Dissertations, Theses, and Capstone Projects

Date of Degree


Document Type


Degree Name





Yunping Jiang

Subject Categories



Markov partitions, martingale, uniformly quasisymmetric


The main subject studied in this thesis is the space of all uniformly quasisymmetric circle endomorphisms preserving the Lebesgue measure. Although many of our arguments work for any degree d≥2, our proof will be mainly written for degree 2 maps.

We will introduce a sequence of Markov partitions of the unit circle by using preimages of the fixed point of such circle endomorphism f. The uniform quasisymmetry condition is equivalent to the bounded nearby geometry condition of the Markov partitions. In Chapter 2 of this thesis, for each f, we use the Lebesgue invariant condition and the bounded geometry property to construct a martingale sequence {Xf,k} which has a L1 limiting function Xf on the dual symbolic space. We also show that the limiting martingale is invariant under symmetric conjugacy. The classical Hilbert transform introduces an almost complex structure on the space of all uniformly quasisymmetric circle endomorphisms that preserve the Lebesgue measure. This is presented in Chapter 3. In Chapters 4 & 5, we study locally constant limiting martingales and the related rigidity problems. A locally constant limiting martingale is the limit of a martingale sequence {Xk} of length n for some n≥0, i.e. the limiting martingale X=Xk for some n. We prove the rigidity problem for martingale sequence of length n≤4. That is, there is a unique way to construct a sequence of Markov partitions if the given limiting martingale Xf is equal to Xf,n for some n≤4. One of the consequences is that if two martingale sequences {Xf,k} and {Xg,k} have the same limit and both have length n≤4, where f and g are two uniformly quasisymmetric circle endomorphisms preserving the Lebesgue measure, then f=g. Another consequence is that if {Xf,k} has length n≤4, then there is no other map in the symmetric conjugacy class of f that preserves Lebesgue measure. In the class of uniformly symmetric circle endomorphisms, we prove that q(z)=z2, which has martingale sequence {Xq,k=2} for any k, is the only map whose limiting martingale is locally constant. Finally, we construct an analytic expanding circle endomorphism which preserves the Lebesgue measure and is a quasisymmetirc conjugate of q(z)=z2, i.e. f=hqh-1. We show that the conjugacy h is symmetric at one point but not symmetric on the whole unit circle.

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