The
mathematical
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
disciplines of
combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
and
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
s interact in a number of ways. The
ergodic theory
Ergodic theory is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, "statistical properties" refers to properties which are expressed through the behav ...
of dynamical systems has recently been used to prove combinatorial theorems about number theory which has given rise to the field of
arithmetic combinatorics
In mathematics, arithmetic combinatorics is a field in the intersection of number theory, combinatorics, ergodic theory and harmonic analysis.
Scope
Arithmetic combinatorics is about combinatorial estimates associated with arithmetic operations ...
. Also
dynamical systems theory
Dynamical systems theory is an area of mathematics used to describe the behavior of complex systems, complex dynamical systems, usually by employing differential equations by nature of the ergodic theory, ergodicity of dynamic systems. When differ ...
is heavily involved in the relatively recent field of
combinatorics on words
Combinatorics on words is a fairly new field of mathematics, branching from combinatorics, which focuses on the study of words and formal languages. The subject looks at letters or symbols, and the sequences they form. Combinatorics on words ...
. Also combinatorial aspects of dynamical systems are studied. Dynamical systems can be defined on combinatorial objects; see for example
graph dynamical system In mathematics, the concept of graph dynamical systems can be used to capture a wide range of processes taking place on graphs or networks. A major theme in the mathematical and computational analysis of GDSs is to relate their structural properties ...
.
See also
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Symbolic dynamics
In mathematics, symbolic dynamics is the study of dynamical systems defined on a discrete space consisting of infinite sequences of abstract symbols. The evolution of the dynamical system is defined as a simple shift of the sequence.
Because of t ...
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Analytic combinatorics
Analytic combinatorics uses techniques from complex analysis to solve problems in enumerative combinatorics, specifically to find asymptotic estimates for the coefficients of generating functions.
History
One of the earliest uses of analyti ...
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Combinatorics and physics Combinatorial physics or physical combinatorics is the area of interaction between physics and combinatorics.
Overview
:"Combinatorial Physics is an emerging area which unites combinatorial and discrete mathematical techniques applied to theoretica ...
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Arithmetic dynamics
Arithmetic dynamics 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 Iterated function, iteration of self-maps of the complex plane or o ...
References
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External links
Combinatorics of Iterated Functions: Combinatorial Dynamics & Dynamical CombinatoricsCombinatorial dynamics
at Scholarpedia
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