Major references

[marcokrt]

I know mphlee, who's much more organized than me, has a list of major publications in the scope of tetration. He could be more help here, lol.

Hahah I hope I was half organized as I'd like to be... My archive is only approximation atm... I had to do a major upgrade of it but had not time. I can share what I have gathered till now, with no claim of completeness. Also I don't claim to understand more then 20% of what follows, nor I claim to have read more than 30% of the following list, but skimmed most of them. Most of the items are in my to read list... some are beyond my math education.


Iteration/Dynamics

1860 Arthur Cayley - On some numerical expansion
1871 Ernst Schroeder - Uber iterirte functionen
1881 Abel
1882 A. Korkine - Sur un probleme d’interpolation. Bulletin des Sciences Math ́ematiques et As-
tronomiques
1884 Koenigs -  Recherches sur le intégrales de certaines équations fonctionnelles

1936 Morgan Ward, F. B. Fuller - The continuous iteration of real functions (Note: classifies all the bounded monoid homomorphism solutions subject to some monotonicity, cites Bennet 1916, points to Korkine 1882 for monoid homorphism iteration, it also proves that each monoid morphism defines a solution to the superfunction problem)
1956 - Erdos, Jabotinsky - On analytic iteration (proves that for complex powerseries \(\sum f_n z^n\) convergent on a disk about zero if there exists a complete complex iteration group then we have an iteration group analytic in the iteration parameter, it claims the existence problem is already settled for \(|f_1|\neq 1\). It claims to solve for \(f_1=1\) and points to Jabotinsky 54, Lewin 60 for "largely open" case of \(f_1\neq 1\) for \(f_1\neq 1\) )
1966 Komatsu - Fractional Power of Operators
1976 Earl Berkson, Horacio Porta - Semigroups of analytic functions and composition operators (note: about infinitesimal generator of a flow/semigroup of hoomorphic mappings)
1978 Lawvere - Categorical dynamics (note: beginning of Lawvere's program to give foundations to continuous dynamics in categorical terms, final goal is to axiomatize continuum mechanics and physiscs)
1980 Lawvere - Toward the description in a smooth topos of the dynamically possible motions and deformations of a continuous-body
1982 Nelson - Homomorphisms of monounary algebras (note: from Lawvere's School, she shows sufficient and necessary condition to the existence of a map of dynamical systems, eg. a superfunction, are presented algebraically)
1982 Lawvere - Introduction to categories in continuum physics
1984 Lawvere - Functorial remarks on the general concept of chaos (note: the concept of dynamical chaos is expressed in purely arrow/composition theoretic language, very fascinating)
1986 Lawvere - Taking categories seriously (note: here Lawvere sums up why category theory provides the natural way to treat dynamics and continuous dynamics: actions, dynamics, spectral analysis, periodic points, metrical concepts, cohesivity, philosophical continuity emerges naturally from the pure and fundamental role of composition)
1990 Kuczma, Choczewski, Ger - Iterative functional equations (note: really the most cited)
1990 Milnor - Dynamics in One Complex Varible
1992 Mrozek - Normal functor and retractors in cat of endomorphisms (note: use of normal functors to construct abstractly Conley indices over dynamical systems)
1996 Semeon Bogatyi - On nonexistence of iterative roots
(1997) Lawvere - Toposes of Laws of motion
1997 Woon - Analytic iteration of Operators
1999 Belitskii, Lyubich - Abel equation and total solvability of linear functional equations


