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On the dynamics of a family of critical circle endomorphisms
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2019 (English)Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesisAlternative title
Om dynamiken av en familj kritiska cirkel-endomorfier (English)
Abstract [en]

In this thesis we study two seperate yet related three parameter-families of continuously differentiable maps from the unit circle to unit circle which have a single critical point. For one of the families we show that there is a set of positive measure of parameters such that there is a set of positive measure for which all points in the latter set, the derivative experiences exponential growth. We do so by applying a similar methodology to what Michael Benedicks and Lennart Carleson used to study the quadratic family. For the other family we attempt to show a similar but weaker result using a similar method, but do not manage to do so. We expound on what difficulties the latter family provides and what features Benedicks and Carleson used for the quadratic family that we do not have available.

Abstract [sv]

I den här uppsatsen studerar vi två olika men relaterede treparameterfamiljer av kontinuerligt differentierbara avbildningar från enhetscirkeln till enhetscirkeln som har exakt en kritisk punkt. For den ena familjen visar vi att det finns en mängd av positivt mått av parametrar sådana att det finns en mängd av positivt mått så att för varje punkt i den senarenämnde mängden erfar derivatan exponentiell tillväxt. Vi uppnår detta genom att använda en metod som liknar den som Michael Benedicks och Lennart Carleson använde för att studera den kvadratiska familjen. För den andra familjen försöker vi visa ett liknande men svagare resultat genom att använda en liknande metodik men misslyckas. Vi diskuterar och förklarar vilka svårigheter den senare familjen ger och vilka egenskaper som Benedicks och Carleson använder sig av hos den kvadratiska familjen som vår familj saknar

Place, publisher, year, edition, pages
2019.
Series
TRITA-SCI-GRU ; 2019:353
Keywords [en]
Dynamical systems, Hyperbolic dynamics, Non-uniform hyperbolicity, Lyapunov exponents
Keywords [sv]
Dynamiska system, Hyperboliska system, Icke-likformig hyperbolicitet, Lyapunovexponenter
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-259743OAI: oai:DiVA.org:kth-259743DiVA, id: diva2:1355018
Subject / course
Mathematics
Educational program
Master of Science - Mathematics
Supervisors
Examiners
Available from: 2019-09-26 Created: 2019-09-26 Last updated: 2019-09-26Bibliographically approved

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