Fixed point and scale invariance

[Einj]

Hello everyone. I'm studying the fixed point of theory in the context of QFT. First of all, let me say what I think I understood about fixed points and then I'll state my question.
Suppose we have a theory with a certain running coupling $\lambda(\mu)$. If we have, for example, an UV fixed point, say $\lambda^*$, this means that when the energy scale increases the coupling will converge towards this value and hence the theory is defined at arbitraty high energy since it remains meaningful.
In the Wilsonian point of view, everytime that we change our energy scale we are introducing a new theory with a new Lagrangian. In this languange a UV fixed point is that Lagrangian towards which every other Lagrangian converge when the energy scale increases.

First of all: is this correct?

Secondly, I found in my places that the theory at the fixed point is scale invariant. Can anyone explain to me why?

Thanks a lot
Cheers

 

[RGevo]

In this example, The fixed point is the position where the beta function for the coupling is zero.

Therefore it is scale independent by definition!

 

[atyy]

In this example, The fixed point is the position where the beta function for the coupling is zero.

Therefore it is scale independent by definition!

RGevo is presumably not at the fixed point? :D