Defining an Integral for a Map x → g(x)

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SUMMARY

The discussion centers on defining an integral for a mapping of the form x → g(x), where x is an element of R². A key conclusion is that an integral in this context is characterized as a non-constant mapping that remains invariant on the forward orbit, effectively serving as a conservative law for the difference equation xn+1 = g(xn). This definition emphasizes the relationship between integrals and the behavior of mappings in dynamical systems.

PREREQUISITES
  • Understanding of R² and its properties
  • Familiarity with mappings and functions in mathematics
  • Knowledge of difference equations
  • Concept of invariance in dynamical systems
NEXT STEPS
  • Research the properties of non-constant mappings in R²
  • Study the concept of invariance in dynamical systems
  • Explore the application of conservative laws in difference equations
  • Learn about integrals in the context of functional analysis
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Mathematicians, students studying dynamical systems, and anyone interested in the theoretical aspects of integrals and mappings in higher dimensions.

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Homework Statement


Could someone define the notion of an integral for a map,

x → g(x), x element of R2
or
xn+1=g(xn

thanks
 
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funcosed said:

Homework Statement


Could someone define the notion of an integral for a map,

x → g(x), x element of R2
or
xn+1=g(xn

thanks


Were you able to find the solution to this problem? Thanks
 
An integral is a non-constant mapping which remains invarient on the forward orbit i.e. it is a conservative law for the difference equation.
 

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