I'm quite new to dynamics, and trying to learn some of the basics with an application to my neck of the woods in mind. I have run across the property in the title a few times, often with little comment (so I suspect it's not terribly deep).
The dynamical systems I have in mind admit Markov partitions, which is used to justify this property in one place (along with 'standard techniques for estimating Hausdorff dimension'). I suppose the problem is that I don't really know how to think about arbitrary closed invariant sets in a system coded by a subshift of finite type.
To ask a pointed question:
Why does a Markov partition ensure this property? (And what are these 'standard techniques for estimating Hausdorff dimension'?)
Would like to make this a comment, but this is turning out to be a bit long...
Are you asking for sufficient conditions? I don't know if those are easy to formulate. In general this is the case for hyperbolic systems, albeit one can construct hyperbolic systems with a nested sequence of proper closed invariant subsets whose Hausdorff dimension tends to the dimension of the phase space, which is why I don't know whether sufficient conditions would be easy to formulate. See here for hyperbolic sets. For examples of aforementioned nested sets, see, for example, Doarte's paper, no. 20 here, and the paper by Gorodetski, no. 18 here, and the paper here -- warning: the last is a shameless self promotion. Of course, one could probably come up with much easier examples, but the above examples are natural in the sense that they arise in highly nontrivial and widely investigated systems.