Complex dynamics, spring 2017

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= Complex dynamics, spring 2017 =

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== News ==

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Today, May 3, the last lecture.

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If you need tutorial for your essay, you can contact Kari Astala by email ([[kari.astala@helsinki.fi>>mailto:kari.astala@helsinki.fi||shape="rect"]]), or see Lauri Hitruhin in person (room C414; his email: [[lauri.hitruhin@helsinki.fi>>mailto:lauri.hitruhin@helsinki.fi||shape="rect"]]).

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The essay should be returned by May 20, to: [[kari.astala@helsinki.fi>>mailto:kari.astala@helsinki.fi||shape="rect"]]

Results of the course exam (on the first part of course) can be found on "Koetulokset / Exam results" -webpage. Average of exam was 18,6 / 24.

== Essays ==

[[Here>>attach:Suggestions for topics.pdf]] are some suggestions for topics for an essay, to pass the second part of the course..

Team-work is also possible for essays; some of the topics can easily be adjusted for this.

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The essay should be roughly about 5-8 pages, depending on the case of course. Naturally, for team works the length should be roughly adjusted by the "team size".

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The essay should be returned by May 20, to: [[kari.astala@helsinki.fi>>mailto:kari.astala@helsinki.fi||shape="rect"]]

**Teacher:** [[Kari Astala>>doc:mathstatHenkilokunta.Astala, Kari]]

**Scope:** 10 cr

**Type:** Advanced studies

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**Teaching: **On Mondays and Wednesdays, at 12 - 14 in room C123. First lecture Monday 16.1.2017

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**Topics:** Complex dynamics (i.e. holomorphic dynamics) belongs to two fields, 'Dynamical Systems' and 'Complex Analysis'. In complex dynamics one studies the discrete-time dynamical system of iterating a complex polynomial P(z). That is, setting z,,n+1,, = P(z,,n,,), n = 0, 1, 2,... , one wants to understand the asymptotic behaviour of z,,n,,, when n → infinity, and see e.g. how this depends on the initial value z(% style="font-size: 12.0px;" %),,0,,(%%). Typical questions are for instance the geometric properties of the associated attractor or of the chaotic part of the system (the Julia set).

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Compared to other dynamical systems, in complex dynamics it is possible to achieve, with tools from complex analysis, a surprisingly detailed understanding of the systems. For complex analysis, complex dynamics allows fascinating applications of the methods and results of the field.

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In brief, the aim of the course is to understand these questions, both local dynamics and, in particular, the geometry of Julia sets and the Mandelbrot set.

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[[image:attach:Julia42.png||height="250"]][[image:attach:Mandelbrot.jpg||height="250"]]

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One can study also dynamics under a rational function R(z) or an entire function f(z). If time allows some topics of these will also be discussed.

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**Prerequisites: **The starting point is knowledge of basic complex analysis, e.g. as covered in the course 'Complex Analysis I'.

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More material and additional results from complex analysis will be necessary, but these will be covered during the course.

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Thus one may think the course also as a replacement or an alternative for the course 'Complex Analysis II'

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== Teaching schedule ==

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Weeks 3-9 and 11-18, Monday and Wednesday 12-14 in room C123.

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Easter holiday 13.-19.4.

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(% style="color: rgb(96,96,96);" %)In the first period, weeks 3-9, there will be exercises every week (for details on these see below).

== Exams ==

The exam for the first part of the course is on Tuesday March 14, at 10.15 - 12.45, room DK117.

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The area of the exam is Chapters 1-5 of the course notes.

(% style="color: rgb(96,96,96);" %)For the second part, the grade is obtained by writing a short essay.

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== Course material ==

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HERE you have the first seven chapters of the course notes plus half of Chapter 8, including lectures of April 26.

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The proof of theorem 7.38 is still missing.

There are several books on this topic; for instance

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* Beardon, //Iteration of Rational Functions//, Springer-Verlag

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* Carlson-Gamelin, //Complex Dynamics//, Springer-Verlag

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* Berteloot - Mayer, //Rudiments de Dynamique Holomorphe//, Lecture notes (In French)

From the following addresses you can get programs to draw Julia sets (Hints by Jan Leino and Valter Uotila):

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* Download "XAOS" from (% style="color: rgb(0,0,0);" %) (%%)[[http:~~/~~/fractalfoundation.org/resources/fractal-software/>>url:http://fractalfoundation.org/resources/fractal-software/||shape="rect"]]

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* Download "Visualisoija" from [[https:~~/~~/www.cs.helsinki.fi/u/jllang/visualoija/>>url:https://www.cs.helsinki.fi/u/jllang/visualoija/||shape="rect"]]

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* [[http:~~/~~/www.mndynamics.com/indexp.html.>>url:http://www.mndynamics.com/indexp.html||shape="rect" class="x_OWAAutoLink"]]

[[Registration>>url:https://weboodi.helsinki.fi/hy/opintjakstied.jsp?html=1&Tunniste=57072||style="font-size: 20.0px;" shape="rect"]]

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(% style="color: rgb(96,96,96);" %)Did you forget to register? (%%)[[What to do?>>url:https://wiki.helsinki.fi/display/mathstatOpiskelu/Kysymys4||style="text-decoration: underline;" shape="rect"]]

== Exercises ==

Exercises are can be found on this page every Wednesday, and exercise classes are on the following Tuesday, at 10-12, room C129.

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Exercise classes are given by Lauri Hitruhin. First exercises Tu 24.1.

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(% style="color: rgb(96,96,96);" %)Based on the number of exercises done you get extra points as follows:

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(% style="color: rgb(96,96,96);" %)25% = +1p, 35% = +2p, 45% = +3p, 55% = +4p, 65% = +5p ja 75% = +6p.

=== Assignments ===

* Exercises 1 Solutions 1

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* Exercises 2 Solutions 2

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* Exercises 3 Solutions 3

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* Exercises 4 Solutions 4

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* Exercises 5 Solutions 5

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* Exercises 6 Solutions 6

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=== Exercise classes ===

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== Course feedback ==

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Course feedback can be given at any point during the course. Click [[here>>url:https://elomake.helsinki.fi/lomakkeet/11954/lomake.html||style="line-height: 1.4285;" shape="rect"]].

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== Some Pictures: ==

Feigenbaum diagram:

[[image:attach:feigenbaum1.png||height="250"]]

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