Generalizing recursion in mapping functions

[Pythagorean]

I have a mapping function:

$$x_{n+1} = \mu (1-x_n)$$

I have some condition that occurs for:

$$\mu (1-x_0) > 1$$ (1)

which is:

$$x_0 < 1- \frac{1}{\mu}$$

but via the map function, there's an initial condition that leads to the above solution:

**UNDER CONSTRUCTION, ERROR FOUND**

 

[Pythagorean]

Well... "solving" my error just confuses me more. Mapping functions with a shift in them are really unintuitive to me. So, from the top:

Given:

$$x_{n+1} = \mu (1-x_n)$$

Some special condition occurs at:

$$x_{n+1} = \mu (1-x_n)$$

Which, in terms of initial condition, means that if:

$$x_0 < 1- \frac{1}{\mu}$$

than the condition will be met. HOWEVER, there are also initial conditions that will map to the above space. How do I find them? For instance, I can reverse and apply the order of operations in the mapping functions (inverse scale, then shift opposite):

$$x_0 < \frac{1- \frac{1}{\mu}}{\mu}+1$$

but this gets unwieldy as I try to go back more and more iterations. Is there an inductive way to represent this "reverse mapping" series as a funciton of n (the number of iterations the mapping function requires). Or am I going about this all wrong?

 

[Stephen Tashi]

HOWEVER, there are also initial conditions that will map to the above space.

It's unclear what you mean by mapping a condition to space. What space?

 

[Pythagorean]

the region defined by
$$x_0 < 1- \frac{1}{\mu}$$

 

[Stephen Tashi]

I don't understand what you mean by "mapping" the condition to the space since it is the condition that defines the space.

(If your are trying to ask a question about the attractors in the domain of iterated functions, it would be best to use the standard terminology for that subject - or give a link to a page that explains your question.)

 

[Pythagorean]

Well, I am, but I was trying to just focus on the micro-issue I'm having. I just want to find the basin of attraction for that region I defined, but I am not sure if its really an attractor (it goes to infinity; this is the right side of the tent map for mu>2).

Anyway, I will spend more time on it and if I don't get it, I will reformulate the question in more detail later.