The embedding problem in topological dynamics and Takens' theorem

Abstract: We prove that every $\mathbb{Z}^{k}$-action $(X,\mathbb{Z}^{k},T)$ of mean dimension less than $D/2$ admitting a factor $(Y,\mathbb{Z}^{k},S)$ of Rokhlin dimension not greater than $L$ embeds in $(([0,1]^{(L+1)D})^{\mathbb{Z}^{k}}\times Y,\sigma\times S)$, where $D\in\mathbb{N}$, $L\in\mathbb{N}\cup{0}$ and $\sigma$ is the shift on the Hilbert cube $([0,1]^{(L+1)D})^{\mathbb{Z}^{k}}$; in particular, when $(Y,\mathbb{Z}^{k},S)$ is an irrational $\mathbb{Z}^{k}$-rotation on the $k$-torus, $(X,\mathbb{Z}^{k},T)$ embeds in $(([0,1]^{2^kD+1})^{\mathbb{Z}^k},\sigma)$, which is compared to a previous result by the first named author, Lindenstrauss and Tsukamoto. Moreover, we give a complete and detailed proof of Takens' embedding theorem with a continuous observable for $\mathbb{Z}$-actions and deduce the analogous result for $\mathbb{Z}^{k}$-actions. Lastly, we show that the Lindenstrauss--Tsukamoto conjecture for $\mathbb{Z}$-actions holds generically, discuss an analogous conjecture for $\mathbb{Z}^{k}$-actions appearing in a forthcoming paper by the first two authors and Tsukamoto and verify it for $\mathbb{Z}^{k}$-actions on finite dimensional spaces.

• The paper demonstrates that any Z^k-action with mean dimension less than m/2 and bounded Rokhlin dimension can be embedded in a Hilbert cube shift.
• It employs a dense Gδ set of functions and advanced combinatorial techniques to extend Takens' theorem to higher rank systems.
• The results refine embedding criteria for complex dynamical systems, offering practical insights for both theoretical analysis and real-world applications.

The Embedding Problem in Topological Dynamics and Takens' Theorem

Introduction

The paper investigates the intersection of topological dynamics, mean dimension theory, and Takens' theorem. The core focus is on embedding problems for topological dynamical systems, specifically addressing actions of higher rank abelian groups like $\mathbb{Z}^k$. The authors explore the conditions under which a given system can be embedded into a universal cubical shift, extending previous work in the area and providing new results on Rokhlin dimension and mean dimension.

Embedding in Topological Dynamics

The embedding problem described in the paper seeks to determine when a system $(X, \mathbb{Z}^k, T)$ can be embedded into a Hilbert cube shift, denoted $(([0,1]^{d})^{\mathbb{Z}^{k}}, \sigma)$, where $\sigma$ represents a shift operation. Here, the complexity of these systems is captured through the concepts of mean dimension and Rokhlin dimension. A fundamental result demonstrated is that any $\mathbb{Z}^k$-action with mean dimension less than $m/2$ and a factor system of bounded Rokhlin dimension will embed in a suitably chosen high-dimensional cubical shift.

Framework of Takens' Theorem

The paper provides a detailed proof of Takens' theorem's applicability to $\mathbb{Z}^k$-actions, involving the reconstruction of dynamical systems from time series data. The goal is to ensure that the observable can create an embedding into a higher-dimensional space using delay coordinate mappings, thus reconstructing the attractor of the dynamics. This general framework allows for embedding even when systems exhibit complex, non-linear behavior and only partial observations are available.

Rokhlin Dimension and Mean Dimension in Embeddings

Rokhlin dimension serves as a critical tool in proving embeddability results, particularly because its definition accommodates the complexities of $\mathbb{Z}^k$-actions. The authors use this concept to improve previous embedding results. For systems admitting factors with irrational rotations (irrational $\mathbb{Z}^{k}$-rotations), the paper significantly reduces the dimensionality necessary for successful embedding, compared to previously known results in the context of aperiodic systems.

Detailed Proof Strategy

In proving embeddability, especially for finite dimensional spaces, the methodology involves crafting a dense $G_\delta$ set of functions such that embeddings can be found within this space. This is accomplished through various combinatorial and topological arguments, leveraging mean dimension calculations and properties of continuous functions over compact spaces.

Conclusion

The results of this paper provide a deeper understanding of embeddings in topological dynamics, pushing the boundaries of when and how complex systems can be embedded into simpler, universal models. The interplay between mean dimension, Rokhlin dimension, and observable embeddings offers fertile ground for further exploration, particularly in extending the theoretical results to practical, real-world systems modeled by higher rank abelian actions. Such insights pave the way for future research to refine these embedding techniques and explore their implications across different domains of dynamical systems theory.