Student Seminars

A series of analysis seminars, with an emphasis on the interests of the group, normally takes place on Tuesday afternoons at 14:30 HKT in LSB 219. All are welcome.

Upcoming

2024

• A reading workshop on Tuesdays at 14:30 HKT in LSB 202.

Previous

2023

• 11/07–11/21 Tuesdays LSB 202
  Zhou Feng 冯洲

  • Hochman, Michael & Shmerkin, Pablo. Local entropy averages and projections of fractal measures. Ann. of Math. (2) , Vol. 175, No. 3 p. 1001-1059. 2012.

  • Summary

    We have explored the powerfulness of lifting Lipschitz codings to morphisms between trees, which is a language to express the authors' genuine observation on the continuity of the average of entropies. As a first simple application, we follow Section 6 to present the lower semicontinuity of dimensions of Bernoulli convolutions. Then we move to Sections 7 and 8. After modeling the local dynamics using CP-chains (Markov chains), again we lift the orthogonal projections to tree morphisms, and prove the lower semicontinuity of the dimensions of ergodic CP-chains. This finishes one of the two major parts in the paper.

  • An entropy-like upper bound. Notes for Section 6 and Sections 7 & 8.

• 10/03–10/31 Tuesdays LSB 219
  Cai-Yun Ma 麻彩云

  • Hochman, Michael & Shmerkin, Pablo. Local entropy averages and projections of fractal measures. Ann. of Math. (2) , Vol. 175, No. 3 p. 1001-1059. 2012.

  • Notes for Section 4 and Section 5.

• 09/19 Tuesday LSB 219
  Edouard Daviaud, Zhou Feng and Cai-Yun Ma

  • Hochman, Michael & Shmerkin, Pablo. Local entropy averages and projections of fractal measures. Ann. of Math. (2) , Vol. 175, No. 3 p. 1001-1059. 2012.

  • The outline of our workshop

2022

Seminar Topics 2022

• 11/15 Tuesday LSB 202
  Yuhao Xie

  • Jordan, Thomas; Pollicott, Mark & Simon, Károly. Hausdorff dimension for randomly perturbed self affine attractors. Comm. Math. Phys. , Vol. 270, No. 2 p. 519-544. 2007.

  • Outline

    We discuss how to calculate the dimension of ergodic measures in the symbolic spaces under the self-affine projections.

• 11/1 Tuesday LSB 202
  Zhou Feng

  • Bishop, Christopher J. & Peres, Yuval. Fractals in probability and analysis. Vol. 162. Cambridge Studies in Advanced Mathematics Cambridge University Press, Cambridge p. ix+402. 2017.

  • Summary

    We finish the Exercises 1.51, 1.52 and 1.53. Moreover, we show that the Hausdorff dimension of the set in Exercise 1.52 is still $0$ after replacing the $\lim$ with $\limsup$ or $\liminf$ in the definition.

  • Note.

• 10/18 Tuesday LSB 202
  Yuji Li

  • Bishop, Christopher J. & Peres, Yuval. Fractals in probability and analysis. Vol. 162. Cambridge Studies in Advanced Mathematics Cambridge University Press, Cambridge p. ix+402. 2017.

  • Summary

    We mainly talked about Levy's modulus of continuity of Brownian motion and also some Holder continuity property due to Levy's construction of Brownian motion.

• 9/20 - 9/27 Tuesdays LSB 202
  Caiyun Ma

  • Bishop, Christopher J. & Peres, Yuval. Fractals in probability and analysis. Vol. 162. Cambridge Studies in Advanced Mathematics Cambridge University Press, Cambridge p. ix+402. 2017.

  • Deliu, Anca; Geronimo, J. S.; Shonkwiler, R. & Hardin, D.. Dimensions associated with recurrent self-similar sets. Math. Proc. Cambridge Philos. Soc. , Vol. 110, No. 2 p. 327-336. 1991.

  • Outline

    Exercise 4.2 Hausdorff and Minkowski dimensions of McMullen Set $K(D)$ agree if and only if $D$ has uniform horizontal fibers.

    Exercise 4.3 Show that if $D$ has uniform horizontal fibers, then $K(D)$ has positive finite Hausdorff measure in its dimension.

