Lyapunov Exponents to Predict the Behavior of the Product of Random Matrices
In this thesis, we investigate the asymptotic behavior of products of random matrices through Lyapunov exponents. Our theoretical framework is grounded in Kingman’s Subadditive Ergodic Theorem, from which we derive the Furstenberg-Kesten Theorem and Oseledets’ Theorem in two dimensions. These result...
- Autores:
- Tipo de recurso:
- Fecha de publicación:
- 2025
- Institución:
- Universidad del Rosario
- Repositorio:
- Repositorio EdocUR - U. Rosario
- Idioma:
- eng
- OAI Identifier:
- oai:repository.urosario.edu.co:10336/45741
- Acceso en línea:
- https://repository.urosario.edu.co/handle/10336/45741
- Palabra clave:
- Exponentes de Lyapunov
Matrices aleatorios
Caos
Teoría ergódica
Lyapunov Exponents
Random Matrices
Chaos
Ergodic Theory
- Rights
- License
- Attribution-NonCommercial-NoDerivatives 4.0 International
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dc.title.none.fl_str_mv |
Lyapunov Exponents to Predict the Behavior of the Product of Random Matrices |
dc.title.TranslatedTitle.none.fl_str_mv |
Exponentes de Lyapunov para predecir el comportamiento del producto de matrices aleatorias |
title |
Lyapunov Exponents to Predict the Behavior of the Product of Random Matrices |
spellingShingle |
Lyapunov Exponents to Predict the Behavior of the Product of Random Matrices Exponentes de Lyapunov Matrices aleatorios Caos Teoría ergódica Lyapunov Exponents Random Matrices Chaos Ergodic Theory |
title_short |
Lyapunov Exponents to Predict the Behavior of the Product of Random Matrices |
title_full |
Lyapunov Exponents to Predict the Behavior of the Product of Random Matrices |
title_fullStr |
Lyapunov Exponents to Predict the Behavior of the Product of Random Matrices |
title_full_unstemmed |
Lyapunov Exponents to Predict the Behavior of the Product of Random Matrices |
title_sort |
Lyapunov Exponents to Predict the Behavior of the Product of Random Matrices |
dc.contributor.advisor.none.fl_str_mv |
Artigiani, Mauro Martínez, Cristian |
dc.subject.none.fl_str_mv |
Exponentes de Lyapunov Matrices aleatorios Caos Teoría ergódica |
topic |
Exponentes de Lyapunov Matrices aleatorios Caos Teoría ergódica Lyapunov Exponents Random Matrices Chaos Ergodic Theory |
dc.subject.keyword.none.fl_str_mv |
Lyapunov Exponents Random Matrices Chaos Ergodic Theory |
description |
In this thesis, we investigate the asymptotic behavior of products of random matrices through Lyapunov exponents. Our theoretical framework is grounded in Kingman’s Subadditive Ergodic Theorem, from which we derive the Furstenberg-Kesten Theorem and Oseledets’ Theorem in two dimensions. These results provide the tools to quantify exponential growth rates and directional behavior in random matrix products. To visualize our theoretical conclusions, we present a series of simulations that illustrate the emergence of Lyapunov exponents and their predictive power in practical settings. |
publishDate |
2025 |
dc.date.accessioned.none.fl_str_mv |
2025-06-19T14:33:07Z |
dc.date.available.none.fl_str_mv |
2025-06-19T14:33:07Z |
dc.date.created.none.fl_str_mv |
2025-06-09 |
dc.type.none.fl_str_mv |
bachelorThesis |
dc.type.coar.fl_str_mv |
http://purl.org/coar/resource_type/c_7a1f |
dc.type.spa.none.fl_str_mv |
Trabajo de grado |
dc.identifier.uri.none.fl_str_mv |
https://repository.urosario.edu.co/handle/10336/45741 |
url |
https://repository.urosario.edu.co/handle/10336/45741 |
dc.language.iso.none.fl_str_mv |
eng |
language |
eng |
dc.rights.*.fl_str_mv |
Attribution-NonCommercial-NoDerivatives 4.0 International |
dc.rights.coar.fl_str_mv |
http://purl.org/coar/access_right/c_abf2 |
dc.rights.acceso.none.fl_str_mv |
Abierto (Texto Completo) |
dc.rights.uri.*.fl_str_mv |
http://creativecommons.org/licenses/by-nc-nd/4.0/ |
rights_invalid_str_mv |
