hours
of face-to-face activities
6 credits
hub: PARMA
course unit
in ITALIAN

## Learning objectives

This course means to introduce basic examples of discrete dynamical systems and the theoretical framework to study more general examples. It is an excellent review of topics studied in previous courses.

## Prerequisites

Basic differential geometry, point set topology, real and complex analysis.

## Course unit content

Introduction to discrete dynamical systems. It can be useful also for students in Physics.

## Full programme

-----------------------------------------------Introduction and examples
Introduction+Rotations
Contracting maps+ Expanding maps of the circle
Linear maps of R^n
The solenoid + Real quadratic maps
--------------------------------------------- Dinamica simbolica
Symbolic Dynamics 1
Symbolic Dynamics 2
-------------------------------------Structural stability
Topological Conjugacy and Structural stability
Topological classification of expanding maps of the circle
---------------------------------------Ergodic Theory
Ergodic Theory 1 Recurrence
Ergodic Theory 2 Ergodic Theorems
Ergodic Theory 3 Invariant Measures
--------------------- Low dimensional dynamics
Circle homeomorphisms (KH 7.1)
Circle diffeomorphisms
Sharkovski's Theorem
Classification of fixed points for diffeos of R^2
-----------------------Local Hyperbolicity
Hyperbolic sets and invariant cone fields
Stable and Unstable Manifold
Stable and Unstable Manifold II, Inclination Lemma
------------------------------------------
Teoremi di approssimazione
Dinamica Complessa
Dinamica Complessa II

## Bibliography

1-Lecture Notes
2-Brin, Michael; Stuck, Garrett Introduction to dynamical systems. Cambridge University Press, Cambridge, 2015.
3- Katok, Anatole; Hasselblatt, Boris Introduction to the modern theory of dynamical systems. Cambridge University Press, Cambridge, 1995

## Teaching methods

lectures or online classes in case of covid emergency

## Assessment methods and criteria

Either a set of exercises to be done at home or an expository talk on a subject to be decided with the lecturer.

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