    Exercise 4.5 Compute the dimension of a self-affine set of finite type if we assume the number of rectangles in corresponding rows is the same for each pattern.

  • Summary

    In computing the box dimension one applies compositions of affine maps to a square containing the attractor and one uses the number of boxes in this covering to get estimates on the minimum number of boxes needed to cover the attractor at various levels. One of the main problems is to know where the attractor is in order to count only those boxes that are needed.

    One method is to project the attractor onto a convenient axis and relate the dimension of the projected attractor to the dimension of the original set.

  • Note.

• 9/13 Tuesday LSB 202
  Yuhao Xie

  • Bishop, Christopher J. & Peres, Yuval. Fractals in probability and analysis. Vol. 162. Cambridge Studies in Advanced Mathematics Cambridge University Press, Cambridge p. ix+402. 2017.

  • Outline

    We will talk about a lower bound for $\dim_{p} E \times F$, and then give an example that the lower bound can be strict.

• 7/26 Tuesday Zoom
  Caiyun Ma

  • Falconer, K. J.. Generalized dimensions of measures on self-affine sets. Nonlinearity , Vol. 12, No. 4 p. 877-891. 1999.

  • Note.

• 5/10 Tuesday Zoom
  Yuhao Xie

  • Lagarias, Jeffrey C. & Wang, Yang. Self-affine tiles in R^n. Adv. Math. , Vol. 121, No. 1 p. 21-49. 1996.

  • Note.

  • Outline

    We will introduce some properties of the homogenous Self-Affine IFS whose affine dimension is equal to the dimension of the space. These results are from Lagarias and Yang.

• 5/3 Tuesday Zoom
  Zhou Feng

  • Kenyon, Richard. Projecting the one-dimensional Sierpinski gasket. Israel J. Math. , Vol. 97 p. 221-238. 1997.

  • Note; Dimension for IFS with exact-overlaps.

  • Summary

    We observe that the projections of every planar homogeneous self-similar set with rational (mod $\pi$) rotation angle are self-similar sets.

    Next we focus on the projections of 1-dim Sierpinski gasket. Let $\tau \in \mathbb{R}$. Let $E_{\tau}$ be the attractor of IFS:

    $$\begin{aligned} \{ 1/3 x, 1/3 x + 2/3, 1/3 x + \tau / 3\}.\end{aligned}$$

    The following results are known.

    • If $\tau \notin \mathbb{Q}$, then $\mathcal{L}(E_{\tau}) = 0$ and $\dim_{H} E_{\tau} = 1$.

    • If $\tau = 2 p/q \in \mathbb{Q}$ with $\gcd(p,q) = 1$ and $p + q \equiv 0 \pmod 3$, then $\mathcal{L} (E_{\tau}) = 2 / q$ and $E_{\tau} = \overline{E_{\tau}^{\circ}}$.

    • If $\tau = 2p/q \in \mathbb{Q}$ with $\gcd(p,q) = 1$ and $p + q \not \equiv 0 \pmod 3$, then $\dim_{H} E_{\tau} < 1$ and $\dim_{H} E_{\tau}$ can be computed by an algorithm (in finite steps).

    The Open Set Condition only holds in the second case. The Exact Overlaps only happen in the third case.

    In particular for the Ex 2.7 in B-P, we compute that $\dim_{H} E_{1/2} = \log_{3}(\frac{\sqrt{5}+3}{2})$. Then we quickly finish Ex 2.15 and 2.16.

  • A joint note in progress: Basics on Moran structures

• 4/26 Tuesday Zoom
  Yuji Li

  • Bishop, Christopher J. & Peres, Yuval. Fractals in probability and analysis. Vol. 162. Cambridge Studies in Advanced Mathematics Cambridge University Press, Cambridge p. ix+402. 2017.

  • Note.

• 4/12 Tuesday Zoom
  Tianhan Yi

  • Falconer, K. J.. On the Hausdorff dimensions of distance sets. Mathematika , Vol. 32, No. 2 p. 206-212 (1986). 1985.

  • Note.

  • Outline

    Hausdorff dimension of distance set, B-P Ex1.53, 3.3, 3.8.

