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A fractal set has a Hausdorff dimension. In some cases, we may generate a fractal by iterating $f,$ and let the fractal be the set of starting points $x$ such that $|f^{\circ n}(x)|$ is bounded as $n$ grows. (The julia set and the sierpinski triangle are such sets, if one allows $f$ to be a Hutchinson operator).

We may also have an invariant measure, $\mu,$ that is, $\mu(A) = \mu(f^{-1}(A)).$

The support of $\mu$ is the fractal set.

My question is: is there a way to modify the "dimension" notion to take this invariant measure into account somehow? Some parts of the fractal might be more dense, and thusly should "contribute more" to the dimension.

An idea would be to use box-counting, but instead of just counting if it is occupied or not, one uses the invariant measure on the box instead. Has this been studied?

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There are a wide variety of notions of dimension of a measure. Your basic intuition is completely correct: for a dynamical system, the dimension of a natural invariant measure provides more relevant information than the dimension of the invariant set, since the system may spend more time in certain parts of the space.

For sufficiently homogeneous measures, all reasonable notions of dimension will agree. By ``sufficiently homogeneous'' I mean something very precise: that $$ C^{-1} \, r^s \le \mu(B(x,r)) \le C\, r^s $$ for some constant $C\ge 1$, some $s\ge 0$ and all points $x$ in the support of $\mu$. Of course the dimension in this case is $s$. Such measures are often called Ahlfors-regular, and an example is the natural measure on the middle-thirds Cantor set.

For more general measures, the local dimension is one of the most important concepts and has already been mentioned: $$ \dim(\mu,x)=\lim_{r\to 0}\frac{\log \mu(B(x,r))}{\log r}. $$ But this is really a function of the point $x$ (and not even, as the limit in the definition may not exist, although one can always speak of upper and lower local dimensions).

There are several ways to globalize the information given by the local dimensions. Perhaps the easiest is to take the essential supremum/infimum of the upper/lower local dimensions. This results in four global concepts of dimensions, known as upper/lower packing/Hausdorff dimensions of the measure. They turn out (somewhat surprisingly) to be closely connected to the dimensions of the sets the measure ``sees''. For example, the upper Hausdorff dimension of a probability measure $\mu$ (that is, the essential supremum of the lower local dimensions), is the same as the infimum of the Hausdorff dimension of $A$ over all Borel sets $A$ of full measure.

A finer study is provided by the multifractal spectrum of a measure $\mu$: for each $\alpha$, we form the level set $E_\alpha$ of all points $x$ where $\dim(\mu,x)=\alpha$. Then we try to understand how the size of $E_\alpha$ depends on $\alpha$, for example by studying the function $\alpha\to \dim_H(E_\alpha)$.

There are (many!) other useful concepts of dimension which are not directly related to local dimension. In computing lower bounds for the Hausdorff dimension, the potential method is widely applicable: if a measure $\mu$ satisfies that the energy integral $$ I_s(\mu) = \int \frac{d\mu(x)\, d\mu(y)}{|x-y|^s} $$ is finite, then the support of $\mu$ has Hausdorff dimension at least $s$. So it makes sense to think of $\sup\{s: I_s(\mu)<\infty\}$ as a notion of dimension of $\mu$. This is often called the (lower) correlation dimension, and is one instance of a more general family of dimensions indexed by a real number $q$ (correlation dimension corresponds to $q=2$, and has several alternative definitions, perhaps pointing to its importance).

Yet another notion of dimension has a dynamical underpinning. Given a probability measure $\mu$ say on the unit cube $[0,1]^d$, we may consider the entropy $H_k(\mu)$ of $\mu$ with respect to the partition into dyadic cubes of side length $2^{-k}$. We then define the entropy (also called information) dimension of $\mu$ as $$ \lim_{k\to\infty} \frac{H_k(\mu)}{k\log 2}. $$

This is just a sample of the diverse zoo of dimensions of a measure. Which ones to use depends on the context and what you are able to compute/prove.

Coming back to invariant measures, it is very often the case that the local dimension exists and takes a constant value at almost every point. Such measures are called exact dimensional, and have the property that lower and upper Hausdorff dimension, as well as entropy dimension, are all equal to this almost sure value. (But correlation dimension may be strictly smaller, and the multifractal spectrum may still be very rich; in other words, even though attained on a set of measure zero, other local dimensions may still be relevant).

Proving that measures invariant under certain class of dynamics are exact dimensional may be very challenging. Eckmann and Ruelle conjectured in 1984 that hyperbolic measures ergodic a $C^{1+\delta}$ diffeomorphism are exact dimensional. This was proved by Barreira, Pesin and Schmeling in 1999; the paper appeared in Annals.

For invariant measures, there is often a strong connection between their dimension and other dynamical characteristics (at least generically). The conformal expanding case is the easiest: in this case one has the well-known formula ``dimension=entropy/Lyapunov exponent". The nonconformal situation is much harder, but still a lot of deep research has been done, for example Ledrappier-Young theory.

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Thank you very much! This was a great answer! – Per Alexandersson May 20 '11 at 22:06
Great answer indeed! – John B Dec 21 '15 at 13:13

Hausdorff dimension of a measure is studied, yes.

The mathematical texts should treat this.
Falconer, Fractal Geometry 2nd edition p. 209
Edgar, Integral, Probability, and Fractal Measures p. 123

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The way it's usually done is as follows: $$ \dim_H \mu = \inf \{ \dim_H Z \mid \mu(Z) = 1 \}. $$ You can also study box dimension of measures, but there you take an infimum over all sets $Z$ with $\mu(Z) \geq 1-\epsilon$, and then a limit as $\epsilon \to 0$.

In addition to the books Gerald mentions, you can find a comprehensive discussion of this in Dimension Theory in Dynamical Systems by Yakov Pesin, and a more introductory discussion in Chapter 4 of Lectures on Fractal Geometry and Dynamical Systems by Yakov Pesin and Vaughn Climenhaga.

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There's a very good notion of "local" dimension of a measure at a point $x$: $$ \dim_x(\mu) = \lim_{r\rightarrow 0}\frac{\log\mu(B_r(x))}{\log r}$$

where $B_r(x)$ is the ball of radius $r$ centered at $x$. (Intuitively, we expect that in a $d$-dimensional space, the volume of a ball is proportional to the $d$th power of the radius, which immediately leads to this definition.) In general, the local dimension isn't defined everywhere and depends on $x$ when it is, but under certain conditions, it is constant $\mu$-almost everywhere, in which case it makes sense to call it the dimension of the measure.

In many cases, the measure $\mu$ is more interesting than its support, and the dimension defined thusly will reflect this. For example, consider a stochastic map on the interval $[0,1]$ that maps it affinely onto $[0,1/4]$, $[1/4,3/4]$, or $[3/4,1]$, each with probability 1/3. If $\mu$ is the invariant measure, then the support of $\mu$ is the whole interval, but you can check that $\dim_x(\mu)=\frac{3\log 3}{5\log 2}$ for $\mu$-a. e. $x$ (if I didn't screw it up). You can also verify that there are plenty of exceptional $x$ for which $\dim_x(\mu)$ is something else or undefined.

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Falconer's other book: The geometry of fractal sets contains a somewhat detailed discussion. The main surprising result is that the dimension of measure is smaller than the measure of its support. This follows from what is called the thermodynamic formalism.

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The result that you mention is not surprising since it is in fact false in general. There are many examples for which the dimension of a measure is equal to that of its support. – John B Dec 19 '15 at 16:57

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