# Dimension topologique, moyenne dimension et th\'eor\`emes de plongement

## TL;DR

This paper discusses conditions under which topological dynamical systems can be embedded into high-dimensional cubical shifts, focusing on mean and periodic dimensions and providing new embedding criteria.

## Contribution

It establishes new embedding theorems linking mean and periodic dimensions to the ability to embed systems into cubical shifts.

## Key findings

- Embedding is possible if mean dimension and periodic dimension are less than d/2.
- For product systems, embedding depends on the lim inf of periodic dimensions.
- Provides conditions for embedding based on dimension constraints.

## Abstract

According to a conjecture of Lindenstrauss and Tsukamoto, a topological system $(X,T)$ embeds in the $d$-dimensional cubical shift $(([0,1]^d)^\mathbb{Z},$shift) if its mean dimension and periodic dimension verify mdim$(X,T)<d/2$ and perdim$(X,T)<d/2$. If $(X,T)=(\prod\limits_{i\in \mathbb{N}}X_i,\prod\limits_{i\in \mathbb{N}}T_i)$ ($(X_i, T_i)$ dynamical systems), and $\liminf\limits_{n\to +\infty} {\rm perdim}(X^{(n)},T^{(n))}) < \frac{d}{2}$, then $(X,T)$ embeds in $(([0,1]^d)^\mathbb{Z},$shift).