Variable Dimensionality Manifolds

Question

I am curious about different notions of dimensionality, particularly as it relates to manifolds.

For instance, there is the standard fixed dimensionality of a pure smooth manifold, the Hausdorff dimension, various different notions of Fractal dimension, and other notions of dimension (e.g. [1], [2]). Some allow for fractional dimensionality; my question is whether this is possible somehow for manifolds (or rather for some manifold-like objects).

One interesting note in the manifold wiki article mentions the notion of manifolds where the dimensionality changes, by e.g. disjoint unioning of a sphere and line. However, the connected components must have the same dimension, I believe.

But is there a notion of generalized manifold with dimensionality that can be "smoothly varying", in some sense?

My question is partly motivated by the idea that one can easily "imagine" such a construct (e.g. a surface that forms a long cone that thins out into a line). Of course, this would not be a manifold (indeed, in computer science, discrete "manifolds", e.g. meshes or point sets, that do this are called "non-manifold"), but perhaps there is a generalized notion that admits this and analysis?

The most obvious issue is local coordinates are obviously integral in nature. But is there no way to parameterize fractals locally (which are of fractional dimension)?

Related questions:

• Can a fractal be a manifold? If so, will its boundary (if it exists) be strictly one dimension lower?
• Can a fractal be a manifold? (Relation to invariant manifolds)

Answer 1

To be honest, I still do not understand what your question is. Here is a couple of concepts which might be related to what you are thinking about. Neither one is specific to manifolds, they are defined for general topological, resp. metric, spaces. In order to avoid pathological examples, I will work with closed subsets $X$ of an $n$-dimensional Euclidean space $E^n$.

Definition 1. The local topological dimension of $X$ at $x$, denoted $dim(X,x)$ is $$ \lim_{r\to 0} dim(\bar{B}(x,r)\cap X), $$ where $dim$ is the Lebesgue covering dimension and $\bar{B}(x,r)$ is the closed ball of radius $r$ centered at $x$.

By the very definition, this dimension takes only integer values.

Definition 2. The local Hausdorff dimension of $X$ at $x$, denoted $Hdim(X,x)$ is $$ \lim_{r\to 0} Hdim(\bar{B}(x,r)\cap X), $$ where $Hdim$ is the Hausdorff dimension.

In view of monotonicity properties of covering dimension and Hausdorff dimension, we have that:

1. $\forall x\in X$, $dim(X,x)\le dim(X)$ and $dim(X,x)$ is upper semicontinuous as a function of $x$.

2. $\forall x\in X$, $Hdim(X,x)\le Hdim(X)$ and $Hdim(X,x)$ is upper semicontinuous as a function of $x$.

Example. $X$ equals the union of the $xy$-plane and the $z$-axis in $R^3$. Then $Hdim(X,p)=dim(X,p)=2$ for $p\in X$ whenever $p=(x,y,0)$ and $Hdim(X,p)=dim(X,p)=1$ for $p\in X$ whenever $p=(0,0,z)$, $z\ne 0$. The same in your example of the union of a 2-dimensional cone and a ray: Neither function will be continuous.

Neither local dimension function is, in general, continuous, as a function of $x$. If $X$ is connected then $dim(X,x)$ is continuous as a function of $x$ if and only if it is constant, equal $dim(X)$. In contrast, there are examples of connected (fractal) $X$ such that $Hdim(X,x)$ is continuous but nonconstant.

As for "discrete manifolds", this is just a wild goose chase: All standard notion of dimension for (nonempty) discrete spaces yield the same answer, namely zero. Nevertheless, I suspect that there is a nontrivial analogue of Hausdorff dimension for discrete subsets of $E^n$, it should be defined similarly to "persistent homology", which is a nontrivial homology theory assigned to discrete metric spaces. Compare this discussion.