關於不變測度的投影

On the projections of invariant measures

王柏安(Po-An Wang)

指導教授 ： 陳國璋

國立清華大學/理學院/數學系/碩士(2014年)

https://doi.org/10.6843/NTHU.2014.00333

摘要

不變測可以由它們的投影來研究。特別是關於不變測度的質心的研究近幾年受到不少關注。陳國璋和董遜(2010[3]) 發現一套方法使得不同的Tent映射的軌道有相同的質心。這可以看做是許多不同的不變測度有重疊的投影在f(x)=x這方向上。這驅使我們考慮:什麼情況下，不同的兩個轉化下所對應的不變測度能擁有重疊的投影在某個方向上?這篇論文目的即是討論此問題。

關鍵字

不變測度 ； 重疊投影

並列摘要

Invariant probability measures can be studied through their projections.
In particular, the study of barycenters of these invariant measures has received
much attention in resent years. K.C.Chen and X.Dong (2010 [3]) found a
way to construct different orbits with the same barycenter for the tent map. It can
be seen as that many invariant measures have overlapped projection along the
direction f(x) = x. This motivates us to consider: when can invariant measures
corresponding to two distinct transformations have overlapped projection
along some direction? This thesis aims to discuss this problem.

並列關鍵字

invariant measure ； overlapped projection

參考文獻

[1] B. Hunt and E. Ott,Optimal periodic orbits of chaotic systems occur at low period,Phy. Rev. E 54,(1996)329-337.

[3] K.C.Chen and X.Dong, On the barycenter of the tent map., Proc. Amer. Math. Soc.,(2010).

[4] O.Jenkinson, Geometric barycentre of invariant measures for the circle maps, Ergodic Theory Dynam. Systems 21, (2001).

[5] O.Jenkinson., Frequency locking on the boundary of barycentre set ,Experiment. Math. 2000, 309-317., (1999)

[7] R. Bamon and I. Malta and M.J. Pacifico, Changing rotation intervals of endomorphisms of the circle., Invent. Math. 83, (1986),257-264.