arXiv · 2009.04757
This is a survey on Sarnak's Conjecture
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Joanna Kułaga-Przymus, Mariusz Lemańczyk. 2020-09-10. Sarnak's Conjecture from the Ergodic Theory Point of View. https://arxiv.org/abs/2009.04757
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We show that there are integers $N$ and odd $D$ such that no algorithm can decide, from the rational coefficients of a homogeneous polynomial vector field $F$ in dimension $N$ of degree $D$, whether the origin is Lyapunov stable for $\dot Y=F(Y)$. This proves a conjecture of V. I. Arnold. We provide a Lean formalization of the proof. The smallest dimension $N$ for which we were able to prove undecidability is $N = 5$. An analogous undecidability result holds for global asymptotic stability and global exponential stability for non-homogenous vector fields. For dimension $N=2$ and homogenous vector fields, we establish decidability for all commonly used stability notions, building heavily on existing results.
For any $t\in\mathbb{R}$, consider the Affine twist map $\rm{Aff_t}:\mathbb{T}^2\to\mathbb{T}^2$ given by $$\rm{Aff_t}(x,y)=(x+y \text{ mod 1}, y+t \text{ mod 1}).$$ The map $\rm{Aff_t}$ clearly possesses an invariant foliation by horizontal curves. If $t$ is rational, each leaf is periodic, and if $t$ is irrational, the orbit of each leaf is dense in the torus. In both cases, the vertical rotation set of the proper lift of $\rm{Aff_t}$ to the vertical cylinder is reduced to $\{t\}.$ Now, for $k\in\mathbb{R},$ define $$f_{k,t}(x,y):=(x+y+k\sin (2πx)\text{ mod 1},y+k\sin (2πx)+t\text{ mod 1}).$$ We show that: 1) when $t=p/q$ for integers $p$ and $q>0$, KAM theory implies the existence of a constant $k_{p/q}>0$ such that for $|k|<k_{p/q}$, the vertical rotation set of an adequate lift of $f_{k,p/q}$ is just $\{p/q\}.$ 2) when $t$ is irrational, for any $k\neq 0$ the vertical rotation set of the adequate lift of $f_{k,t}$ is a non-degenerate interval which contains $t$ in its interior. In other words, it is not easy to build area-preserving twist maps whose vertical rotation sets are reduced to a single irrational number. From Theorem A of \cite{eujul}, such a map needs to have an invariant foliation by Lipschitz graphs over the horizontal coordinate. In particular, all its iterates must satisfy a twist condition. This is precisely what does not hold for $f_{k,t}$, for all non-zero values of $k$.
We develop a topological framework for Engel expansions that treats both directions of the correspondence between points of $(0,1]$ and nondecreasing digit sequences. We endow the sequence space with the product topology to study the evaluation map, and we fix a nonterminating digit algorithm to study the digit coding map. We also record the correspondence between cylinder sets and fundamental intervals, and give an application to Baire category results for functions of the digits.