I did a bad abuse of notation. Also one of my claim was a little bit wrong because of this abuse.
I'll give you the wanted definitions and some good pictures, like the one you used, to aid our intuition.
i just need 24h.
[edit 23 semptember 1am] I'm sorry, I need more time for the pictures. In a couple of hours.
Btw \({\rm gen}(X)\) are the states without a preimage. The eternal generators \({\rm ete}(X)\) are equivalence classes of bi-infinite sequences of states that are "linked by \(f\) ". The generated part of \(X\) is generated by \({\rm gen}(X)\) , i.e. closed under application of \(f\) . The eternal part of \(X\) is made by the states that have at least one infinite chain of preimage.
Tonight I'll make a proper answer with the pictures.
I'll give you the wanted definitions and some good pictures, like the one you used, to aid our intuition.
i just need 24h.
[edit 23 semptember 1am] I'm sorry, I need more time for the pictures. In a couple of hours.
Btw \({\rm gen}(X)\) are the states without a preimage. The eternal generators \({\rm ete}(X)\) are equivalence classes of bi-infinite sequences of states that are "linked by \(f\) ". The generated part of \(X\) is generated by \({\rm gen}(X)\) , i.e. closed under application of \(f\) . The eternal part of \(X\) is made by the states that have at least one infinite chain of preimage.
Tonight I'll make a proper answer with the pictures.
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)