Chapter 10.7, Problem 3E

Interpretation Introduction

Interpretation:

To show if g is a solution of the functional equation then $\text{μg}\left(\frac{\text{x}}{\text{μ}}\right)$ with same value of $\text{α}$

Concept Introduction:

• Renormalization is based on the self-similarity of the figtree- the twigs look like the earlier branches, except they are scaled down in both x and r directions. The figtree structure shows an endless repetition of the same dynamical processes, a ${\text{2}}^{\text{n}}\text{-cycle}$ is created, it becomes superstable and loses stability in a period doubling bifurcation.

• Self-similarity mathematically expressed as, comparing $\text{f}$ with second iterate ${\text{f}}^{\text{2}}$ at corresponding values of r and then renormalize one map into other.

• The function $\text{f}$ can be renormalized by taking its second iterate, rescale $\text{x}\to \frac{\text{x}}{\text{α}}$ and shift the value of r to next superstable value.

• The functional equation for $\text{g(x)}$ is,

  ${\text{g(x) = αg}}^{\text{2}}\left(\frac{\text{x}}{\text{α}}\right)$

  Here, $\text{α}$ is universal scale factor and $\text{g(x)}$ is defined in terms of itself.