I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

## Important Tutorial Announcement

If you can’t make the tutorial on account of a clash please email me with the module code of the module the tutorial is clashing with.

## Question 13 (c) from Tutorial

Solution: We say that $y$ is eventually fixed point of $g$ if some (finite) iterate of $y$, say $g^{N}(x)$ is a fixed point.

Now suppose that $y$ is eventually fixed, at say $x_f=g^N(y)$ so that the orbit of $y$ is

$\text{orb}(y)=\{y,g(y),\dots,g^N(y)=x_f,x_f,x_f,\dots\}$.

Now by part (b) the orbit of $x_f$ under $g^{-1}$ is

$\text{orb}(x_f)=\{x_f,x_f,\dots,y\}$.

However by part (a), $x_f$ is also a fixed point for $g^{-1}$ so it follows that

$\text{orb}(x_f,)=\{x_f,x_f,\dots,x_f\}\Rightarrow x_f=y$,

that is $y$ is a fixed point of $g$ $\bullet$

## Week 3

On Monday we proved two facts about periodic orbits (on the  bottom of p.12 and the top of p.13 in the course notes)

On Wednesday we learnt how to find the period-2 points of a polynomial mapping. Finding periodic points, say period-2 points means finding points $x\in S$ such that if we apply the iterator function twice, then we get back to $x$:

$f(f(x))=f^2(x)=x$.

Solving this equation is not necessarily that easy but we proved that if $f:S\rightarrow S$, then the fixed-point factor-theorem applies: $f(x)-x$ divides into $f^2(x)-x$ and this helps immensely.

We also learnt how to find eventually fixed points.

## Week 4

In Week 4 we will study  attracting fixed points.

## Exercises

I have emailed ye a copy of the exercises and ye should be able to look at questions

• 10, 12(17), 16 – 18, 20 – 22
• 13 is hard
• 14 & 15 were done in Monday’s lecture

As there are a lot of questions it might make sense to allocate so much time and say do (A)s first, then (B)s then (C)s or whatever.

## Test Postponement and Other CA Information

To give ye adequate time to prepare, the test will take place on February 12 in Week 6. Everything up to but not including section 3.4 in the typeset notes is examinable: we will have this covered by Februaray 3 but probably January 29. I have emailed ye a copy of a sample test.

The Concept MCQ will still take place in Week 8. I have decided not to give ye a sample and I might make it a half hour test rather than an hour. The homework will be given to you towards the end of the semester and I will give ye three weeks to do it. It will probably be on complex numbers and won’t be as long as last year’s homework.

You will be given marks for the best two out of Test, Concept MCQ and Homework.

## Math.Stack Exchange

If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly. For example this question about where the OP didn’t understand why roots of $f(x)-x$ are roots of $f^2(x)-x$.