# The universal metric properties of nonlinear transformations

@article{Feigenbaum1979TheUM, title={The universal metric properties of nonlinear transformations}, author={Mitchell J. Feigenbaum}, journal={Journal of Statistical Physics}, year={1979}, volume={21}, pages={669-706} }

AbstractThe role of functional equations to describe the exact local structure of highly bifurcated attractors ofxn+1 =λf(xn) independent of a specificf is formally developed. A hierarchy of universal functionsgr(x) exists, each descriptive of the same local structure but at levels of a cluster of 2r points. The hierarchy obeysgr−1(x)=−αgr(gr(x/α), withg=limr → ∞ gr existing and obeyingg(x) = −αg(g(x/α), an equation whose solution determines bothg andα. Forr asymptoticgr ∼ g − δ−rh* where δ > 1…

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## References

SHOWING 1-10 OF 10 REFERENCES

Quantitative universality for a class of nonlinear transformations

- Mathematics
- 1978

AbstractA large class of recursion relationsxn + 1 = λf(xn) exhibiting infinite bifurcation is shown to possess a rich quantitative structure essentially independent of the recursion function. The…

Iteration of endomorphisms on the real axis and representation of numbers

- Mathematics
- 1978

We study a class of endomorphisms of the set of real numbers x of the form: ~ 2014~ ~/M, ~ E [0, 2]. The function f is continuous, convex with a single maximum but otherwise arbitrary; A is a real…

Universal metric properties of bifurcations of endomorphisms

- Mathematics
- 1979

Endomorphisms of the real axis with one extremum have some universal metric properties which depend only on their analytic dependence near the extremum (bifurcation velocity, reduction parameter). It…

Universal properties of maps on an interval

- Mathematics
- 1980

We consider itcrates of maps of an interval to itself and their stable periodic orbits. When these maps depend on a parameter, one can observe period doubling bifurcations as the parameter is varied.…

On Finite Limit Sets for Transformations on the Unit Interval

- MathematicsJ. Comb. Theory, Ser. A
- 1973

J. Stat. Phys

- J. Stat. Phys
- 1978

Pomeau, Iterations of Endomorphisms on the Real Axis and Representation of Numbers, Saclay

- 1977

Phys. Rep

- Phys. Rep
- 1974

J. Combinatorial Theory

- J. Combinatorial Theory
- 1973

Bifurcations et Groupe de Renormalisation

- Bifurcations et Groupe de Renormalisation