Arithmetic dynamics[1] is a field that amalgamates two areas of mathematics, dynamical systems and number theory. Part of the inspiration comes from complex dynamics, the study of the iteration of self-maps of the complex plane or other complex algebraic varieties. Arithmetic dynamics is the study of the number-theoretic properties of integer, rational, p-adic, or algebraic points under repeated application of a polynomial or rational function. A fundamental goal is to describe arithmetic properties in terms of underlying geometric structures.
Global arithmetic dynamics is the study of analogues of classical diophantine geometry in the setting of discrete dynamical systems, while local arithmetic dynamics, also called p-adic or nonarchimedean dynamics, is an analogue of complex dynamics in which one replaces the complex numbers C by a p-adic field such as Qp or Cp and studies chaotic behavior and the Fatou and Julia sets.
The following table describes a rough correspondence between Diophantine equations, especially abelian varieties, and dynamical systems:
Diophantine equations
Dynamical systems
Rational and integer points on a variety
Rational and integer points in an orbit
Points of finite order on an abelian variety
Preperiodic points of a rational function
^Silverman, Joseph H. (2007). The Arithmetic of Dynamical Systems. Graduate Texts in Mathematics. Vol. 241. New York: Springer. doi:10.1007/978-0-387-69904-2. ISBN 978-0-387-69903-5. MR 2316407.
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Arithmeticdynamics is a field that amalgamates two areas of mathematics, dynamical systems and number theory. Part of the inspiration comes from complex...
Classically, discrete dynamics refers to the study of the iteration of self-maps of the complex plane or real line. Arithmeticdynamics is the study of the...
mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is...
factors Formula for primes Factorization RSA number Fundamental theorem of arithmetic Square-free Square-free integer Square-free polynomial Square number Power...
these analogies, coining the term arithmetic topology for this area of study. Arithmetic geometry Arithmeticdynamics Topological quantum field theory...
In number theory, the aliquot sum s(n) of a positive integer n is the sum of all proper divisors of n, that is, all divisors of n other than n itself....
The number 6174 is known as Kaprekar's constant after the Indian mathematician D. R. Kaprekar. This number is renowned for the following rule: Take any...
mapping from some algebraic variety to itself. The related theory of arithmeticdynamics studies iteration over the rational numbers or the p-adic numbers...
numbers and cycles of length two that represent amicable pairs. Arithmeticdynamics Weisstein, Eric W. "Aliquot Sequence". MathWorld. Sloane, N. J. A...
Mathematically, the number 1 has unique properties and significance. In normal arithmetic (algebra), the number 1 is the first natural number after 0 (zero) and...
their digit sums with the digit sums of their prime factorizations. Arithmeticdynamics Casting out nines Checksum Digital root Hamming weight Harshad number...
has media related to Collatz conjecture. 3x + 1 semigroup Arithmeticdynamics Modular arithmetic Residue-class-wise affine group It is also known as the...
and dynamical systems Shift space Shift of finite type Complex dynamicsArithmeticdynamics Hadamard, J. (1898). "Les surfaces à courbures opposées et leurs...
these equations. Diophantine geometry is part of the broader field of arithmetic geometry. Four theorems in Diophantine geometry that are of fundamental...
associated with arithmetic operations such as addition, subtraction, multiplication and division. ArithmeticdynamicsArithmeticdynamics is the study of...
has the result 0, and consequently, division by zero has no meaning in arithmetic. As a numerical digit, 0 plays a crucial role in decimal notation: it...
seen_numbers.add(number) number = pdi_function(number) return number == 1 Arithmeticdynamics Fortunate number Harshad number Lucky number Perfect digital invariant...
Mixing (mathematics) Almost periodic function Symbolic dynamics Time scale calculus Arithmeticdynamics Sequential dynamical system Graph dynamical system...
In recreational mathematics, arithmetic billiards provide a geometrical method to determine the least common multiple and the greatest common divisor of...
maximum value at a60 with 972,463 digits, before reaching 1 at a157. Arithmeticdynamics Collatz conjecture Recurrence relation Pickover, Clifford A. (1992)...
numbers into deficient, perfect, or abundant, in his Introduction to Arithmetic (circa 100 CE). However, he applied this classification only to the even...
visual novel adventure game Nine Hours, Nine Persons, Nine Doors. Arithmeticdynamics Base 9 Casting out nines Digit sum Divisibility rule Hamming weight...
Rashed, Roshdi (1994). The development of Arabic mathematics: between arithmetic and algebra. Vol. 156. Dordrecht, Boston, London: Kluwer Academic Publishers...
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