Definition
Mean dimension is a topological invariant of dynamical systems that quantifies the average number of parameters per unit time required to describe the orbits of the system. Formally, for a continuous action of the group ℤ (or ℝ) on a compact metric space $X$, the mean dimension $\operatorname{mdim}(X,T)$ is defined as the limit, if it exists, of the normalized covering dimension of increasingly fine iterates of the system.
Overview
Introduced by Mikhail Gromov in 1999 and further developed by Elon Lindenstrauss and Benjamin Weiss, mean dimension extends the classical notion of topological dimension to infinite‑dimensional dynamical contexts where traditional invariants such as topological entropy may be zero. It is particularly useful for studying systems with infinite entropy, symbolic extensions, and embedding problems. Mean dimension can be interpreted as the “average” dimension of the space when observed over long time intervals.
Etymology/Origin
The term combines “mean,” indicating an average or typical value, with “dimension,” referring to the covering (Lebesgue) dimension of topological spaces. The concept emerged from Gromov’s work on metric geometry and dynamical systems, where he sought a quantitative measure of complexity beyond entropy.
Characteristics
| Property | Description |
|---|---|
| Additivity | For product systems $(X_1\times X_2, T_1\times T_2)$, $\operatorname{mdim}(X_1\times X_2)=\operatorname{mdim}(X_1)+\operatorname{mdim}(X_2)$. |
| Monotonicity | If $Y$ is a closed invariant subset of $X$, then $\operatorname{mdim}(Y)\le \operatorname{mdim}(X)$. |
| Invariance under Conjugacy | Conjugate dynamical systems have identical mean dimensions. |
| Relation to Entropy | Systems with positive topological entropy have mean dimension at least zero; many zero‑entropy systems can still have positive mean dimension. |
| Computation | Often obtained via open covers, metric entropy analogues, or embedding theorems (e.g., into Hilbert cubes). Exact values are known for shift spaces, toral automorphisms, and certain smooth actions. |
| Zero Mean Dimension | Systems with finite topological dimension or expansive actions typically have $\operatorname{mdim}=0$. |
Related Topics
- Topological dimension – the classical covering dimension of a topological space.
- Topological entropy – a measure of orbit complexity; complementary to mean dimension.
- Metric mean dimension – a variant defined using metrics rather than open covers.
- Symbolic dynamics – shift spaces where mean dimension distinguishes infinite‑alphabet shifts.
- Embedding theorems – results (e.g., Lindenstrauss–Weiss) concerning embedding dynamical systems into shift spaces of prescribed dimension.
- Dynamical systems – the broader field encompassing continuous actions on compact spaces.