Theory of soft iterated composition (flow)

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Contents

Author

mel viso


License

Released under the GNU Free Documentation License.

Preface

This theory comes from my time as student for a degree in physics. The exposition don't follows the standard notation. Please note that this page only expose results, not demonstrations. All the expressions included in this page have passed computational tests. The experimental validity of such expressions seems to be more general than what we can hope. I'm no time for determining the exact conditions for the validity of the exposed results. If someone wants to determine these conditions or another kind of collaboration, the help is welcomed.

Notation

Let Image:im000.png with Image:im001.png. Then, Image:im002.png we denote by Image:im003.png the n times composition of the function, that is : Image:im004.png

When possible, we can use real, complex or vectorial variables for the index of composition, as we explain later, instead of the integer values. These cases are the target of this theory. When these cases happens, we use the same notation as for integer indices.


Soft iterated composition for real functions

Let Image:im005.png be a real function and suppose that there it exists some real function p for what

Image:im006.png


happens with some norm (like uniform norm). If that function exists (can be non unique) we named it as compositional logarithm of f, and we denotate


Image:im007.png


properties of compositional logarithm of real functions

  • Logarithmic meaning :
    Image:im002.png ; Image:im008.png


  • Characteristic equation : If f is differentiable and the composition index can be real, the following equation must be satisfied:


Image:im009.png


  • Reconstruction series : In the above circumstances the following series, when convergents, reconstructs the composition for arbitrary real (or even complex) index:


Image:im010.png


  • Compositional index function : If it exist the function H defined by


Image:im011.png


we named it as compositional index of f. It verifies the equation Image:im012.png. When H is biyective, we can direct write Image:im013.png


  • Fixed point approximation : For a fixed point (f(x0)=x0) we have the properties


Image:im014.png


Image:im015.png


With these properties and the characteristic equation, Taylor's series can be obtained for the compositional logarithm.


  • Asymptotic approximations : Let f be a biyective function Image:im005.png for what it exists a function Image:im016.png whose composition can be easily extended to real values and verifiying that


Image:im017.png


for some x in R. For each closed set of points x verifiying this property, we can write the composition of the function f for arbitrary real composition index as


Image:im018.png

examples

Asymptotic approximations

In the sourceforge's page MandelZoom you can find a example of asymptotic aproximation for the composition of complex functions with the form Image:im019.png using Image:im020.png as composition extender in the out set. The following image shows the aspect for the soft iterated composition translated to soft colour changes.


Image:im021.png

Notes & questions

  • Conmutative property for function members of the same compositional family can be used to develop expansion series even with rational or real coefficients. For univariate real polinomials of N degree, expansions with highest power K will match a ln(K)/ln(N) composition order.
  • For compositions inside real N dimensional space, the compositional logarithm is replaced by a NxN matrix, the order of composition is not a real value but a N-dim vector, and there are at least N independent, conmutative functions corresponding to each index of this vector.
  • ¿Can be extended this kind of composition to operators in the Hilbert space? If true, we can expect a Hilbert operator as order of composition, a dense collection of conmutative base operators.
  • The superposition principle used in physics is always present in a composition family. ¿How deep is the relation between them?
  • Althought I have sucessfuly applied conmutativity as requiriment to obtain polinomial expansions of fractional composition orders, ¿what are the conditions for a function that allows the equivalence between conmutativity with another and to be composition related to this one?
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