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Furstenberg's conjecture and measure rigidity for some classes of non-abelian affine actions on tori
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2015 (English)Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesisAlternative title
Furstenbergs förmodan och måttrigiditet för några klasser av icke-abelska affina verkningar på torusar (Swedish)
Abstract [en]

In 1967 Furstenberg proved that the set {2n3mα(mod 1) | n, m ∈N} is dense in the circle for any irrational α. He also made the following famous measure rigidity conjecture: the only ergodic measures on the circle invariant under both x —> 2x and x —> 3x are the Lebesgue measure and measures supported on a finite set. In this thesis we discuss both Furstenberg’s theorem and his conjecture, as well as the partial solution of the latter given by Rudolph. Following Matheus’presentation of Avila’s ideas for a proof of a weak version of Rudolph’s theorem, we prove a result on extending measure preservation from a semigroup action to a larger semigroup action. Using this result we obtain restrictions on the set of invariant measures for certain classes of non-abelian affine actions on tori. We also study some general properties of affine abelian and non-abelian actions and we show that analogues of Furstenberg’s theorem hold for affine actions on the circle.

Abstract [sv]

1967 bevisade Furstenberg att mängden {2n3mα(mod 1) | n, m ∈N} är tät i cirkeln för alla irrationella tal α. Furstenberg ligger även bakom följande berömda förmodan: de enda ergodiska måtten påcirkeln som är invarianta under både x 􏰀—> 2x och x 􏰀—> 3x är Lebesguemåttet och mått med ändligt stöd. I det här examensarbetet behandlar vi Furstenbergs sats, Furstenbergs förmodan och Rudolphs sats. Vi följer Matheus presentation av Avilas idéer för ett bevis av en svag variant av Rudolphs sats och vi bevisar att en måttbevarande semigruppverkan under vissa antaganden kan utökas till en semigruppverkan av en större semigrupp. Med hjälp av detta resultat erhåller vi begränsningar av mängden av mått invarianta under vissa klasser av icke-abelska affina verkningar påtorusen. Vi studerar även allmänna egenskaper hos affina abelska och icke-abelska verkningar och vi visar att satser analoga med Furstenbergs sats håller för affina verkningar påcirkeln.

Place, publisher, year, edition, pages
2015.
Series
TRITA-MAT-E, 2015:53
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:kth:diva-172037OAI: oai:DiVA.org:kth-172037DiVA: diva2:845399
Subject / course
Mathematics
Educational program
Master of Science - Mathematics
Examiners
Available from: 2015-08-12 Created: 2015-08-11 Last updated: 2015-08-12Bibliographically approved

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