• 3/29 Tuesday Zoom
  Yuhao Xie

  • Bishop, Christopher J. & Peres, Yuval. Fractals in probability and analysis. Vol. 162. Cambridge Studies in Advanced Mathematics Cambridge University Press, Cambridge p. ix+402. 2017.

  • Note.

  • Outline

    We do some exercises about box dimension in the chapter five of Mattila’s book. And also introduce Example 1.3.3 and 1.7.1 in Bishop and Peres’ book.

• 3/22 Tuesday Zoom
  Zhou Feng

  • Bishop, Christopher J. & Peres, Yuval. Fractals in probability and analysis. Vol. 162. Cambridge Studies in Advanced Mathematics Cambridge University Press, Cambridge p. ix+402. 2017.

  • Note.

  • Summary

    We finished Exercises 1.2, 1.5, 1.38, 1.39, 1.42, 1.44, 1.45, 1.46 and 1.47 in the reference. During this progress, we have learned:

    • a trick to combine a sequence of scales in a desired way.

    • the power of dimension formula for homogeneous Moran sets. See the section about sets defined via digits in Basics on Moran structures. In this joint note, we also record a dimension formula for perturbed symmetric Cantor sets .

    • a combination of two ways of defining sets via digits.

    • a principle of choosing best Bernoulli measures on sets defined by digit densities.

• 3/15 Tuesday Zoom
  Yuji Li

  • Bishop, Christopher J. & Peres, Yuval. Fractals in probability and analysis. Vol. 162. Cambridge Studies in Advanced Mathematics Cambridge University Press, Cambridge p. ix+402. 2017.

  • Note.

• 3/1 — 3/8 Tuesdays Zoom
  Caiyun Ma

  • Feng, Dejun; Wen, Zhiying & Wu, Jun. Some dimensional results for homogeneous Moran sets. Sci. China Ser. A , Vol. 40, No. 5 p. 475-482. 1997.

  • Outline 3/1

    Theorem 1. Suppose $E \in \mathcal{M}\left(J,\left\{n_{k}\right\},\left\{c_{k}\right\}\right)$. Then we have

    $$\begin{aligned} \liminf_{k \rightarrow \infty} \frac{\log n_{1} n_{2} \cdots n_{k}}{-\log c_{1} c_{2} \cdots c_{k+1} n_{k+1}} \leqslant \operatorname{dim}_{\mathrm{H}} E \leqslant \liminf_{k \rightarrow \infty} \frac{\log n_{1} n_{2} \cdots n_{k}}{-\log c_{1} c_{2} \cdots c_{k} }.\end{aligned}$$

    Theorem 2. For any $E\in \mathcal{M}\left(J,\left\{n_{k}\right\},\left\{c_{k}\right\}\right)$, we have

    $$\begin{aligned} \limsup_{k \rightarrow \infty} \frac{\log n_{1} \cdots n_{k}}{-\log c_{1} c_{2} \cdots c_{k}} \leqslant \operatorname{dim}_{\mathrm{P}} E \leqslant \overline{\operatorname{dim}_{\mathrm{B}}} E \leqslant \limsup_{k \rightarrow \infty} \frac{\log n_{1} \cdots n_{k+1}}{-\log c_{1} c_{2} \cdots c_{k}+\log n_{k+1}}.\end{aligned}$$

  • Feng, Dejun; Rao, Hui & Wu, Jun. The net measure properties of symmetric Cantor sets and their applications. Progr. Natur. Sci. (English Ed.) , Vol. 7, No. 2 p. 172-178. 1997.

  • Outline 3/8

    Theorem 1. Let $E$ be the symmetric Cantor set determined by the sequences $\left\{n_{k}\right\}_{k} \geqslant 1$, $\left\{c_{k}\right\}_{k \geqslant 1}$, Then

    $$\begin{aligned} c \underline{\lim} _{k \rightarrow \infty} \prod_{i=1}^{k} n_{i} c_{i}^{s} \leqslant H^{s}(E) \leqslant \underline{\lim }_{k \rightarrow \infty} \prod_{i=1}^{k} n_{i} c_{i}^{s},\end{aligned}$$

    where $0 \leqslant s \leqslant 1$, and $c$ is an absolute positive constant.

  • Note 3/1; Note 3/8.

