For the general coherence, it is advised to study all the material 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 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) homework 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 topics:
Question 1,3,4: 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 for these questions is the material from the Hasselblatt-Katok book (all results AND proofs):
HK Chapter 7: sections 7.1, 7.2, 7.3, 7.4
(except section 7.1.4, but inclusive of 7.2.4 and 7.4.3)
Question 2: Circle homeomorphisms. Relevant material for this question is material from the Hasselblatt-Katok book (all results AND proofs):
HK Chapter 4: sections 4.3 and 4.4 (except 4.4.2, 4.4.3, 4.4.5)
and from the Brin and Stuck book:
BS Chapter 7: section 7.1 (largely overlaps with material of [HK]).
Please note that thus, as I hinted at during the lectures, you need not study the material on topological entropy.
Relevant preparation from previous M3A23 exams:
2009: questions 1,2,3(a)&(b)
2008: questions 2,3,4(c),5
2007: questions 2,3,4,5
For links to the 2007 and 2008 exams, see http://www2.imperial.ac.uk/~jswlamb/DynamIC/m3a23/
The 2011/2012 progress tests are of course also relevant.
GOOD LUCK!
The emphasis of the exam will lie on testing your knowledge 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) homework 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 topics:
Question 1,3,4: 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 for these questions is the material from the Hasselblatt-Katok book (all results AND proofs):
HK Chapter 7: sections 7.1, 7.2, 7.3, 7.4
(except section 7.1.4, but inclusive of 7.2.4 and 7.4.3)
Question 2: Circle homeomorphisms. Relevant material for this question is material from the Hasselblatt-Katok book (all results AND proofs):
HK Chapter 4: sections 4.3 and 4.4 (except 4.4.2, 4.4.3, 4.4.5)
and from the Brin and Stuck book:
BS Chapter 7: section 7.1 (largely overlaps with material of [HK]).
Please note that thus, as I hinted at during the lectures, you need not study the material on topological entropy.
Relevant preparation from previous M3A23 exams:
2009: questions 1,2,3(a)&(b)
2008: questions 2,3,4(c),5
2007: questions 2,3,4,5
For links to the 2007 and 2008 exams, see http://www2.imperial.ac.uk/~jswlamb/DynamIC/m3a23/
The 2011/2012 progress tests are of course also relevant.
GOOD LUCK!