Attribution-NonCommercial-NoDerivatives 4.0 International Abierto (Texto Completo) http://creativecommons.org/licenses/by-nc-nd/4.0/ http://purl.org/coar/access_right/c_abf2 |
dc.format.extent.none.fl_str_mv |
40 pp |
dc.format.mimetype.none.fl_str_mv |
application/pdf |
dc.publisher.none.fl_str_mv |
Universidad del Rosario |
dc.publisher.department.none.fl_str_mv |
Escuela de Ingeniería, Ciencia y Tecnología |
dc.publisher.program.none.fl_str_mv |
Programa de Matemáticas Aplicadas y Ciencias de la Computación - MACC |
publisher.none.fl_str_mv |
Universidad del Rosario |
institution |
Universidad del Rosario |
dc.source.bibliographicCitation.none.fl_str_mv |
Robert G. Bartle. The Elements of Integration and Lebesgue Measure. 1st ed. Wiley Clas- sics Library v.92. Hoboken: John Wiley & Sons, Incorporated, 1995. 1 p. isbn: 9780471042228 9781118164488. Walter Rudin. Real and complex analysis. 3. ed., internat. ed., [Nachdr.] McGraw-Hill international editions Mathematics series. New York, NY: McGraw-Hill, 2013. 416 pp. isbn: 978-0-07-100276-9 978-0-07-054234-1. Donald L. Cohn. Measure Theory: Second Edition. Birkhäuser Advanced Texts Basler Lehrbücher. New York, NY: Springer New York, 2013. isbn: 978-1-4614-6955-1 978-1- 4614-6956-8. doi: 10.1007/978-1-4614-6956-8. url: https://link.springer.com/ 10.1007/978-1-4614-6956-8 (visited on 08/14/2024). Elias M. Stein and Rami Shakarchi. Real analysis: measure theory, integration, and Hilbert spaces. Princeton lectures in analysis v. 3. Princeton, N.J: Princeton University Press, 2005. 402 pp. isbn: 9780691113869. Manfred Einsiedler and Thomas Ward. Ergodic Theory: with a view towards Number The- ory. London: Springer London, 2011. isbn: 978-0-85729-020-5 978-0-85729-021-2. doi: 10.1007/978- 0- 85729- 021- 2. url: https://link.springer.com/10.1007/978- 0-85729-021-2 (visited on 08/14/2024). Daniel W. Stroock. Probability theory: an analytic view. 2nd ed. Cambridge New York: Cambridge University Press, 2011. 1 p. isbn: 978-0-521-76158-1 978-0-511-97424-3 978-1- 139-01188-4 Sheldon M. Ross. Introduction to probability models. Tenth edition. Amsterdam Boston: Academic Press, an imprint of Elsevier, 2010. 1 p. isbn: 978-0-12-375686-2 978-0-12- 375687-9. Dimitri P. Bertsekas and John N. Tsitsiklis. Introduction to probability. 2nd ed. Optimiza- tion and computation series. Belmont: Athena scientific, 2008. isbn: 978-1-886529-23-6. Patrick Billingsley. Probability and measure. 3rd ed. Wiley series in probability and math- ematical statistics. New York: Wiley, 1995. 593 pp. isbn: 978-0-471-00710-4. Paul E. Pfeiffer. Probability for Applications. Springer Texts in Statistics. New York, NY: Springer New York, 1990. 679 pp. isbn: 978-1-4615-7676-1. doi: 10.1007/978-1-4615- 7676-1. Queen Mary University of London. Lecture 2: Second Language Acquisition. https:// qmplus . qmul . ac . uk / pluginfile . php / 2470175 / mod _ resource / content / 13 / L2 - 2022.pdf. Accessed: 2025-04-21. 2022. Alessandro Rinaldo. Lecture Notes: February 20, 2018. https://www.stat.cmu.edu/ ~arinaldo/Teaching/36752/S18/Scribed_Lectures/Feb20.pdf. Scribed lecture notes for 36-752: Advanced Probability Overview, Spring 2018, Carnegie Mellon University. 2018. David A. Stephens. The Borel-Cantelli Lemma. https://www.math.mcgill.ca/dstephens/ OldCourses/556-2006/Math556-BorelCantelli.pdf. Lecture notes for Math 556: Math- ematical Statistics I, McGill University. 2006. Jeffrey S. Rosenthal. A first look at rigorous probability theory. 2. ed., reprinted. New Jersey: World Scientific, 2010. 219 pp. isbn: 978-981-270-371-2. Marcelo Viana. Lectures on Lyapunov exponents. Cambridge studies in advanced math- ematics 145. Cambridge ; New York: Cambridge University Press, 2014. 202 pp. isbn: 978-1-107-08173-4. Peter Walters. An introduction to ergodic theory. 1. softcover printing. Graduate texts in mathematics 79. New York: Springer, 2000. 