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Consider $\mathcal{T}=S^{1}\times D^{2}$, where $S^{1}=[0,1]\mod 1$ and $D^{2}=\{(x,y)\in\mathbb{R}^{2}|x^{2}+y^{2}\le 1\}$. Fix $\lambda\in(0,\frac{1}{2})$ and define the map $F:\mathcal{T}\to\mathcal{T}$ by $$F(\phi,x,y)=\left(2\phi,\lambda x+\frac{1}{2}\cos 2\pi\phi,\lambda y+\frac{1}{2}\sin 2\pi\phi\right)$$

Then define the Lyapunov exponent $\chi(x,v)$ by $$\chi(x,v)=\overline{\lim_{n\to\infty}}\frac{1}{n}\log\|df^{n}(x)v\|$$.

So I wrtie $$\chi(\phi,x,y,v)=\overline{\lim_{n\to\infty}}\frac{1}{n}\log\|dF^{n}(\phi,x,y)v\|$$

The problem is that I calculate $\frac{\partial}{\partial y\partial x\partial\phi}F(\phi,x,y)=0$. I think that's the wrong way of going about it anyway.

I also have that $F^{n}(\mathcal{T})\cap\{\phi=constant\}$ consists of $2^{n}$ disks of radius $\lambda^{n}$, but I need help with calculating $dF^{n}(\mathcal{T})$.

Alp Uzman
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Jason Born
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2 Answers2

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Your working definition for a Lyapunov exponent is the (forward) Lyapunov exponent of the derivative cocycle of $F$:

Since $F:\mathcal{T}\to\mathcal{T}$ is a diffeomorphism, we have the differential $dF:T\mathcal{T}\to T\mathcal{T}$, and we can consider its iterations. If we fix an initial point $p:=(\phi,x,y)\in\mathcal{T}$,

$$T_p\mathcal{T}\stackrel{d_p F}{\to} T_{F(p)}\mathcal{T}\stackrel{d_{F(p)}F}{\to}T_{F^2(p)}\mathcal{T}\stackrel{d_{F^2(p)}F}{\to}T_{F^3(p)}\mathcal{T}\stackrel{d_{F^3(p)}F}{\to}\cdots\stackrel{d_{F^{n-1}(p)}F}{\to}T_{F^{n}(p)}\mathcal{T}\stackrel{d_{F^n(p)}F}{\to}\cdots,$$

and by the chain rule the composition of the first $n$ differentials is $d_p(F^n)$. Hence the forward Lyapunov exponent is

\begin{align} \chi^+:\;&T\mathcal{T}\to[-\infty,\infty]\\ &(p,v_p)\mapsto \limsup_{n\to\infty}\dfrac{\log \Vert d_p(F^n)v_p\Vert}{n}. \end{align}

So all you need is to compute the Jacobian of $F$ at an arbitrary point $p$.

By the way, this map is a quite popular example in dynamics, e.g., see the cover of Hasselblatt & Katok's A First Course in Dynamics: with a Panorama of Recent Developments or Brin & Stuck's Introduction to Dynamical Systems.

Alp Uzman
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How did you get derivative of the third order? You have only to compute all the first order derivatives, make a matrix out of them and compute its norm. Nothing more.

demitau
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