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. ## Review Tutorial/Lecture

Wednesday 1 May at 10:00 in WGB G014.

## Feedback

Thank you for your feedback today.

I would make the following comments:

The Homework is hard but fair: ye are final year students. Also I have repeatedly said that I am willing to answer questions about it.

It has everything to do with what we are doing in class (iterator functions, fixed points, orbits, etc.) and ye are supposed to know about the other topics from other modules.

I agree that it might require a lot of thought for 12.5% but when you are finished with it I have no doubt whatsoever that your understanding of the material can only be increased.

More exercise sheets? Agreed — although I didn’t see much evidence of us doing the too-few questions that I was posing weekly.

Regarding getting your tests back: ye have an option to view them but I need to keep them I’m afraid.

## Tutorials

Summer 2012: Question 4 (e),  (f)

Autumn 2012: Question 4 (d), (f)

## 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 on the Tent Mapping.

## Week 12

We summarised our work on roots of unity and illustrated the following result… parts 4 & 5 are not examinable so don’t worry that it wasn’t covered in lectures. In fact as it turned out the proof was a lot harder than I thought… it is not true for all $\alpha$ — only for almost all irrational $\alpha$ when $f(z)=z^2$ as far as I know ! The following proposition is true though.

### Proposition

Suppose that $f:\mathbb{C}\rightarrow\mathbb{Z}$ is a power mapping $f(z)=z^n$ for some $n\geq 2$.

Then the dynamical system $(\mathbb{C},d)$ exhibits the following behaviours:

1. If $z_0\in\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$ then $z_n\rightarrow 0$.
2. If $z_0\in\mathbb{C}\backslash\bar{\mathbb{D}}:=\{z\in\mathbb{Z}:|z|>1\}$ then $z_n\rightarrow\infty$
3. If $z_0=e^{q\,2\pi i}$ with $q\in\mathbb{Q}$ then $z_0$ is eventually periodic.

Suppose now that $f(z)=z^2$. Then

4.    I $z_0=e^{\alpha\,2\pi i}$ with $\alpha\in\mathbb{R}\backslash\mathbb{Q}$ for, in binary, $\alpha=0.0101100011000001010100110101011111\dots_2$

Then $z_0$ has a dense orbit.

(In fact, $(\mathbb{T},z^2)$ is a chaotic mapping.)

5.     Suppose now that $n>2$  and we are again looking at $f(z)=z^n$. Then there exists an $\beta\in\mathbb{R}\backslash\mathbb{Q}$ such that the orbit of $z_0=e^{\beta\cdot 2\pi i}$ is not dense in $\mathbb{T}$.

Proof: The proof of statement 4. is essentially the same as that of that of the doubling mapping (the bottom of p.55 top of p.56 in the notes).

The proof of 5. goes as follows:

Let $n>2$ and let $O_n$ be the (proper) subset of $[0,1]$ of irrational numbers containing only $0$s and $1$s in their base- $n$ expansion. If we take $\beta\in O_n$ we then have that $\{n^N\beta\}\in O_n\text{ for all }N\in\mathbb{N}$

and so $S$ can not be dense in $[0,1]$ when $n>2$ $\bullet$