There is one more style of notation I'd like to introduce.
I won't use it that often in this course, but it's quite standard; it's likely you'll see it elsewhere,
so, I think I should at least cover it briefly.
OK. So, again we have an iterated function, which is just a loop.
We input x, apply f to it, get an output and then we repeat.
So x1, the first value in the itinerary, is just f applied to my seed or initial condition.
To get x2, I apply f to x1. To get the next value in the orbit, the next value in the itinerary, I just apply the function to the current value.
Here is another way to think about this. To get x2, I start with the seed and I do f to it twice.
So I start with x0, do the function to it once, get x1. Twice, I get x2.
We can write that in a formula as follows.
So this notation means to get x2, I start with my seed x0, apply f to it and apply f to it again.
In other words, x2 is f applied twice to x0.
Similarly, x3, the third iterate, I obtain by starting with the seed and applying f to it 3 times. I can write that as follows.
So, x3 is x0 after f acts on it again and again. So this is x0 to which f has been applied three times.
Writing all these fs in a row is a little bit cumbersome. So, there is a shorthand notation for this.
So this, f with a three to the parenthesis is understood to mean f done three times. So this is x0 after f has acted on it three times.
And in general, this notation, f with an n up in superscript and parenthesis means f applied n times to x.
It's doing f n times in a row to whatever the input was.