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On sequences resulting from iteration of modified quadratic and palindromic mappings

1997
*
Theoretical Computer Science
*

Blanchard and Fabre (1994) have considered description

doi:10.1016/s0304-3975(96)00294-0
fatcat:5plp4fxjebg2bhqvfsadufcfuq
*of*symbolic systems*by*sequences*obtained**by**iteration**of*two*mappings*$1, and d" defined*by*g,(u) = u [u], and L/, J u[u]:, m here the*word*[u], ... Our result is*obtained**by*investigation*of*two wider*classes**of*sequences. * ... I n such case the limit*of*the*iteration**of*m*on*w is an*infinite**word*denoted*by*m"'(w). ...##
###
Page 2090 of Mathematical Reviews Vol. , Issue 87d
[page]

1987
*
Mathematical Reviews
*

Subramanian (Madras)
87d:68081
Pansiot, Jean-Jacques (F-STRAS-CC) 87d:68082

*On**various**classes**of**infinite**words**obtained**by**iterated**mappings*. ... This is a very nice survey article*on**infinite**words*generated*by**various*types*of*L-systems. ...##
###
Pell and Clapeyron Words as Stable Trajectories in Dynamical Systems
[article]

2017
*
arXiv
*
pre-print

We show that,

arXiv:1707.09333v2
fatcat:ebvgcekia5hr7e6j3emuktetae
*of*the ten physically-relevant*classes**of**one*-dimensional quasicrystal, precisely two can appear as stable, attracting trajectories in dynamical systems, which we term the*infinite*Pell and ... Clapeyron*words*. ... While the value x ∞ converged to depends*on*µ, it always lies*on*the right*of*the*map*, so is described*by*the*word*R. ...##
###
Page 5080 of Mathematical Reviews Vol. , Issue 87i
[page]

1987
*
Mathematical Reviews
*

This defines an equivalence

*on*the set*of*structures and gives an identification*of*these equivalence*classes*with the universal Teichmiiller space which is well known to be*infinite*-dimensional. ... The for- mula*obtained*is hm(f) = f 5°; A771 dm where, in their*words*, “7; denotes, roughly speaking, the dimension*of*m ‘in the direction*of*the subspace E,’”. ...##
###
Time Quasilattices in Dissipative Dynamical Systems

2018
*
SciPost Physics
*

We show that there are precisely two admissible time quasilattices, which we term the

doi:10.21468/scipostphys.5.1.001
fatcat:4omlfjf4avd4lkhdhf2iyjbdky
*infinite*Pell and Clapeyron*words*, reached*by*a generalization*of*the period-doubling cascade. ... These tilings*of*the time axis, with two unit cells*of*different durations, can be generated as cuts through a periodic lattice spanned*by*two orthogonal directions*of*time. ... Funding information The author acknowledges support from a Lindemann Trust Fellowship*of*the English-Speaking Union and the Astor Junior Research Fellowship*of*New College, Oxford. ...##
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Richness of Chaos in the Absolute Newhouse Domain

2011
*
Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)
*

We show that universal

doi:10.1142/9789814324359_0122
fatcat:fszm6ivhezewldnssihbt4ehk4
*maps*(i.e. such whose*iterations*approximate every possible dynamics arbitrarily well) form a residual subset in an open set in the space*of*smooth dynamical systems. ... The result implies that many dynamical systems emerging in natural applications may,*on*a very long time scale, have quite unexpected dynamical properties, like coexistence*of*many non-trivial hyperbolic ... We will call the*maps*f m,ψ*obtained**by*such procedure renormalized*iterations**of*f . ...##
###
Iterative arrays with a wee bit alternation
[chapter]

1999
*
Lecture Notes in Computer Science
*

An

doi:10.1007/3-540-48321-7_13
fatcat:kmattudennhehm576vz3pmj34u
*iterative*array is a line*of*interconnected interacting finite automata. ... The number*of*alternating state transitions is regarded as a limited resource which depends*on*the length*of*the input. ... Finally we*obtain**infinite*hierarchies*of*properly included language families varying the amount*of*allowed alternation. ...##
###
A Note on Palindromicity

1999
*
Letters in Mathematical Physics
*

Two results

doi:10.1023/a:1007678316583
fatcat:ebq2762nwza7nech4mxr46gsza
*on*palindromicity*of*bi-*infinite**words*in a finite alphabet are presented. ... These give rise to diagonal tight-binding models in*one*dimension with purely singular continuous spectrum. ... This work was supported*by*the German Science Foundation (DFG). ...##
###
Page 6630 of Mathematical Reviews Vol. , Issue 88m
[page]