250 pp. isbn: 9780387951522. Artur Avila and Jairo Bochi. “On the Subadditive Ergodic Theorem”. In: Ergodic The- ory and Dynamical Systems 29.1 (Jan. 2009). Publisher: Cambridge University Press,pp. 1–16. doi: 10 . 1017 / S014338570800067X. url: https : / / doi . org / 10 . 1017 / S014338570800067X. Hillel Furstenberg and Harry Kesten. “Products of Random Matrices”. In: The Annals of Mathematical Statistics 31.2 (June 1960), pp. 457–469. issn: 0003-4851. doi: 10.1214/ aoms/1177705909. url: http://projecteuclid.org/euclid.aoms/1177705909 (visited on 08/14/2024). Walter Rudin. Functional analysis. 2nd ed. International series in pure and applied math- ematics. New York: McGraw-Hill, 1991. 424 pp. isbn: 9780070542365. Jean H. Gallier. Geometric methods and applications: for computer science and engineer- ing. 2nd ed. Texts in applied mathematics 38. New York: Springer, 2011. 680 pp. isbn: 9781441999603 9781441999610. Geon Ho Choe. Computational ergodic theory. Algorithms and computation in mathemat- ics v. 13. Berlin: Springer, 2005. 453 pp. isbn: 978-3-540-23121-9. Walter Gander. Algorithms for the QR-Decomposition. Tech. rep. 80-021. Re-typed in LATEX in October 2003, In memory of Prof. H. Rutishauser. Zürich, Switzerland: Seminar für Angewandte Mathematik, Eidgenössische Technische Hochschule, 1980. url: https: //people.inf.ethz.ch/gander/papers/qrneu.pdf. |
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Artigiani, Mauro7b7040ca-f33d-4b80-bcbf-806c727430d5-1Martínez, Cristian73c3e106-5afb-4ee5-8229-c1f263876f29-1Bermúdez Guzmán, JulianaProfesional en Matemáticas Aplicadas y Ciencias de la ComputaciónProfesional en Matemáticas Aplicadas y Ciencias de la ComputaciónPregrado47eddce6-d6d7-44b2-86d5-d869861a22c6-12025-06-19T14:33:07Z2025-06-19T14:33:07Z2025-06-09In this thesis, we investigate the asymptotic behavior of products of random matrices through Lyapunov exponents. Our theoretical framework is grounded in Kingman’s Subadditive Ergodic Theorem, from which we derive the Furstenberg-Kesten Theorem and Oseledets’ Theorem in two dimensions. These results provide the tools to quantify exponential growth rates and directional behavior in random matrix products. To visualize our theoretical conclusions, we present a series of simulations that illustrate the emergence of Lyapunov exponents and their predictive power in practical settings.In this thesis, we investigate the asymptotic behavior of products of random matrices through Lyapunov exponents. Our theoretical framework is grounded in Kingman’s Subadditive Ergodic Theorem, from which we derive the Furstenberg-Kesten Theorem and Oseledets’ Theorem in two dimensions. These results provide the tools to quantify exponential growth rates and directional behavior in random matrix products. To visualize our theoretical conclusions, we present a series of simulations that illustrate the emergence of Lyapunov exponents and their predictive power in practical settings.40 ppapplication/pdfhttps://repository.urosario.edu.co/handle/10336/45741engUniversidad del RosarioEscuela de Ingeniería, Ciencia y TecnologíaPrograma de Matemáticas Aplicadas y Ciencias de la Computación - MACCAttribution-NonCommercial-NoDerivatives 4.0 InternationalAbierto (Texto Completo)http://creativecommons.org/licenses/by-nc-nd/4.0/http://purl.org/coar/access_right/c_abf2Robert G. Bartle. The Elements of Integration and Lebesgue Measure. 1st ed. Wiley Clas- sics Library v.92. Hoboken: John Wiley & Sons, Incorporated, 1995. 1 p. isbn: 9780471042228 9781118164488.Walter Rudin. Real and complex analysis. 3. ed., internat. ed., [Nachdr.] McGraw-Hill international editions Mathematics series. New York, NY: McGraw-Hill, 2013. 