DynamiqueDirectionnelle < CellularAutomata

CellularAutomata

CellularAutomata.DynamiqueDirectionneller1.1 - 10 Oct 2006 - 19:34 - MathieuSabliktopic end

Dynamique directionnelle

Definitions:

Let $x\in \mathcal{A}^{\mathbb{Z}}$ , let $\alpha\in\mathbb{R}$ , let $\mathbb{K}=\mathbb{N}$ or $\mathbb{Z}$ and let $\epsilon >0$ $B_{\Sigma}(x,\epsilon)=\{y\in\Sigma : d_C(x,y)<\epsilon\},$

$E^{\alpha}_{\Sigma}(x,\epsilon,\mathbb{K})=\{y\in\Sigma : d_C(\sigma^{\lfloor n\alpha\rfloor}\circ F^n(x),\sigma^{\lfloor n\alpha\rfloor}\circ F^n(y))<\epsilon, \forall n\in\mathbb{K}})\}.$

The set $Eq^{\alpha}_{\mathbb{K}}(\Sigma,F)$ of $\mathbb{K}$ -equicontinuous points of slope $\alpha$ is defined by

$x\in Eq^{\alpha}_{\mathbb{K}}(\Sigma,F) \Longleftrightarrow \forall \epsilon>0, \exists\delta >0, B_{\Sigma}(x,\delta)\subset E^{\alpha}_{\Sigma}(x,\epsilon,\mathbb{K}).$

$(\Sigma,F)$ is $\mathbb{K}$ -equicontinuous of slope $\alpha$ if

$\forall \epsilon>0, \exists \delta>0, \forall x\in\Sigma, \ B_{\Sigma}(x,\delta)\subset E^{\alpha}_{\Sigma}(x,\epsilon,\mathbb{K}).$

$(\Sigma,F)$ is $\mathbb{K}$ -sensitive of slope $\alpha$ if

$\exists \epsilon>0, \forall \delta>0, \forall x\in\Sigma , \ \exists y\in B_{\Sigma}(x,\delta)\setminus E^{\alpha}_{\Sigma}(x,\epsilon,\mathbb{K}).$

$(\Sigma,F)$ is $\mathbb{K}$ -expansive of slope $\alpha$ if

$\exists \epsilon>0, \forall x\in\Sigma, \ E^{\alpha}_{\Sigma}(x,\epsilon,\mathbb{K})=\{x\}.$

Since the CA are defined on the lattice $\mathbb{Z}$, the information can come from the right or the left. So it is possible to break up the concept of expansivity into right-expansivity and left-expansivity:

$(\Sigma,F)$ is $\mathbb{K}$ -right-expansive of slope $\alpha$ if there exists $\epsilon >0$

$E_{\Sigma}^{\alpha}(x,\epsilon,\mathbb{K})\cap E_{\Sigma}^{\alpha}(y,\epsilon,\mathbb{K})=\emptyset$ for all $x,y\in\Sigma$ such that $x_{[0,+\infty]}\ne y_{[0,+\infty]}$ .

$(\Sigma,F)$ is $\mathbb{K}$ -left-expansive of slope $\alpha$ if there exists $\epsilon>0$

$E_{\Sigma}^{\alpha}(x,\epsilon,\mathbb{K})\cap E_{\Sigma}^{\alpha}(y,\epsilon,\mathbb{K})=\emptyset$ for all $x,y\in\Sigma$ such that $x_{[-\infty,0]}\ne y_{[-\infty,0]}$ .

Thus the CA $(\Sigma,F)$ is $\mathbb{K}$ -expansive of slope $\alpha$ if it is $\mathbb{K}$ -left-expansive and $\mathbb{K}$ -right-expansive of slope $\alpha$ .

Theorem:

Let $(\mathcal{A}^{\mathbb{Z}},F)$ be a CA, let $\Sigma\subset\mathcal{A}^{\mathbb{Z}}$ be a transitive subshift, let $\mathbb{K}=\mathbb{N}$ or $\mathbb{Z}$ and let $\alpha\in\mathbb{R}$ . We are in one of the following case:

$Eq^{\alpha}_{\mathbb{K}}(\Sigma,F)=\Sigma$ $\Longleftrightarrow$ $(\Sigma,F)$ is $\mathbb{K}$ -equicontinuous of slope $\alpha$ ;

$\emptyset\ne Eq^{\alpha}_{\mathbb{K}}(\Sigma,F)\ne\Sigma$ $\Longleftrightarrow$ $(\Sigma,F)$ is not $\mathbb{K}$ -sensitive of slope $\alpha$ $\Longleftrightarrow$ $(\Sigma,F)$ has a $\mathbb{K}$ -blocking word of slope $\alpha$ ;

$(\Sigma,F)$ is $\mathbb{K}$ -sensitive of slope $\alpha$ but is not $\mathbb{K}$ -expansive of slope $\alpha$ ;

$(\Sigma,F)$ is $\mathbb{K}$ -expansive of slope $\alpha$ .

Theorem (Equicontinuous directions):

Let $(\mathcal{A}^{\mathbb{Z}},F)$ be a CA of neighborhood $\mathbb{U}=[r,s]$ and let $\Sigma\subset\mathcal{A}^{\mathbb{Z}}$ be a strongly mixing subshift. We are in one of the following cases:

$\mathbf{A}=\mathbb{R}$ , which is equivalent to $(\Sigma,F)$ is nilpotent;

Exemple: AcNilpotent^?

there exist $\alpha',\alpha''\in [-s,-r]$ , with $\alpha'<\alpha''$ , such that

$]\alpha',\alpha''[\subset\mathbf{A}(\Sigma,F)\subset[\alpha',\alpha'']\subset [-s,-r]$ ;

Exemples:

there exists $\alpha\in[-s,-r]$ such that $\aln(\gs,F)=\{\alpha\}$ ;

Exemples: SifT^?

$\mathbf{A}(\gs,F)=\emptyset$ .

Exemples: AcPermutatifs^?
Latex rendering error!! DVI file was not created.

to top

End of topic
Skip to action links | Back to top

You are here: CellularAutomata > DynamicClassification > DynamiqueDirectionnelle

to top