Expansions in Complex Bases

2007 ◽  
Vol 50 (3) ◽  
pp. 399-408 ◽  
Author(s):  
Vilmos Komornik ◽  
Paola Loreti
Keyword(s):  
Number Theory ◽  
Dynamical Systems ◽  
Greedy Algorithm ◽  
Finite Automata ◽  
Complex Case ◽  
Seminal Paper

AbstractBeginning with a seminal paper of Rényi, expansions in noninteger real bases have been widely studied in the last forty years. They turned out to be relevant in various domains of mathematics, such as the theory of finite automata, number theory, fractals or dynamical systems. Several results were extended by Daróczy and Kátai for expansions in complex bases. We introduce an adaptation of the so-called greedy algorithm to the complex case, and we generalize one of their main theorems.

Dynamical Systems in Number Theory

1982 ◽  
pp. 157-177
Author(s):  
I. P. Cornfeld ◽  
S. V. Fomin ◽  
Ya. G. Sinai
Keyword(s):  
Number Theory ◽  
Dynamical Systems

Dynamical Systems, Number Theory and Applications

10.1142/9695 ◽  
2015 ◽  
Author(s):  
Thomas Hagen ◽  
Florian Rupp ◽  
Jürgen Scheurle
Keyword(s):  
Number Theory ◽  
Dynamical Systems

scholarly journals Fixed Points and Fermat: A Dynamical Systems Approach to Number Theory

2000 ◽  
Vol 107 (5) ◽  
pp. 422-428
Author(s):  
Michael Frame ◽  
Brenda Johnson ◽  
Jim Sauerberg
Keyword(s):  
Number Theory ◽  
Dynamical Systems ◽  
Fixed Points ◽  
Systems Approach

GROUPS WITH INDEXED CO-WORD PROBLEM

2006 ◽  
Vol 16 (05) ◽  
pp. 985-1014 ◽  
Author(s):  
DEREK F. HOLT ◽  
CLAAS E. RÖVER
Keyword(s):  
Dynamical Systems ◽  
Large Class ◽  
Word Problem ◽  
Word Problems ◽  
Automorphism Groups ◽  
Finite Automata ◽  
Restricted Class ◽  
Closure Properties ◽  
Free Products ◽  
Thompson Groups

We investigate co-indexed groups, that is groups whose co-word problem (all words defining nontrivial elements) is an indexed language. We show that all Higman–Thompson groups and a large class of tree automorphism groups defined by finite automata are co-indexed groups. The latter class is closely related to dynamical systems and includes the Grigorchuk 2-group and the Gupta–Sidki 3-group. The co-word problems of all these examples are in fact accepted by nested stack automata with certain additional properties, and we establish various closure properties of this restricted class of co-indexed groups, including closure under free products.

scholarly journals Fixed Points and Fermat: A Dynamical Systems Approach to Number Theory

2000 ◽  
Vol 107 (5) ◽  
pp. 422 ◽  
Author(s):  
Michael Frame ◽  
Brenda Johnson ◽  
Jim Sauerberg
Keyword(s):  
Number Theory ◽  
Dynamical Systems ◽  
Fixed Points ◽  
Systems Approach

scholarly journals O-Minimal Invariants for Discrete-Time Dynamical Systems

2022 ◽  
Vol 23 (2) ◽  
pp. 1-20
Author(s):  
Shaull Almagor ◽  
Dmitry Chistikov ◽  
Joël Ouaknine ◽  
James Worrell
Keyword(s):  
Number Theory ◽  
Dynamical Systems ◽  
Discrete Time ◽  
Program Verification ◽  
Transcendental Number ◽  
Linear Recurrence ◽  
Open Problems ◽  
Linear Dynamical Systems ◽  
Recurrence Sequences ◽  
Positivity Problems

Termination analysis of linear loops plays a key rôle in several areas of computer science, including program verification and abstract interpretation. Already for the simplest variants of linear loops the question of termination relates to deep open problems in number theory, such as the decidability of the Skolem and Positivity Problems for linear recurrence sequences, or equivalently reachability questions for discrete-time linear dynamical systems. In this article, we introduce the class of o-minimal invariants , which is broader than any previously considered, and study the decidability of the existence and algorithmic synthesis of such invariants as certificates of non-termination for linear loops equipped with a large class of halting conditions. We establish two main decidability results, one of them conditional on Schanuel’s conjecture is transcendental number theory.

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