Chapter 10.3, Problem 8E

Interpretation Introduction

Interpretation:

Consider the map of the unit interval into itself. An orbit $\left\{{\text{x}}_{\text{n}}\right\}$ is said to be dense if it eventually gets arbitrarily close to every point on the interval. Such an orbit has to hop around rather crazily. More precisely, given any $\text{ε >0}$ and any point $\text{p}\in \left[\text{0,1}\right]$, the orbit $\left\{{\text{x}}_{\text{n}}\right\}$ is dense if there is some finite $\text{n}$ such that $\left|{\text{x}}_{\text{n}}\text{- p}\right|\text{ < ε}$.

To construct explicitly a dense orbit for the decimal shift map ${\text{x}}_{\text{n+1}}=10{\text{x}}_{\text{n}}(\mathrm{mod}1)$.

Concept Introduction:

• ➢ The decimal shift map of the unit interval the fixed point is in the rational and irrational form which defined the stability of the map.

• ➢ The decimal shift map of the unit interval $\left[\text{0,1}\right]$ is ${\text{x}}_{\text{n+1}}\text{ = f}\left({\text{x}}_{\text{n}}\right)$.