• 2/15 Tuesday Zoom
  Tianhan Yi

  • Einsiedler, Manfred & Ward, Thomas. Ergodic theory with a view towards number theory. Vol. 259. Graduate Texts in Mathematics Springer-Verlag London, Ltd., London p. xviii+481. 2011.

  • Note.

• 2/8 Tuesday Zoom
  Caiyun Ma

  • Bishop, Christopher J. & Peres, Yuval. Fractals in probability and analysis. Vol. 162. Cambridge Studies in Advanced Mathematics Cambridge University Press, Cambridge p. ix+402. 2017.

  • Outline

    We consider the digits set that each digit occurs with a certain frequency. The resulting sets are dense in $[0,1]$, so we need only consider their Hausdorff dimension and packing dimension.

    Let

    $$\begin{aligned} A_{p}=\left\{x=\sum_{n=1}^{\infty} x_{n} 2^{-n}: x_{n} \in\{0,1\}, \lim _{j \rightarrow \infty} \frac{1}{j} \sum_{k=1}^{j} x_{k}=p\right\}.\end{aligned}$$

    Thus $A_{p}$ is the set of real numbers in $[0,1]$ in which a 1 occurs in the binary expansion with asymptotic frequency $p$. We show that

    Theorem 1. For any $p\in (0,1)$,

    $$\begin{aligned} \operatorname{dim}\left(A_{p}\right)=h_{2}(p)=-p \log _{2} p-(1-p) \log _{2}(1-p).\end{aligned}$$

    where $\dim$ denotes the Huasdorff dimension or packing dimension.

  • Note.

• 1/25 Tuesday LSB 202
  Yuhao Xie

  • Bishop, Christopher J. & Peres, Yuval. Fractals in probability and analysis. Vol. 162. Cambridge Studies in Advanced Mathematics Cambridge University Press, Cambridge p. ix+402. 2017.

  • Outline

    Let $S\subset \mathbb{N}$, define $A_{S}:=\{\sum_{k=1}^{\infty}x_{k}2^{-k}:k\in S, x_{k}\in\{ 0,1\} \}$. Calculate $\dim_{H}A_{S}$ and $\dim_{P}A_{S}$.

2021

• 11/23 — 11/30 Tuesdays LSB 202
  Yuhao Xie

  • Hochman, Michael & Shmerkin, Pablo. Local entropy averages and projections of fractal measures. Ann. of Math. (2) , Vol. 175, No. 3 p. 1001-1059. 2012.

• 10/19 — 11/9 Tuesdays LSB 202
  Zhou Feng

  • Breuillard, E. & Varjú, P. P. Cut-off phenomenon for the ax+b Markov chain over a finite field. ArXiv:1909.09053. 2019.

  • Outline

    We will study the cut-off phenomenon for a type of Markov chain over finite fields. The target Markov chain also has a self-similar structure.

  • Summary

    We first established a concrete setup and some basic intuition on the concerned Markov chain. With the concepts of the mixing times and the cut-off phenomenon, we stated the main theorems and discussed the delicate dependencies of the parameters. Next we introduced the key observation connecting the sum of $\ell_2$ norms and the expectation of the number of the roots of a random polynomial in $\mathbb{F}_p$.

    After reviewing the Fourier transform on $\mathbb{F}_{p}$, we established the Konyagin's $O(\log p \log\log p)$ upper bound on the mixing time. The proof follows Konyagin's orginal idea and relies on the Dobrowolski's lower bound on Mahler measures.

    Based on the convolution structure of the envolving measures and the key observation, we applied the Fourier transform and the small-average-control-num-of-large-fibers arguments to obtain an upper bound on the mixing time for general $p$ but not all multipliers with high orders. To prove the $\ell_2$ cut-off phenomenon, a nested version of the aformmentioned argument was conducted via Markov inequality. The complete proof depends an upper bound of an expectation where the GRH came into the game through the key observation.

• 9/28 — 10/12 Tuesdays LSB 202
  Yuji Li

  • Bishop, C. J. & Peres, Y. Fractals in probability and analysis. Cambridge University Press, Cambridge, 2017, 162, ix+402

  • Bourgain, J. Ruzsa's problem on sets of recurrence. Israel J. Math., 1987, 59, 150-166

  • Outline

    Properties of Poincare sets and its connection towards van der Corput set.

