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:
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.
Will there be a revision class for this course?
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There is no revision class planned but I am most happy to answer questions from students, in group or individually, about the contents of this course. All information relevant for the exam is provided in the instructions posted on this blog.
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