There has been "some" debate on the notion of fractal (as an illustration, see for example the discussion in this link). One of the possible notions includes relating Hausdorff dimension and packing dimension. While I really don't care about what could be any reasonable definition of fractal, I find it peculiar that people might have introduced such a notion without apparently understanding whether packing measure and packing dimension occur "naturally" say in dynamical systems.

To my best understanding, packing dimension occurs naturally on the boundary of some Kleinian groups, but I know no other "natural" occurrence. So my question is:

Are there any other "natural" occurrences of packing dimension and/or packing "measure"? My personal preference would be in dynamical systems but it can be in other fields.

Quoting from here: "If not, then why should we have a definition involving such a notion? A notion should be somewhat motivated, say in the context of dynamics or of some other area. Let me note that Hausdorff measure and Hausdorff dimension seem to occur "naturally" much more."

Added December 21: Let me point out that Patterson-Sullivan measures should relate to the question, and the same goes for the study of the behavior at the critical exponent.


1 Answer 1


I tend to disagree with your thesis that Hausdorff measure and dimension is "more natural" than packing measure and dimension. It is true that Hausdorff dimension is far more widely used, but I attribute this to history (it was defined some 60 years earlier) and the fact that both notions often agree so people often default to the Hausdorff case.

Both notions of measure/dimension are dual in many ways, so often it makes sense to consider them together:

  1. Hausdorff dimension is defined in terms of optimal coverings, while packing dimension is defined in terms of optimal packings.
  2. When using measures to estimate dimension (as is often the case), Hausdorff dimension corresponds to small mass at infinitely many scales, while packing dimension corresponds to small mass at all sufficiently small scales. For example, if $\mu(A)>0$ and $$\liminf_{r\to 0} \mu(B(x,r))r^{-s} < +\infty,$$ then $\dim_H(A)\ge s$, while if the same holds with $\limsup$, then $\dim_P(A)\ge s$.
  3. There is no universal formula for the Hausdorff (or packing!) dimension of a cartesian product, but there are optimal inequalities involving both notions: $$ \dim_H(A)+\dim_H(B)\le \dim_H(A\times B)\le \dim_H(A)+\dim_P(B)\le \dim_P(A\times B) $$

Moreover, there are many instances where packing measure/dimension is the "correct" one. Off the top of my head, I give three rather different examples:

  • A dynamical example: there are many self-similar sets for which the Hausdorff measure in its dimension is zero, but for which the packing measure is positive, thus the natural measure to consider on self-similar sets is packing, not Hausdorff, measure. See [Peres, Yuval; Simon, Károly; Solomyak, Boris. Self-similar sets of zero Hausdorff measure and positive packing measure. Israel J. Math. 117 (2000), 353--379].
  • A probabilistic example: let $B$ be one-dimensional Brownian motion. An $a$-fast time of $B$ is a point $t$ such that $$ \limsup_{h\downarrow 0} \frac{|B(t + h) − B(t)|}{2 h \log(1/h)} \ge a $$ What sets $E$ contain $a$-fast points? The answer depends precisely on the packing dimension of $E$. See [Khoshnevisan, Davar; Peres, Yuval; Xiao, Yimin. Limsup random fractals. Electron. J. Probab. 5 (2000), no. 5, 24 pp.].
  • A geometric example Let $E$ be a set which contains the boundary of an axes-parallel square with center in every point of the plane. With "circle" instead of "square boundary", it was proved by Marstrand and Bourgain that such a set must have positive Lebesgue measure. But with squares the situation is very different: $E$ may have Hausdorff dimension $1$ (same as a single square boundary!) This pathological behavior is not hard to see, but it suggests Hausdorff dimension is the "wrong" notion for this problem. Indeed, $E$ must have packing dimension at least $7/4$, and this is sharp. See [Nagy, Dániel; Keleti, Tamás and Shmerkin, Pablo. Squares and their centers. http://arxiv.org/abs/1408.1029].
  • 1
    $\begingroup$ Thank you very much for your reply. I knew the two first works. But let me insist on the "dynamical systems" aspect, thus reformulating a bit: Are there any other "natural" occurrences of packing dimension and/or packing "measure" in the context of dynamical systems? I tend to disagree with you that having positive packing measure is a good argument; even the Patterson-Sullivan procedure goes precisely by avoiding this problem, somehow in a blunt manner, but quite to the point, and so I also don't like my own example of Kleinian groups. By the way, I'm quite fond of your work. $\endgroup$
    – John B
    Dec 21, 2015 at 21:57
  • $\begingroup$ For dynamical objects Hausdorff and packing dimensions often (though not always) agree. Although I don't like the definition of fractal as the equality of Hausdorff and packing dimension, this equality does say something about the geometry of the object - very roughly, that its size is the same at all small scales. So I would say that throughout dynamics packing dimension occurs naturally! $\endgroup$ Dec 22, 2015 at 11:31

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