| Code |
15614
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| Year |
1
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| Semester |
S1
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| ECTS Credits |
8
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| Workload |
TP(60H)
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| Scientific area |
Mathematics
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Entry requirements |
Does not have.
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Learning outcomes |
In this Curricular Unit an introduction to Measure Theory and Integration is made and the study of chaotic dynamical systems is developed through invariant measures, recurrence properties and entropy.
At the end of the Curricular Unit students should:
i. be able to define measure spaces and recognize some of their main features;
ii. know the fundamentals of Lebesgue integration and understand the main results about Lp spaces;
iii. know how to identify invariant measure(s) of a dynamical system and make use of it as an analysis tool;
iv. know the main ergodic theorems and their corollaries;
v. be able to define, interpret and develop the concept of entropy of a dynamical system;
vi. know and be able to work with different basic models of discrete dynamical systems;
vii. use Ergodic Theory to formulate and solve problems in different areas of Mathematics and Science.
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Syllabus |
1. Measure
1.1 Measurable spaces.
1.2 Measure spaces
1.3 Lebesgue measure
2. Integration
2.1 Measurable munctions
2.2 Lebesgue integral
2.3 The Riemann integral and the Lebesgue integral
2.4 Lp Spaces
2.5. Radon-Nikodym Theorem
2.6 Fubini's Theorem
3 Invariant measures for dynamical systems
3.1 Discrete dynamical systems
3.2 Invariant measures
3.3 Examples
3.4 Poincaré's Recurrence Theorem and Kac's Theorem
3.5 Induced transformations
3.6 Existence of invariant measures
4. Ergodic systems
4.1 Ergodic measures
4.2 Ergodic theorems and applications
4.3 Multiple recurrence theorems and applications
4.4 Mixing systems
5. Entropy
5.1 Entropy of a partition
5.2 Entropy of a dynamical system
5.3 Kolmogorov-Sinai Theorem and Shannon-McMillan-Breiman Theorem
5.4 Applications
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Main Bibliography |
Mañé, R: (1983). Introdução à Teoria Ergódica. IMPA.
Parthasarathy, K. R. (2005). Introduction to Probability and Measure (Texts and Readings in Mathematics Book 33).Hindustan Book Agency.
Silva, C. E. (2008). Invitation to ergodic theory. Providence (R.I.): American mathematical Society.
Viana, M & Oliveira, K. (2019). Fundamentos da Teoria Ergódica. SBM.
Walters, P. (2000). An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79) (Softcover Repri ed.).Springer.
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Teaching Methodologies and Assessment Criteria |
The teaching of this Curricular Unit is based on the presentation by the teacher of the contents and respective bibliography, followed by the resolution of exercises and open discussion of related problems. Periodically, lists of exercises/problems are presented to students, which are corrected and submitted for discussion. A topic on Ergodic Theory is given to each student to develop and present in the classroom.
The assessment consists of a global written test (T), five (lists of) exercises (E) and a work to be presented (A) on atopic to be proposed by the teacher. The final evaluation will be given by calculating CF=0.5*T+0.25*E+0.25*A, where the partial ratings T, E and A are expressed on a 0-20 scale.
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Language |
Portuguese. Tutorial support is available in English.
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