2001 Carracedo, Alix - The Theory of Fractional Powers of Operators (note: vast theory)
2003 Marco Abate - Discrete local holomorphic dynamics (note: survey about complex dynamics about a fixed point)
2005 Marco Abate - Index theorems for holomorphic self-maps (note: generalizing index theorems, eg. Lefschetz theorem, from self max with isolated fixpoints to maps with fixpoints forminf positive dimension)
2005 Muller, Schleicher - How to add a non-integer number of terms, and how to produce unusual infinite summations
2006 Hilberdink - Orders of Growth of real functions (note: using the concept of growth rate a natural criterion for the uniqueness of fractional iterates of continuous functions is studied)
2007 Muller, Schleicher - Fractional Sums and Euler-like Identities
2008 Marco Abate - Discrete holomorphic local dynamical systems (note: survey with big bibliography)
2009 Cheritat - Parabolic implosion a mini-course (Amazing illustrations!!!)
2009 Curtright, Zachos - Evolution profiles and functional equations
2009 Robbins - On analytic iteration and iterated exponentials
2009 Trappmann, Kouznetsov - Uniqueness of Analytic Abel Functions without fixed points
2011 Muller, Schleicher - How to add a non-integer number of terms - From Axioms to New Identities
2011 G A Edgar - Fractional Iteration of series and transseries (note: points back to Korkine 1882 for iteration of powerseries and to Cayley 1860 about discussion on real iteration groups, i.e. fractional iteration)
2011 Trappmann, Kouznetsov - Uniqueness of holomorphic abel functions
2012 Giunti,  Mazzola - Dynamical systems on monoids - toward a general theory of deterministic systems and motion (note: basic framework of dynamics over arbitrary monoids of time, it should point to Lawevere but it doesn't!)
2013 Behrisch, Kerkhoff, Poschel, Shneider, Siegmund- Dynamical systems in Categories (note: dynamical systems can be translated as coalgebras/algebras over monads/comonads, really serious and deep article)
2014/03 Daniel Geisler - Bell polynomials of iterated functions
2014 Domoradzki, Stawiska - LUCJAN EMIL BOTTCHER AND HIS MATHEMATICAL LEGACY (note: seems worth to check, some unpubblished material is presented along with historical survey of Bottcher life and mathematical achievements)
2015 Marco Abate - Fatou flowers and parabolic curves (note: survey of multivariable generalizations of the Leau-Fatou flower theorem)
2015 Aschenbrenner, Bergweiler - Julia equation and differential trascendence (note: Jabotinsky ilog is shown to be differentially trascendental over the ring of entire functions)
2016/08 Geisler - The Existence and Uniqueness of the TaylorSeries of Iterated Functions (note: proof attempt, retracted claim?)
2017 Kouznetsov - Superfunctions (note: monography)
2019 Rogers - Toposes of discrete monoid Actions (note: characterization of topoi of actions over arbitrary monoids of time, up to equivalence)
2020 Rogers, Hamelaer - Monoid properties as invariants of toposes of monoid actions (note: super deep, how the semigroup property of the time monoid translates into properties of all the dynamical systems over that monoid, topos theoretically)
2021 Arnauld Maret - RTG Seminar
2021/04 Nixon - Infinite composition and complex dynamics - Generalizing Schroeder and abel functions
2021/05 Rogers - Toposes of topological monoid actions (note: same enterprise but jump from discrete time to topological, ie continuous, monoids of time)
2021/12 Rogers - Toposes of monoid actions, phd Thesis (note: from Olivia Caramello's School, big thesis summing up all the Topos theoretic recasting of dynamical systems theory over arbitrary monoids of time... this was the phd thesis I had to do if I wasn't so fucking stupid to lose my chance at the university...)
2022 Jacopo Garofali - Dynamical Sheaves, phd thesis (note: from Olivia Caramello's School, recasting of holomorphic dynamics in the language of topos theory)


Iteration of exponential/tetration

1986 Clenshaw, Lozier, Olver, Turner - Generalized exponential and logarithmic functions (note: study of solution to inverse abel equation of exp, application to computer arithmetic)
1990 Clement Frappier - Iterations of a kind of exponentials (note: a kind of nested exp)
1991 Daniel Geisler -  Algebraic Exponential Dynamics
1991 Peter Walker - Infinitely differentiable generalized logarithmic and exponential functions

2009/01 Kouznetsov, Trappmann - Protrait of the four regular super-exponentials
2009/04 Trappmann, Kouznetsov - 5+ methods for real analytic tetration (note: older version)
2009 Trappmann, Kouznetsov - Uniqueness of Holomorphic superlogarithms
2010/06 Trappmann, Kouznetsov - 5+ methods for real analytic tetration
2010 Kouznetsov - Tetrational as special function
2010 Kouznetsov, Trappmann - portrait of the four regular super-exponentials to base sqrt2
2011 Trappmann - The intuitive logarithm
2014/10 Aldrovandi - Tetration an iterative approach
2016 Paulsen - Finding the natural solution to \(f(f(x))=\exp(x)\)
2017 Cowgill - Exploring tetration in the complex plane
2017 Paulsen, Cowgill - Solving \(F(z+1)=b^{F(z)}\) in the complex plane
2020 Helms - Determining of and finding patterns in n-perios of exp-function
2020 Ripa - The congruence speed formula
2021/02 Nixon - A tetration function by unconventional means [v2]
2021/04 Nixon - The limits of a family of asymptotic solutions to the tetration equation [v1]
2021/05 Ueda - Extension of tetration to real and complex heights


Hyperoperations

(1915) Bennet - Note on an Operation of the Thrid Grade
(1947) Goodstein - Transfinite ordinals in recursive number theory
(1947) Robinson - primitive recursive functions
(1953) Arcidiacono - Sulla Estensione delle operazioni aritmetiche
(1969) Doner, Tarski - An extended arithmetic of ordinal numbers
(1975) Raspletin - Hyperoperations
(1989) Rubtsov - A component R0 in R3
(1989) Rubtsov - Algorithms ingredients in a set of algebraic operations
(1990) Rubtsov - A complement of a set of real numbers and his application in cybernetics
(1990) Rubtsov - A hypothetical reflexive complement of a set of real numbers
(1990) Rubtsov - An image by derivative obtained by replacement of operations
(1994) Rubtsov - Integro-differential objects of a new nature
(1995) Muller - Reihenalgebra - What comes beyond exponentiation
(199 Rubtsov - New Mathematical Objects

