I totally forgot Devaney's book! It's amazing. It was on my todo list since ages. Now I'm eating it like bread... many things are super clear now...
I think that after it I can begin something more juicy and than straight to Milnor... maybe before september If I manage to read carefully I can parse 60% of this forum's posts.
I'll be back on this post when I've finished Devaney... I hope I'll be able to find time for some exercises too
Off topic, reading Devaney on symbolic dynamics I just remember that Lawere proposed an algebraic definition of chaos. Let \({\bf X}=(X,f)\) be a dynamical system. Let \(A^\mathbb N=(A^{\mathbb N},\sigma)\) be the shift space over the alphabet \(A\). Given an observable \(\phi:X\to A\) that assign to every state a measurement, there always exists a canonical dynamical systems map \({\bar \phi}:{\bf X}\to A^\mathbb N\) assigns what Devaney calls "itinerary" (of observations). \[x \mapsto (\phi(x),\phi(f^1(x)),\phi(f^2(x)),..., \phi(f^n(x)),...)\]
The observable \(\phi\) is said chaotic if \(\bar \phi\) is surjective. It means that the observable is so weak that we can observe every possible behavior... we call this chaotic behavior.
The reason I asked the question is because I wanted to have a mental picture of the dynamics of exponentiation in term of graphs/phase protrait.
I think that after it I can begin something more juicy and than straight to Milnor... maybe before september If I manage to read carefully I can parse 60% of this forum's posts.
I'll be back on this post when I've finished Devaney... I hope I'll be able to find time for some exercises too
Off topic, reading Devaney on symbolic dynamics I just remember that Lawere proposed an algebraic definition of chaos. Let \({\bf X}=(X,f)\) be a dynamical system. Let \(A^\mathbb N=(A^{\mathbb N},\sigma)\) be the shift space over the alphabet \(A\). Given an observable \(\phi:X\to A\) that assign to every state a measurement, there always exists a canonical dynamical systems map \({\bar \phi}:{\bf X}\to A^\mathbb N\) assigns what Devaney calls "itinerary" (of observations). \[x \mapsto (\phi(x),\phi(f^1(x)),\phi(f^2(x)),..., \phi(f^n(x)),...)\]
The observable \(\phi\) is said chaotic if \(\bar \phi\) is surjective. It means that the observable is so weak that we can observe every possible behavior... we call this chaotic behavior.
The reason I asked the question is because I wanted to have a mental picture of the dynamics of exponentiation in term of graphs/phase protrait.
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)
