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 on , we say has exact dimension (denoted if for -almost every ,
–as radius approach to , the volume of the ball behaves more and more like , but no further restriction on the rate it approaches .
To illustrate the concept, we look at the following:
Given a measure-preserving system and a partition ,
Notation: For any , denotes the set containing .
i.e. pulling back the partition by , note that is itself a partition of , with elements having same measures as .
Given two petitions , let denote the smallest common refinement of and .
The entropy of a partition is defined as
–Since Log approaches as 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 .
The measure-theoretic entropy of the system w.r.t. is defined as:
Observe that for partitions if each element of is divided by elements of into equal pieces, then and this is as large as it can get.
Hence –as remarked above, . i.e. we have is at most .
*If the sets in 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 , the smaller the entropy gets. When just permutes the s, entropy is, of course, .
Let be ergodic, for almost all ,
I would not try to dig into the proof of this here…
(Note that ergodicity is clearly necessary since we can have fixing and ‘evenly mixes’ , for points in , the limit converges to , for points not in , the limit converges to …I’m not sure if it suffices to require 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.
–Although blows up to infinity as gets finer, but there is a limited amount of growth, i.e. when the partition is fine, it’s harder for to intersect each . Hence one should believe that is (at least normally) finite.
Consider the angle doubling map . Let be an invariant measure, then . 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 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 as defined above (invariant measure of angle doubling map), is it true that any projection of has exact dimension?