Really nice Gottfried... swimming against the attraction field... interesting.
We need an expert opinion on this. I'm sure is some trivial proposition one can find in any book on dynamics... but I'm too ignorant. I'm sure James can settle this in 1 second.
To think about it... if the orbit crosses the basin of attraction of the fixed point... maybe it is not necessary that from that moment each successive point of the orbit has a distance from the fixed point that is monotone decreasing.
Conjecture 2: let \(f:X\to X\), \(X\) a metric space, \(p\in X\) attracting and \(I\subseteq X\) its basin of attraction. Let \(\chi:[0,\infty)\to X\) be a \(x_0\)-initialized superfucntion that is continuous, i.e. \(\chi(0)=x_0\) and \[\chi(t+1)=f(\chi(t))\]
if exists a \(r\in [0,\infty)\) s.t. \(\chi({r})\in I\) is in the basin of attraction of \(p\) then
\[D(t):=d(\chi(t+r),p)\]
is monotone decreasing. I.e. for \(r<t\) we have \(\chi(t)\) gets closer and closer to \(p\) and never oscillates back.
I'd say that this is false and that an orbit that crosses the basin is doomed to reach the fixed point but it can reach it in oscillating ways...I conjecture...
if this is true there will be moments where the orbit is swimming against the attraction field... and we have no problems...
At this point it can be interesting to understand how the map \(f:X\to X\) maps \(p\) based loops \(\gamma:{}S^1\to X\) to loops \(f\circ \gamma:{}S^1\to X\) since each \(p\)-initialized superfunction \(\chi:[0,\infty)\to X\) is completely determined by it's first loop \(\gamma:[0,1)\to \to X\) and all the other loops are determined by iteration \(\chi|_{[n,n+1)}=f^n\circ \gamma:[n,n+1) \to X\).
Question. For example... if \(f\) is continuous... then is \(\gamma\) homotopic to \(f\circ \gamma\)?
It would be interestin to study the map that each superfucntion induces on the fundamental group of \(X\): the map \({\boldsymbol\chi}:\mathbb N \to \pi_1(X) \)
\[{\boldsymbol\chi}(n)=[f^n\circ \chi|_{[0,1)}]\]
We need an expert opinion on this. I'm sure is some trivial proposition one can find in any book on dynamics... but I'm too ignorant. I'm sure James can settle this in 1 second.
To think about it... if the orbit crosses the basin of attraction of the fixed point... maybe it is not necessary that from that moment each successive point of the orbit has a distance from the fixed point that is monotone decreasing.
Conjecture 2: let \(f:X\to X\), \(X\) a metric space, \(p\in X\) attracting and \(I\subseteq X\) its basin of attraction. Let \(\chi:[0,\infty)\to X\) be a \(x_0\)-initialized superfucntion that is continuous, i.e. \(\chi(0)=x_0\) and \[\chi(t+1)=f(\chi(t))\]
if exists a \(r\in [0,\infty)\) s.t. \(\chi({r})\in I\) is in the basin of attraction of \(p\) then
\[D(t):=d(\chi(t+r),p)\]
is monotone decreasing. I.e. for \(r<t\) we have \(\chi(t)\) gets closer and closer to \(p\) and never oscillates back.
I'd say that this is false and that an orbit that crosses the basin is doomed to reach the fixed point but it can reach it in oscillating ways...I conjecture...
if this is true there will be moments where the orbit is swimming against the attraction field... and we have no problems...
At this point it can be interesting to understand how the map \(f:X\to X\) maps \(p\) based loops \(\gamma:{}S^1\to X\) to loops \(f\circ \gamma:{}S^1\to X\) since each \(p\)-initialized superfunction \(\chi:[0,\infty)\to X\) is completely determined by it's first loop \(\gamma:[0,1)\to \to X\) and all the other loops are determined by iteration \(\chi|_{[n,n+1)}=f^n\circ \gamma:[n,n+1) \to X\).
Question. For example... if \(f\) is continuous... then is \(\gamma\) homotopic to \(f\circ \gamma\)?
It would be interestin to study the map that each superfucntion induces on the fundamental group of \(X\): the map \({\boldsymbol\chi}:\mathbb N \to \pi_1(X) \)
\[{\boldsymbol\chi}(n)=[f^n\circ \chi|_{[0,1)}]\]
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)