Slabo i jako miješanje u ergodskoj teoriji

master's thesis

• Author: Iva Barada
• Mentor: Siniša Slijepčević (mentor)
• Committee member: Siniša Slijepčević (predsjednik povjerenstva)
• Committee member: Igor Pažanin (član povjerenstva)
• Committee member: Boris Muha (član povjerenstva)
• Committee member: Bojan Basrak (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2015-09-25, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu obrađen je pojam ergodičnosti s naglaskom na miješanje (jako i slabo miješanje). Od samog početka imali smo zadatak odgovoriti na pitanja o konvergenciji dugoročnog prosječnog ponašanja sustava, odnosno jednakosti usrednjenog vremena i prostora. Kako bi to bilo moguće prije svega trebali smo uvesti niz pojmova posebice iz teorije mjere te se pobliže upoznati s dinamičkim sustavima. U konačnici smo odgovorili na pitanja koja su bila postavljena na početku ovog rada. Iskazima i dokazima teorema prolazili smo kroz veliki dio 19. i 20. stoljeća. Tako smo na putu dokazivanja konvergencije dugoročnog prosječnog ponašanja sustava iskazali von Neumannov usrednjen ergodski teorem u Hilbertovom prostoru te usrednjen ergodski teorem u Banachovom prostoru. Konvergencija dugoročnog prosječnog ponašanja sustava dokazana je Birhoffovim teoremom iz 1931. godine, za čiji dokaz je bio potreban maksimalan ergodski teorem. Na putu dokazivanja jednakosti usrednjenog vremena i prostora upoznali smo pojam povratnosti, kao povratak točke u skup nakon nekog vremena. Von Neumann i Birhoffov su nam pružili kvantitativnu analizu povratnosti, a Poincareovim teoremom povratnosti opisana je njena kvalitativna analiza. Ergodičnost je svojstvo sustava za koje vrijedi da svaka invarijantna izmjeriva funkcija na cijelom skupu je konstanta gotovo svugdje. Jednakost prostornog i vremenskog usrednjenja smo dokazali pod pretpostavkom da sustav ima svojstvo ergodičnosti. Konačno, upoznali smo se s jakim i slabim miješanjem te njihovim svojstvima. Najvažnije što smo dokazali je da jako miješanje je „jače” svojstvo od slabog miješanja te ergodičnosti. Jako miješanje smo prikazali i u području Markovljevih lanaca, gdje je to svojstvo ekvivalentno ireducibilnosti i povratnosti matrice prijelaza. S druge strane ekvivalentne su sljedeće tvrdnje: transformacija je slabo miješanje; Kartezijev produkt transformacija je slabo miješanje; Kartezijev produkt transformacija je ergodičan. Na kraju rada, prikazali smo dva primjera ergodične teorije u MATLABu. Prvi primjer ima svojstvo jakog miješanja, a u drugom je istaknuto svojstvo ergodičnosti.

Abstract (english)

This thesis presented concept of ergodicity with focus on strong and weak mixing. From the very beginning we had the task to answer questions about convergence of long-term average behavior of system, i.e. equality of the time mean and space mean. In order to accomplish that, we needed to introduce numerous terms especially from measure theory and familiarize ourselves with the dynamic systems. At the end we successfully answered on all questions that were asked at the beginning of this paper. By statements and proofs we went through big part of 19th and 20th century. On the way to prove the convergence of long-term average behavior of the system, we have stated von Neumann averaged ergodic theorem in Hilbert space and averaged ergodic theorem in Banach space. The convergence of long-term average behavior of the system was proved with Birkhoff’s theorem from 1931. for which proof we needed maximal ergodic theorem. We also introduced concept of recurrence as a return of point into a set after some period of time. Von Neumann and Birkhoff have provided us with quantitative analysis of recurrence, while Poincare’s theorem of recurrence gave us qualitative analysis. Ergodicity is characteristic of a system for which applies that each invariant measurable function is constant i.e. we proved coincidence of time mean and space mean under the assumption that the system has the property of ergodicity. Finally, we represented the idea of strong and weak mixing and their properties. The most important thing we proved is the fact that strong mixing implies weak mixing and ergodicity. We presented strong mixing in Markov chain theory where this property is equivalent to irreducibility and recurrence of transition matrix. On the other hand, it is equivalent that transformation is weak mixing and that. At the end of the thesis, we showed two examples of ergodic theory in MATLAB. The first example has property of strong mixing, while the second one is ergodic.

• Keywords: ergodičnost; jako i slabo miješanje; konvergencija dugoročnog prosječnog ponašanja sustava; von Neumannov teorem; Birhoffov teorem; Poincareov teorem; MATLAB
• Keywords (english): ergodicity; strong and weak mixing; convergence of long-term average behavior of system; von Neumann theorem; Birkhoff’s theorem; Poincare’s theorem; MATLAB
• Language: croatian
• URN:NBN: urn:nbn:hr:217:365074
• Study programme: Title: Finance and Business Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Access restricted to students and staff of home institution
• Created on: 2017-05-22 11:04:59