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On sequences resulting from iteration of modified quadratic and palindromic mappings

Anton Černý
1997 Theoretical Computer Science  
Blanchard and Fabre (1994) have considered description 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).  ... 
doi:10.1016/s0304-3975(96)00294-0 fatcat:5plp4fxjebg2bhqvfsadufcfuq

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]

Felix Flicker
2017 arXiv   pre-print
We show that, 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.  ... 
arXiv:1707.09333v2 fatcat:ebvgcekia5hr7e6j3emuktetae

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

Felix Flicker
2018 SciPost Physics  
We show that there are precisely two admissible time quasilattices, which we term the 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.  ... 
doi:10.21468/scipostphys.5.1.001 fatcat:4omlfjf4avd4lkhdhf2iyjbdky

Richness of Chaos in the Absolute Newhouse Domain

Dmitry Turaev
2011 Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)  
We show that universal 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 .  ... 
doi:10.1142/9789814324359_0122 fatcat:fszm6ivhezewldnssihbt4ehk4

Iterative arrays with a wee bit alternation [chapter]

Thomas Buchholz, Andreas Klein, Martin Kutrib
1999 Lecture Notes in Computer Science  
An 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.  ... 
doi:10.1007/3-540-48321-7_13 fatcat:kmattudennhehm576vz3pmj34u

A Note on Palindromicity

Michael Baake
1999 Letters in Mathematical Physics  
Two results 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).  ... 
doi:10.1023/a:1007678316583 fatcat:ebq2762nwza7nech4mxr46gsza

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]

Jeffrey C. Lagarias
2021 arXiv   pre-print
This paper is an overview and survey 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.  ... 
arXiv:2111.02635v1 fatcat:xbenrwvnnvcxhooykcvex5ayay

The 3x+1 Problem: An Annotated Bibliography, II (2000-2009) [article]

Jeffrey C. Lagarias
2012 arXiv   pre-print
The 3x+1 problem concerns 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.  ... 
arXiv:math/0608208v6 fatcat:a3wdbqlvz5hqfimoxzpjmwbdyi

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).  ... 


2012 International journal of algebra and computation  
We investigate the relationship between the 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  ... 
doi:10.1142/s0218196712500385 fatcat:hnirmvrrvbherocz4vu7e3owvu

The Collatz conjecture and De Bruijn graphs

Thijs Laarhoven, Benne de Weger
2013 Indagationes mathematicae  
We study variants 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  ... 
doi:10.1016/j.indag.2013.03.003 fatcat:p2qdwz4uc5fmxnd4t2rvr4mzsa
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