For Question 1, we want to consider the following figure
For part a, in the capacity dimension calculation for the set in the above figure, what is epsilon?
Epsilon, in the capacity dimension calculation, is the side length of each box
As labeled here, each box is 0.25 wide, so the answer to this is 0.25
For part b, we want to know what N(epsilon) is
Recall that N(epsilon), in this calculation, is the number of boxes of size epsilon needed to cover the set
Notice here, if you count the blue boxes, you need 12 boxes to cover the Julia set, so the answer is 12
For the remainder of the questions, we値l be considering this plot
This is a log log plot of 1/epsilon versus X(epsilon)
For Question 2, we want to consider the power law X(epsilon) scales like 1/epsilon to the nu
We know that this power law holds in the scaling region of this curve
Using this information, how would we then approximate nu, given this curve?
We can approximate nu, given this power law and this plot, using the slope of the line fitted to the curve in region B
For Question 3, considering the same figure, on this type of plot region B is called a scaling region
For Question 4, we want to delve into what is causing the three different regions in this figure
Region B is a scaling region, and the shape of this part of the curve is due to the power law relationship
This is the actual region we care about in this plot
Regions A and C are numerical side effects of the algorithm
Region A is being caused by epsilon being too large
In this case, the entire set is being covered by a single epsilon ball
In region C, on the other hand, epsilon is too small
And in this case each point is being covered by a single epsilon ball
Regions A and C are numerical artifacts of the algorithm, and are not important to this calculation
Region B, the so-called scaling region of this power law relationship, is usually what you care about when dealing with this kind of plot