Thanks for this commentary. It will be really useful in the future. Now it did help make my mind a lil bit.
A quick search on keywords led me to this recent paper that may bring some useful bibliographic pointer.
-2022, Ferreira - A note on forward iteration of inner
functions
To be honest I got a ton of interesting pointers... since it came to my mind that there is a term for dealing with iterated function systems and is non-autonomous dynamical systems, opposed to autonomous ones. Autonomous systems are just representation of time semigroups, i.e. semigroup homomorphisms. While non-autonomous systems are representations of categories... i.e. functors... just like your omega notation.
-2018, Bracci et al. - BACKWARD ORBITS AND PETALS OF SEMIGROUPS OF HOLOMORPHIC
SELF-MAPS OF THE UNIT DISC
-2018, Bracci et al. - ASYMPTOTIC BEHAVIOR OF ORBITS OF HOLOMORPHIC
SEMIGROUPS
-2020, Bracci, Roth - SEMIGROUP-FICATION OF UNIVALENT SELF-MAPS OF THE UNIT DISC
-2022, Benini et al. - The Denjoy–Wolff set for holomorphic sequences,
non-autonomous dynamical systems and wandering domains
I'm always positively surprised by how much of different keywords there are on almost the same topic. Also is descriptive of how much I'm still ignorant on the subject after all these years wandering in the literature.
I remember asking myself few months ago, when I discovered the theorem, if the Denjoy–Wolff point was important or useful. Idk if I ever heard of it on this forum before but I'm sure its a pretty standard result.
A quick search on keywords led me to this recent paper that may bring some useful bibliographic pointer.
-2022, Ferreira - A note on forward iteration of inner
functions
To be honest I got a ton of interesting pointers... since it came to my mind that there is a term for dealing with iterated function systems and is non-autonomous dynamical systems, opposed to autonomous ones. Autonomous systems are just representation of time semigroups, i.e. semigroup homomorphisms. While non-autonomous systems are representations of categories... i.e. functors... just like your omega notation.
-2018, Bracci et al. - BACKWARD ORBITS AND PETALS OF SEMIGROUPS OF HOLOMORPHIC
SELF-MAPS OF THE UNIT DISC
-2018, Bracci et al. - ASYMPTOTIC BEHAVIOR OF ORBITS OF HOLOMORPHIC
SEMIGROUPS
-2020, Bracci, Roth - SEMIGROUP-FICATION OF UNIVALENT SELF-MAPS OF THE UNIT DISC
-2022, Benini et al. - The Denjoy–Wolff set for holomorphic sequences,
non-autonomous dynamical systems and wandering domains
I'm always positively surprised by how much of different keywords there are on almost the same topic. Also is descriptive of how much I'm still ignorant on the subject after all these years wandering in the literature.
I remember asking myself few months ago, when I discovered the theorem, if the Denjoy–Wolff point was important or useful. Idk if I ever heard of it on this forum before but I'm sure its a pretty standard result.
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)