• 9/14 Tuesday LSB 202
  Tianhan Yi

  • Einsiedler, M. & Ward, T. Ergodic theory with a view towards number theory: Chp11. Springer-Verlag London, Ltd., London, 2011, 259, xviii+481.

  • Outline

    Mautner Phenomenon and Howe-Moore Theorem for $SL_2(\mathbb{R})$.

• 9/7 Tuesday LSB 239
  Yuhao Xie

  • Falconer, K. J. & Marsh, D. T. On the Lipschitz equivalence of Cantor sets. Mathematika, 1992, 39, 223-233

  • Outline

    We will talk about a necessary condition for the lipschitz equivalence of two dust-like self-similar sets. This classic result is by Falconer.

• 8/17 — 8/31 Tuesdays LSB 239
  Caiyun Ma

  • Feng, D.-J. Lyapunov exponents for products of matrices and multifractal analysis. I. Positive matrices. Israel J. Math., 2003, 138, 353-376.

  • Summary

    Let $(\Sigma, \sigma)$ be a full shift space on an alphabet consisting of $m$ symbols and let $M: \Sigma \rightarrow L^{+}\left(\mathbb{R}^{d}, \mathbb{R}^{d}\right)$ be a continuous function taking values in the set of $d \times d$ positive matrices. Denote by $\lambda_{M}(x)$ the Lyapunov exponent of $M$ at $x$.

    Proposition 1. The set $\mathcal{L}_M$ of possible Lyapunov exponents denoted is just an interval.

    Theorem 1. For any $\alpha\in \mathcal{L}_M$,

    $$\begin{aligned} \dim_H\left\{x \in \Sigma:\lambda_{M}(x)=\alpha\right\}=\dim_P\left\{x \in \Sigma: \lambda_{M}(x)=\alpha\right\}.\end{aligned}$$

    The pressure function of $M$ is defined by

    $$\begin{aligned} P_{M}(q)=\lim _{n \rightarrow \infty} \frac{1}{n} \log \sum_{\omega \in \Sigma_{n}} \sup _{x \in[\omega]}\left\|M(x) M(\sigma x) \ldots M\left(\sigma^{n-1} x\right)\right\|^{q}, \quad q \in \mathbb{R}.\end{aligned}$$

    The matrix function $M$ induces a map $M_{*}: \mathcal{M}_{\sigma}(\Sigma) \rightarrow \mathbb{R} \cup\{-\infty\}$ given by

    $$\begin{aligned} M_{*}(\mu)=\lim _{n \rightarrow \infty} \frac{1}{n} \int \log \left\|M(y) M(\sigma y) \ldots M\left(\sigma^{n-1} y\right)\right\| d \mu(y), \quad \mu \in \mathcal{M}_{\sigma}(\Sigma).\end{aligned}$$

    Theorem 2. For any $\alpha\in \mathcal{L}_M$,

    $$\begin{aligned} \dim_{H} E_{M}(\alpha) &=\frac{1}{\log m} \inf _{q \in \mathbb{R}}\left\{-\alpha q+P_{M}(q)\right\} \\ &=\frac{1}{\log m} \sup \left\{h(\mu): \mu \in \mathcal{M}_{\sigma}(\Sigma), M_{*}(\mu)=\alpha\right\},\end{aligned}$$

    where $h(\mu)$ denotes the measure-theoretic entropy of $\mu$.

• 7/27 — 8/10 Tuesdays Zoom
  Jiaming Wu

  • Einsiedler, M. & Ward, T. Ergodic theory with a view towards number theory. Springer-Verlag London, Ltd., London, 2011, 259, xviii+481. Chp 9.

  • Notes

• 6/22 — 7/20 Tuesdays LSB 222
  Zhou Feng

  • A. Rapaport. Exact dimensionality and Ledrappier-Young formula for the Furstenberg measure. To appear in Transactions of the AMS.

  • Notes

• Last century — 6/15 Tuesdays LSB 222
  Tianhan Yi

  • Benoist, Y. & Quint, J.-F. Random walks on reductive groups. Springer, Cham, 2016, 62, xi+323. Chp 1–5.

  • Notes