416 pp. isbn: 978-0-07-100276-9 978-0-07-054234-1.Donald L. Cohn. Measure Theory: Second Edition. Birkhäuser Advanced Texts Basler Lehrbücher. New York, NY: Springer New York, 2013. isbn: 978-1-4614-6955-1 978-1- 4614-6956-8. doi: 10.1007/978-1-4614-6956-8. url: https://link.springer.com/ 10.1007/978-1-4614-6956-8 (visited on 08/14/2024).Elias M. Stein and Rami Shakarchi. Real analysis: measure theory, integration, and Hilbert spaces. Princeton lectures in analysis v. 3. Princeton, N.J: Princeton University Press, 2005. 402 pp. isbn: 9780691113869.Manfred Einsiedler and Thomas Ward. Ergodic Theory: with a view towards Number The- ory. London: Springer London, 2011. isbn: 978-0-85729-020-5 978-0-85729-021-2. doi: 10.1007/978- 0- 85729- 021- 2. url: https://link.springer.com/10.1007/978- 0-85729-021-2 (visited on 08/14/2024).Daniel W. Stroock. Probability theory: an analytic view. 2nd ed. Cambridge New York: Cambridge University Press, 2011. 1 p. isbn: 978-0-521-76158-1 978-0-511-97424-3 978-1- 139-01188-4Sheldon M. Ross. Introduction to probability models. Tenth edition. Amsterdam Boston: Academic Press, an imprint of Elsevier, 2010. 1 p. isbn: 978-0-12-375686-2 978-0-12- 375687-9.Dimitri P. Bertsekas and John N. Tsitsiklis. Introduction to probability. 2nd ed. Optimiza- tion and computation series. Belmont: Athena scientific, 2008. isbn: 978-1-886529-23-6.Patrick Billingsley. Probability and measure. 3rd ed. Wiley series in probability and math- ematical statistics. New York: Wiley, 1995. 593 pp. isbn: 978-0-471-00710-4.Paul E. Pfeiffer. Probability for Applications. Springer Texts in Statistics. New York, NY: Springer New York, 1990. 679 pp. isbn: 978-1-4615-7676-1. doi: 10.1007/978-1-4615- 7676-1.Queen Mary University of London. Lecture 2: Second Language Acquisition. https:// qmplus . qmul . ac . uk / pluginfile . php / 2470175 / mod _ resource / content / 13 / L2 - 2022.pdf. Accessed: 2025-04-21. 2022.Alessandro Rinaldo. Lecture Notes: February 20, 2018. https://www.stat.cmu.edu/ ~arinaldo/Teaching/36752/S18/Scribed_Lectures/Feb20.pdf. Scribed lecture notes for 36-752: Advanced Probability Overview, Spring 2018, Carnegie Mellon University. 2018.David A. Stephens. The Borel-Cantelli Lemma. https://www.math.mcgill.ca/dstephens/ OldCourses/556-2006/Math556-BorelCantelli.pdf. Lecture notes for Math 556: Math- ematical Statistics I, McGill University. 2006.Jeffrey S. Rosenthal. A first look at rigorous probability theory. 2. ed., reprinted. New Jersey: World Scientific, 2010. 219 pp. isbn: 978-981-270-371-2.Marcelo Viana. Lectures on Lyapunov exponents. Cambridge studies in advanced math- ematics 145. Cambridge ; New York: Cambridge University Press, 2014. 202 pp. isbn: 978-1-107-08173-4.Peter Walters. An introduction to ergodic theory. 1. softcover printing. Graduate texts in mathematics 79. New York: Springer, 2000. 250 pp. isbn: 9780387951522.Artur Avila and Jairo Bochi. “On the Subadditive Ergodic Theorem”. In: Ergodic The- ory and Dynamical Systems 29.1 (Jan. 2009). Publisher: Cambridge University Press,pp. 1–16. doi: 10 . 1017 / S014338570800067X. url: https : / / doi . org / 10 . 1017 / S014338570800067X.Hillel Furstenberg and Harry Kesten. “Products of Random Matrices”. In: The Annals of Mathematical Statistics 31.2 (June 1960), pp. 457–469. issn: 0003-4851. doi: 10.1214/ aoms/1177705909. url: http://projecteuclid.org/euclid.aoms/1177705909 (visited on 08/14/2024).Walter Rudin. Functional analysis. 2nd ed. International series in pure and applied math- ematics. New York: McGraw-Hill, 1991. 424 pp. isbn: 9780070542365.Jean H. Gallier. Geometric methods and applications: for computer science and engineer- ing. 2nd ed. Texts in applied mathematics 38. New York: Springer, 2011. 680 pp. isbn: 9781441999603 9781441999610.Geon Ho Choe. Computational ergodic theory. Algorithms and computation in mathemat- ics v. 13. Berlin: Springer, 2005. 453 pp. isbn: 978-3-540-23121-9.Walter Gander. Algorithms for the QR-Decomposition. Tech. rep. 80-021. Re-typed in LATEX in October 2003, In memory of Prof. H. Rutishauser. 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