How to study for the 2015 exam.

For the general coherence, it is advised to study all the material of the lectures, seriously. However, in order to have you focus on the parts of the course most relevant for the exam, I provide some guidelines for study below.

The emphasis of the exam will lie on testing your knowledge and understanding of the lecture notes. You should be prepared to recall definitions (precisely), and (part of) proofs of important theorems/lemmas/propositions, as well as apply results discussed in the course to answer problems concerning concrete examples. Also, some (parts of) problem sheet or test questions may appear in the exam. The exam questions have several parts, usually (but not always) with increasing difficulty as the question progresses. Please note that you get disproportional credit for answering the simpler questions well, so please when you go through the exam, make sure that you answer those carefully before tackling the harder questions!

The exam will consist of 4 questions concerning the following material:

• Chaos, topological transitivity, topological mixing, density of periodic solutions, sensitive dependence on initial conditions, symbolic dynamics, topological Markov chains, topological (semi-)conjugacy with consequences, and applications to circle maps and interval maps.
  Relevant material is from the Hasselblatt-Katok book (all relevant definitions, results AND proofs): HK Chapter 7: sections 7.1, 7.2, 7.3, 7.4 (except sections 7.1.4, 7.1.5, 7.2.4, 7.3.3, 7.4.4 and 7.4.5) + Note on preservation of topological mixing under semi-conjugacy (blog 14-11-14).
• Fractals and their dimensions.
  Relevant material is from Falconer (2003) book (all relevant definitions and results):
  F03 Chapter 2:  sections 2.1, 2.2, 2.3 (incl proofs)
  F03 Chapter 3:  sections 3.1, 3.2 (incl proofs; useful type of exercise: 3.3-3.8, 3.11, 3.16)
  F03 Chapter 4:  section 4.1 (no proofs; useful type of exercise: 4.1-4.6)
• Iterated Function Systems (IFS) and the Collage Theorem.
  Relevant material for this question is from the Falconer (2003) book (all relevant definitions and results but NO proofs)
  F03 Chapter 9: section 9.1, 9.2, 9.5 (useful type of exercise: 9.3-9.5, 9.8, 9.9)
  and from Michael Barnsley's (2000) book, [B] Chapter III, sections 7,10,11 (until Lemma 11.1) Chapter IV, sections 1, 2, and 4.

How to prepare for the Mastery/Comprehension question (for MSci and MSc students).

The question will relate to sections 5-8 of Chapter IX of Barnsley (1988), see the link [B] Chap IX in the right-hand-side margin, concerning the probabilistic approach to IFSs. You are expected to study the material in detail and to be able to answer questions with precision. A good understanding of the definitions and theorems is important. While it is strongly recommended that you make yourself familiar with all the presented proofs in the text, they need not be learned by heart.

Model answers to exercises in [F03]

Please note that there is a solutions manual for the problems in [F03] provided by the publisher, written by Falconer http://bcs.wiley.com/he-bcs/Books?action=resource&bcsId=8728&itemId=111994239X&resourceId=34235 

Third (and last) test on Friday 12/12, 10-11am, in room 642

The test covers the following material from [F03]:

• Introduction
• Chapters 2, 3 and 4: sections 2.1, 2.2, 2.3, 3.1, 3.2, and 4.1

and the following exercises

• Exercises #4

The prototypical question in this test will be "determine the Hausdorff and/or Box dimension of the following fractal". This requires giving upper and lower bounds 

Preservation of topological mixing under semiconjugacy

Relevant notes can be found here. This result is part of the course, but not included in the [HK] notes.

Second test on Friday 21/11, 10-11am, in room 642

The test will cover the following material:

[HK] 7.3 and 7.4
Note on "Preservation of topological mixing under semiconjugacy" as in the 14/11 post, above.
Exercises #3

The focus here lies on the properties of symbolic dynamics, and its application to one-dimensional and two-dimensional examples (along the lines discussed in the lectures and [HK] 7.3 and 7.4).

Your will receive back the marked scripts for these tests on Friday 28/11.

Exercises #4

[F03]  exercises 2.1-2.16
[F03]  exercises 3.1, 3.3, 3.4, 3.5, 3.6, 3.8, 3.11
[F03]  exercises 4.1-4.6