# On Lyapunov numbers

I'm working on a paper , I almost understood the whole paper , I just didn't understand some examples of that. I try to explain a little about some definitions.

$$(X,f)$$ is a dynamical system , where $$X$$ is a compact metric space with metric $$d$$ and $$f : X \rightarrow X$$ is a continuous map.

I try to define some Lyapunov numbers :

The first Lyapunov number is defined the way like this :
$$L_{r} := \sup \lbrace \varepsilon : \forall x \in X , \forall U_{x} , \exists y \in U_{x} , \exists n \in \mathbb{Z}_{+} \text{such that}: d(f^{n}(x),f^{n}(y)) > \varepsilon \rbrace$$
The third Lyapunov number is defined the way like this :
$$\overline{L_r} := \sup \lbrace \varepsilon : \forall x \in X , \forall U_{x} \quad \exists y \in U_{x} \quad \text{such that}: \limsup_{n \to \infty} d(f^{n}(x) , f^{n}(y) ) > \varepsilon \rbrace$$

And the orbit of a map is defined :

$$\lbrace f^{n}(x) \quad : n\geq 0 \rbrace$$

Now some examples were solved in the paper , but I can not understand how they were solved:

Consider the following map :
$$g : [0,1] \rightarrow [0,1]$$ and $$g(x) = 3((x-\frac{1}{3}) - |x-\frac{1}{3}| + |x-\frac{2}{3}|)$$

In the paper , it is said that $$\frac{1}{2}$$ is a fixed point of the map , so clearly that $$L_{r} =\frac{1}{2}$$.

I can't understand how the $$L_{r}$$ was computed.
This is the link of the paper : On the Lyapunov numbers

• I'm kind of lost in the definitions: why did you use $\ni$? – Simone Ramello Jul 18 at 8:07
• That was my mistake , I edited. – Reza Jul 18 at 8:31

Since $$x_0=1/2$$ is a fixed point then
$$d(f^n(x_0), f^n(y))=d(x_0, f^n(y))=d(1/2, f^n(y))\leq 1/2$$
since we are inside $$[0,1]$$. Therefore $$L_r\leq 1/2$$.
On the other hand the crucial thing you missed from the paper is that $$g$$ is topologically transitive. And so in this scenario there is $$y_0$$ such that the orbit of $$y_0$$ is dense. Therefore $$d(1/2, f^n(y_0))$$ is arbitrarly close (from below) to $$1/2$$. Meaning $$d(f^n(x_0), f^n(y_0))$$ is arbitrarly close to $$1/2$$ and therefore $$L_r\geq 1/2$$.