Chapter 11.1, Problem 6E

Interpretation Introduction

Interpretation:

To show that the decimal shift map has countably many periodic stable orbits.

To show that the map has uncountably many aperiodic orbits.

To show that the eventual fixed point of the map is a point that iterates to a fixed point after a finite number of steps.

Concept Introduction:

• The fixed point of a differential equation is a point where $\text{f(x*) = 0 ; while substitution f(x*) = }\dot{\text{x}}\text{ is used and x\&*\#x00A0;is a fixed point}$.

• Countable and uncountable sets, where the notion created by the cantor will be followed.

  Different infinite sets can be compared as follows, considering the basis set N as the set of natural numbers. If set A has one - to – one correspondence, i.e. every element in set A can be mapped to one and only one element of set N, then set A is called countable, otherwise it is uncountable.