Sarnak conjecture, understand with standard model

Sarnak conjecture is a conjecture lie in the overlap of dynamic system and number theory. It is mainly focus on understanding the behavior of entropy zero dynamic system by look at the correlation of an observable and the Mobius function .

We state it in a rigorous way:

let $ (X,T)$ be a entropy zero topological dynamic system. Let Mobius function be defined as $ \mu(n)=(-1)^t$, where $ $ is the number of different primes occur in the decomposition of $ n$.

Then for any continuous function $ f:X\to R$ and $ x\in X$, observable $ \xi(n)=f(T^n(x))$ is orthogonal to the Mobius function; i.e. ,

$ \lim_{N\to \infty}\frac{1}{N}\sum_{n=0}^{N-1}\mu(n)\xi(n)=o(N).$

I mainly focus on the special cases when dynamic system $ X$ is the skew product on $ T^2$ and when the dynamic system which is a interval exchange in $ [0,1]$.

Skew product

For the first one, $ \Theta=(T,T^2)$,$ T:T^2\longrightarrow T^2 $:
$ T(x)=x+\alpha,T(y)=cx+y+h(x)$
$ y_1(n)=T^{n}(x)=x+n\alpha,y_2(n)=T^n(y)=nx+\frac{n(n-1)}{2}\alpha+y+\sum_{n=1}^{N-1}h(x+i\alpha) $, where $ c=1,-1$.

by Bourgain-Ziegelar-Sarnak theorem we know the difficulties is focus on deal with the exponent

$ S_{p,q}(N)=\sum_{n=1}^N\mu(n)e^{\phi(n)+\sum_{m\in Z}e(mx)\hat H(m)(\frac{e(npm\alpha)-1}{e(m\alpha)-1}- \frac{e(nqm\alpha)-1}{e(m\alpha)-1})}$

for all $ p,q$ is suffice large primes pair.

and a much simper case is the affine map:$ T:(x,y)\to (x+\alpha,cx+y+\beta)$ on $ \mathbb T^2$ and the general case $ T:(x_1,…,x_n)\to A(x_1,…,x_n)$ where A is a upper-triangle matrix with diagonal 1; i.e. $ A=I+B$, B is nilpotent. So the sarnak conjecture in this case is reduce to the Davenport estimate on exponent by B-Z-S theorem:

$ |\sum_{n=0}^{N}e^{2\pi if(n)}|\leq c_A\frac{N}{(log N)^A}$, $ \forall A>0$.

Interval exchange map

For the interval exchange map, we can explain it by a composition of rotation of some part of $ S_1$ step by step and with a renormalization process to glue the neighbor rotations.

Now let us explain a little with this interesting dynamic system. We focus in the simplest nontrivial case, which is the 3-interval exchange map. In this case, just consider the permutation of intervals $ I_1,I_2,I_3$, and it is easy to see there is only one case is nontrivial that is permutation: $ I_1\to I_3,I_2\to I_2,I_3\to I_1$. We explain a little more with other trivial case:

When  $ I_1\to I_2,I_2\to I_3,I_3\to I_1$, the interval exchange map is just a rotation and for which the sarnak conjecture is just come from:

$ |\sum_{n=0}^{N}e^{2\pi in\alpha}\mu(n)|=o(N)$, $ \forall \alpha\in R$.

Which is trivial because $ \sum_{n=0}^{N}e^{2\pi in\alpha}\mu(n)=\frac{1-e^{2\pi iN\alpha}}{1-e^{2\pi i\alpha}}$.

For the case $I_1\to I_2, I_2\to I_1, I_3\to i_3$ the map $ T$ is a rotation on $ I_1\cap I_2$ but it is a identity map on $ I_3$ and the orbits of point only lying one of $I_1\cap I_2, I_3$, lying in which one depend on the original point $ x$ we take is lying in which one.

Now we focus on the most difficult situation. It is annoying but it is the obstacle we must get over to go far. Fortunately it could be explained as in the following picture.


3-Interval exchange map as two rotation map glue with a renormalization map.


Now we explain what happen in the picture, it is mainly say one identity, which explain how to look 3-interval exchange map as a composition of rotation map with a renormalization map to glue them. Rotation is a kind of map we have good understanding but we do not understand very well with the renormalization map which is glue the two endpoints of $ I_2,I_3$ which are not the common endpoint of them. Then you get two circle glue like a “8” , and $ T_2$ is just rotate one of it and make the other one to be invariance.

Now we roughly could think about what is the thing we need to charge with, it is just:

$ \sum_{n=0}^{N}f((T_1\circ R\circ T_1)^n(x))\mu(n)=o(N)$.

Now we do some calculate with this geometric explain of interval exchange map.

Let $ A=I_1, B=I_2\cap I_3$, then $ A\cap B=\emptyset, A\cup B=[0,1]$. And $ |A|=\alpha$, $ 0<\beta<|B|$. the rotation $ T_1:x\to x-\alpha$, $ T_2:x\to x+\beta$.

Standard model

Is there a standard model of entropy zero dynamic system?

This problem seems to be too ambitious. But it occur naturally when I an trying to have a global understand of the Sarnak conjecture.