The following function

$$f(x)=-2 \cos \left(\sqrt{2} \arccos \left(\frac{x-1}{2}\right)\right)+1$$

has interesting property to form a continuous curve with its own integer iterations. The following image illustrates this property:

Here blue is f(x), red is f(f(x)), yellow is f(f(f(x))) and green is f(f(f(f(x)))). It seems that all these functions form a continuous, and, probably, smooth curve.

The question is what is the general criterion for a function to have such property. Can you point some more examples of functions with such property?

P.S. If to use the folowing function $$f(x)=-2 \cos \left(\sqrt{2} \arccos \left(\frac{x-1}{2}\right)\right)+1.1$$ the curve becomes as below with one continuous branch and numerous closed circular branches.

For function $$f(x)=-2 \cos \left(\sqrt{2} \arccos \left(\frac{x-1}{2}\right)\right)+0.9$$ the curve is completely continuous and seems to be smooth.