“Exact dimension” of measures

September 29, 2010

Mike throw this question to me today which I have a huge lack of background knowledge to understand >.< … Since think there are some cool stuff there which I’m unable to find much reference, to prevent me from forgetting anything, I guess it’s a good idea to record and organize the discussion here:

Definition: Given measure \mu on X, we say \mu has exact dimension \alpha (denoted \mbox{Edim} (\mu) = \alpha if for \mu-almost every x,

\displaystyle \lim_{r \rightarrow 0} \frac{\log(\mu(B(x,r)))}{\log(r)} = \alpha

–as radius approach to 0, the volume of the ball behaves more and more like r^\alpha, but no further restriction on the rate it approaches r^\alpha.

To illustrate the concept, we look at the following:

Given a measure-preserving system (X, \mathcal{B}, \mu, T) and a partition \mathcal{P} = \{P_1, P_2, \cdots, P_n \},

Notation: For any x \in X, \mathcal{P}(x) denotes the set P_i \in \mathcal{P} containing x.

T^{-i}(\mathcal{P}) = \{ T^{-i}(P_j) \ | \ 1 \leq j \leq n \} i.e. pulling back the partition by T^{-i}, note that T^{-i}(\mathcal{P}) is itself a partition of X, with elements having same measures as \mathcal{P}.

Given two petitions \mathcal{P}, \ \mathcal{P}', let \mathcal{P} \vee \mathcal{P}' denote the smallest common refinement of \mathcal{P} and \mathcal{P}'.

The entropy of a partition \mathcal{P} is defined as

H(\mathcal{P}) = - \displaystyle \sum_{i=1}^n \mu(P_i) \log(\mu(P_i))

–Since Log approaches -\infty as \mu(P_i) gets small, as we would expect, the entropy measures how ‘fine’ the partition is. The smaller the parts gets, the larger the entropy; To get a sense of the rate of growth as partition gets finer: if one bisects each set in the partition to form a new partition, the entropy would increase by \log(2).

The measure-theoretic entropy of the system w.r.t. \mathcal{P} is defined as:

h(T, \mathcal{P}) = \displaystyle \lim_{n \rightarrow \infty} H( \bigvee_{i=1}^n T^{-i} (\mathcal{P}))

Observe that for partitions \mathcal{P}, \ \mathcal{P}' if each element of \mathcal{P}' is divided by elements of \mathcal{P} into equal pieces, then H(\mathcal{P} \vee \mathcal{P}') = H(\mathcal{P}) + H(\mathcal{P}') and this is as large as it can get.

Hence H( \bigvee_{i=1}^n T^{-i} (\mathcal{P})) \leq n \times H(\mathcal{P}) –as remarked above, H(T^{-1}(\mathcal{P}) = H(\mathcal{P}). i.e. we have h(T, \mathcal{P}) is at most H(\mathcal{P}).

*If the sets in \mathcal{P} intersects as much as they can and divides all parts uniformly smaller, then the entropy will equal to the entropy of the partition, in other cases, the more they ‘coincide’ under the action by T^{-1}, the smaller the entropy gets. When T just permutes the P_is, entropy is, of course, 0.

Shannon-McMillan-Breiman theorem:

Let T be ergodic, for \mu almost all x \in X,

\displaystyle \lim_{n \rightarrow \infty} - \log (\mu( \vee_{i=1}^n T^{-i}\mathcal{P}(x)))/n = h(T, \mathcal{P})

I would not try to dig into the proof of this here…

(Note that ergodicity is clearly necessary since we can have T fixing P_1 and ‘evenly mixes’ P_2, \cdots, P_n, for points in P_1, the limit converges to 0, for points not in P_1, the limit converges to \displaystyle \sum_{i=2}^n \mu(P_i) \log(\mu(P_i))…I’m not sure if it suffices to require T to be ergodic)

The measure-theoretic entropy of a system is defined as the sup over all partitions of the entropy w.r.t. the partition. i.e.

\displaystyle h(T) = \sup_{\mathcal{P}} h(T, \mathcal{P})

–Although H(\mathcal{P}) blows up to infinity as \mathcal{P} gets finer, but there is a limited amount of growth, i.e. when the partition is fine, it’s harder for T^{-1}(P_i) to intersect each P_j. Hence one should believe that H(T) is (at least normally) finite.

Consider the angle doubling map f(x) = 2x ( \ \mbox{mod} \ 1). Let \mu be an invariant measure, then \mbox{Edim}(\mu) = h(\mu, T) / \log(2). i.e. exact dimension exists for each such measure.

This fact can be seen by considering the space of binary sequences, our measure assigns a weight to each initial sequence, take sequence of refining partitions to be just the all initial sequences of length n to obtain the entropy.

Question:1. Find an example of two measures with exact dimension where a projection of the product does not have exact dimension.

2. What about the projection of the product of the same measure?

3. For measure \mu as defined above (invariant measure of angle doubling map), is it true that any projection of \mu \times \mu has exact dimension?

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