Sarnak's conjecture (2010) states that any deterministic (i.e. zero entropy)
dynamical system $(X, T)$ is linearly disjoint from Möbius function
$\mu$, that is, $$\lim_{N\to \infty}\frac{1}{N}\sum_{n=1}^N
\mu(n)f(T^nx)=0,$$ for any continuous function $f$ and any point $x\in X$.
Even though Sarnak conjecture has been proven for various dynamical systems,
it is still widely open in general. Tanja Eisner (2015) proposed a variant
of Sarnak's conjecture, called polynomial Sarnak's conjecture for minimal
systems, which assets that for any minimal deterministic dynamical system
$(X,T)$, one has that $$\lim_{N\rightarrow \infty}\frac{1}{N}
\sum_{n=1}^N\mu(n)f(T^{p(n)}x)=0,$$ for any continuous function $f$,
any polynomial $p:\mathbb{N}\to \mathbb{N}_0$ and any point $x\in X$.
Polynomial Sarnak's conjecture for minimal systems implies Sarnak's
conjecture. In this talk, I will disprove polynomial Sarnak's conjecture
for minimal systems by showing a counter-example. This is a joint work
with Zhengxing Lian.