Combinatorics and dynamical systems
Encyclopedia
The mathematical disciplines of combinatorics
and dynamical system
s interact in a number of ways. The ergodic theory
of dynamical systems has recently been used to prove combinatorial theorems about number theory which has given rise to the field of arithmetic combinatorics
. Also dynamical systems theory
is heavily involved in the relatively recent field of 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
.
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
and dynamical system
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
s interact in a number of ways. The ergodic theory
Ergodic theory
Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
of dynamical systems has recently been used to prove combinatorial theorems about number theory which has given rise to the field of arithmetic combinatorics
Arithmetic combinatorics
Arithmetic combinatorics arose out of the interplay between number theory, combinatorics, ergodic theory and harmonic analysis. It is about combinatorial estimates associated with arithmetic operations...
. Also dynamical systems theory
Dynamical systems theory
Dynamical systems theory is an area of applied mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems. When difference...
is heavily involved in the relatively recent field of combinatorics on words
Combinatorics on words
Combinatorics on words is a branch of mathematics which applies combinatorics to words and formal languages. The study of combinatorics on words arose independently within several branches of mathematics, e.g. number theory, group theory and probability...
. Also combinatorial aspects of dynamical systems are studied. Dynamical systems can be defined on combinatorial objects; see for example graph dynamical system
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 and the global dynamics that result.The work on...
.
See also
- Symbolic dynamicsSymbolic dynamicsIn mathematics, symbolic dynamics is the practice of modeling a topological or smooth dynamical system by a discrete space consisting of infinite sequences of abstract symbols, each of which corresponds to a state of the system, with the dynamics given by the shift operator...
- Analytic combinatoricsAnalytic combinatoricsAnalytic combinatorics is a branch of combinatorics that describes combinatorial classes using generating functions, with formal power series that often correspond to analytic functions....
- Combinatorics and physicsCombinatorics and physicsCombinatorial physics or physical combinatorics is the area of interaction between physics and combinatorics."Combinatorial Physics is an emerging area which unites combinatorial and discrete mathematical techniques applied to theoretical physics, especially Quantum Theory."."Physical combinatorics...
- Arithmetic dynamicsArithmetic dynamicsArithmetic dynamicsis a field that amalgamates two areas of mathematics, dynamical systems and number theory.Classically, discrete dynamics refers to the study of the iteration of self-maps of the complex plane or real line...