Hi -

I'm considering the properties of the regular iteration after fixpoint-shift.

To have easier conditions I look at the function f(x) = x^2 - 0.5 and its iterates.

It has the fixpoints and .

I compute the half-iterate f°0.5(x) by translating f(x) -> g(x - x_a) + x_a and determine the powerseries for g°0.5(x)

The result is interesting; the curve for the half-iterate meets the first fixpoint, but misses the second. Instead, the series diverges in that region.

However, using the more reliable results of f°0.5(x) at other values of x, I can construct a bit of continuation, which gives one winding around (xb,xb).

Naively, I'd expected that the function crossed both fixpoint but it seems, that this needed a completely different powerseries, and thus a modified procedure.

I'm without an idea currently, how to proceed here. Does anyone have a comment?

Here are two plots:

a) an overview. Integer iterates f°-1(x), f°0, f°1,f°2, f°3,f°4 and the regular f°0.5 iteration based on the powerseries representation

b) a detail, f°0.5 and f°1 in the vicinity of (xb,xb) and a continuation based on more reliable results of f°0.5 at other values of x

Gottfried

I'm considering the properties of the regular iteration after fixpoint-shift.

To have easier conditions I look at the function f(x) = x^2 - 0.5 and its iterates.

It has the fixpoints and .

I compute the half-iterate f°0.5(x) by translating f(x) -> g(x - x_a) + x_a and determine the powerseries for g°0.5(x)

The result is interesting; the curve for the half-iterate meets the first fixpoint, but misses the second. Instead, the series diverges in that region.

However, using the more reliable results of f°0.5(x) at other values of x, I can construct a bit of continuation, which gives one winding around (xb,xb).

Naively, I'd expected that the function crossed both fixpoint but it seems, that this needed a completely different powerseries, and thus a modified procedure.

I'm without an idea currently, how to proceed here. Does anyone have a comment?

Here are two plots:

a) an overview. Integer iterates f°-1(x), f°0, f°1,f°2, f°3,f°4 and the regular f°0.5 iteration based on the powerseries representation

b) a detail, f°0.5 and f°1 in the vicinity of (xb,xb) and a continuation based on more reliable results of f°0.5 at other values of x

Gottfried

Gottfried Helms, Kassel