1988
*
Mathematical Reviews
*

88m:68034

*infinitely*many elements*of*S. Further, any square-free*word*w*on*~ that is a prefix*of*a sufficiently long square-free*word*u*on*is also the prefix*of*an*infinite*square-free*word*x € S. ... Intu- itively, such a flowchart can be*obtained*from*one**of*the familiar x-flowchart schemes*by*replacing the go-to functions*by*morphisms in T. ...##
###
Page 5772 of Mathematical Reviews Vol. , Issue 2000h
[page]

2000
*
Mathematical Reviews
*

A wicDTOL system derives exactly

*one**word*from a given*word**by**iterating*the table which is the value*of*the control function*of*the length*of*the given*word*. ... In this paper we survey results*on*syn- chronization languages, in particular,*various*approaches to*obtain*a characterization*of*this language family using closure under a set*of*rewriting rules. ...##
###
The 3x+1 Problem: An Overview
[article]

2021
*
arXiv
*
pre-print

This paper is an overview and survey

arXiv:2111.02635v1
fatcat:xbenrwvnnvcxhooykcvex5ayay
*of*work*on*the 3x+1 problem, also called the Collatz problem, and generalizations*of*it. It gives a history*of*the problem. ... (as*of*2010). (2) How can this problem be hard, when it is so easy to state? ... The author was supported*by*NSF Grants DMS-0500555 and DMS-0801029. ...##
###
The 3x+1 Problem: An Annotated Bibliography, II (2000-2009)
[article]

2012
*
arXiv
*
pre-print

The 3x+1 problem concerns

arXiv:math/0608208v6
fatcat:a3wdbqlvz5hqfimoxzpjmwbdyi
*iteration**of*the*map*T(n) =(3n+1)/2 if n odd; n/2 if n even. The 3x +1 Conjecture asserts that for every positive integer n>1 the forward orbit*of*n includes the integer 1. ... This paper is an annotated bibliography*of*work done*on*the 3x+1 problem published from 2000 through 2009, plus some later papers that were preprints*by*2009. ... The original Collatz*map*(see Klamkin (1963)) given*by*f 41. Stefan Kohl (2007), Wildness*of**iteration**of*certain residue*class*-wise affine*mappings*, Advances in Applied Math. 39 (2007), 322-328. ...##
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Page 5415 of Mathematical Reviews Vol. , Issue 87j
[page]

1987
*
Mathematical Reviews
*

to

*words*, whereas y*maps*equivalence*classes*[w] to equivalence*classes*[o(w)]). ... [w]] denote a*word*[its equivalence*class*] in G; let a parametric expression be an element*of*the free product*of*G and a free group F*on*n generators (which are regarded as variables over G). ...##
###
UNARY FA-PRESENTABLE SEMIGROUPS

2012
*
International journal of algebra and computation
*

We investigate the relationship between the

doi:10.1142/s0218196712500385
fatcat:hnirmvrrvbherocz4vu7e3owvu
*class**of*unary FA-presentable semigroups and*various*semigroup constructions. ... We prove the following: Every unary FA-presentable structure admits an injective unary automatic presentation where the language*of*representatives consists*of*every*word*over a*one*-letter alphabet. ... Acknowledgements The first author was supported*by*the project EP/C523229/1, funded*by*the EPSRC, and later*by**by*the European Regional Development Fund through the programme COMPETE and*by*the Portuguese ...##
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The Collatz conjecture and De Bruijn graphs

2013
*
Indagationes mathematicae
*

We study variants

doi:10.1016/j.indag.2013.03.003
fatcat:p2qdwz4uc5fmxnd4t2rvr4mzsa
*of*the well-known Collatz graph,*by*considering the action*of*the 3n+1 function*on*congruence*classes*. ... We then look at a natural generalization*of*these finite graphs to the 2-adic integers, and show that the isomorphism between these*infinite*graphs is exactly the conjugacy*map*previously studied*by*Bernstein ... So if we consider the (*iterated*) action*of*T*on*a congruence*class**of*some n (mod 2 k ), then we find that the first k elements*of*(x i (n)) are identical for all numbers in this congruence*class*, and ...
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