(2001 09) Geisler - Recurring digits in the ackermann function
(2001 12) Loday - Arithmetree
(2001) Micheal L. Carroll - The natural chain of binary arithmetic operations and generalized derivatives (note: basically part of Rubstov but formally)
(2004 09) Rubtsov, Romerio - Ackermanns function and new arthmetical operations
(2006 07 25) Rubtsov, Romerio - Notes on Hyperoperations - Progress Report - NKS Forum III
(2006 0 Rubtsov, Romerio - Hyperopertions as a tool for science and engineering, for ICM-06
(2007 06) Romerio - NKS Forum - Hyper-operations. Progress Report. Zeration.
(2007 08 09) Tetration FAQ
(2007 11 04) Romerio - Incomplete Towers
(2007) Trappmann - Arborescent Numbers, Hyperoperations,Division trees
(2008 07 10) Tetration REF

(2010) Williams - What Lies Between + and x (and beyond)
(2014 02 17) Reale - Operazione di rango zero e numeri non transitivi
(2014 06 11) Barrette - Hyperoperator manuscript 02
(2014 11) Kouznetsov - Evaluation of Holomorphic ackermanns
(2015 07) MphLee - On fractional ranks Mathematics Stack Exchange
(2016) Crespo, Montàs - Fractional Mathematical Operators and Their Computational Approximation
(2017) Altman - Intermediate arithmetic operations on ordinal numbers
(2018 06) Tezlaf - On ordinal dynamics and the multiplicity of transfinite cardinality
(2019 02) Barrett - The fundamental theorems of Hyperoperations
(2019 03) Leonardis, d'Atri, Caldarola - Byond Knuth notation for Unimaginable numbers
(2019) Caldaro, d'Atri, Maiolo - What are the unimaginable numbers
(2019) Dalthorp - Hyperoperations - introduction to the theory and potential solutions
(2020 07) Rubtsov - Application of hyperoperations for engineering practice
(2020 12) Aguilera, Freund, Rathjen, Weiermann - Ackermann and goodstein go functorial
(2020 12) Salazar - Hyperoperations in exponential fields
(2020) Judijasa - The logarithmic chain complex
(2021 03 13) Geisler - Extension of hyperoperators
(2021 03 03) Nixon - Hyper-operations By Unconventional Means
(2021 05) Jaramillo - Hyperoperations in exponential fields
(2021 05 24) Nixon - A family of bounded and analytic hyper-operators
(2021) Andonov - Constructing a hyperoperation sequence-pisa hyperoperations


Iterated composition

2003 Keen, Lakic - Forward iterated Function Systems (note: points to Gill 198
2005/09 Keen, Lakic - Accumulation constants of iterated function systems with Bloch target
2006/01 Keen, Lakic - Limit points of iterated function systems domains
2014/02 Kyriakos Kefalas - On smooth solutions of non-linear dyynamical systems \(f_{n+1}=u(f_n0)\), PART I (note: cites Hooshmand 06, Walker 91, Szekeres 5
2020/01 Nixon - The compositional integral - A brief introduction
2020/11 Nixon - The compositional integral - The narrow and the complex looking glass

I've many more papers collected but I don't have time to organize them.


EDIT: +added essential bibliography about the algebraic/categorial perspective on dynamics.

Awesome list! I would like to just point out that my real surname is "Ripà" (instead of "Ripa", a very common mistake indeed), mentioning also the last paper that I have written together with the other TetrationForum user "Luca Onnis", entitled "Number of stable digits of any integer tetration", which completes the trilogy on the congruence speed that I started by submitting to Notes on Number Theory and Discrete Mathematics, in 2019, the manuscript entitled "On the constant congruence speed of tetration" (see NNTDM, Vol. 26(3), pp. 245-260, DOI: 10.7546/nntdm.2020.26.3.245-260).
After that, I finally managed to provide an inverse map of this new function (assuming that radix-10 is given by hypothesis), the congruence speed of any integer tetration with the base which is not a multiple of 10, by publishing the mentioned paper entitled "The congruence speed formula" (available also on the arXiv at https://arxiv.org/abs/2208.02622).

Then, me and Luca have finally provided also the direct map of the "congruence speed of tetration" by considering the valuation function (applied to a few very simple manipulations of the given base) of the divisors of the squarefree value of the considered numerical system (i.e., 2 and 5, since 10 = 2 · 5) thanks to the paper titled "Number of stable digits of any integer tetration", which closes the bounds that I previously gave in "The congruence speed formula", by providing extended proofs based on the theorems published in "The congruence speed formula", so I think that it would be the best to mention both "The congruence speed formula" and "Number of stable digits of any integer tetration" (arXiv version: https://arxiv.org/abs/2210.07956) in order to provide the full map of this peculiar property of the integer tetration, named "constant congruence speed".

Just my two cents.