bo198214 Wrote:What also burningly interests me how this looks with other functions that have an attractive fixed point.
I think for our example =x^2+x-1/16)
with fixed points 
this does no more work, that the eigenvalues of the Carleman matrices converge to the powers of the derivative at one fixed point.
Well, I see, that the sequence of sets of eigenvalues of truncations of increasing size of the Carleman-matrix C is not stabilizing. But we have this with the Bb-matrices as well, if we use b outside 1..e^(1/e) - so I don't think we have really a problem here. The canned eigensystem-solver do not work analytically, but with polynomials based on the truncation-size, giving best fitting results for that sizes.
The trace of C is a divergent sum; however, with Cesaro-sum of increasing orders, I get stabilization at ~ 2.0 from order 4.5, where order 1.0 means direct summation. (Surprisingly, Euler-summation seems not to applicable here). This result would back the assumtion of eigenvalues consisting of the infinite set (1,1/2,1/4,1/8,...)
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I've done some eigenanalysis using the fixpoint-shifts, where fixpoint 1 is t0 = -1/4 and fixpoint 2 is t1=+1/4.
Call C the transposed Carleman-matrix for your function f(x), such that
\sim * C = V(f(x))\sim )
as usual.
The triangular matrices U0 and U1 are generated by the functions g0(x) and g1(x), which represents the fixpoint-shifted versions, such that
 = g_0(x')'' )
or
\sim * U_0 = V(f(x)')\sim )
where
and
or
 = g_1(x')'' )
or
\sim * U_1 = V(f(x)')\sim )
where
and
(' and " indicate the appropriate shifting).
Then:
U0 has the eigenvalues
 )
U1 has the eigenvalues
 )
With the assumtion, that these eigenvalues hold also as possible solutions for C, we need to show, that we can find infinitely many invariant vectors X_k such that
If I generate such eigenvectors via U0 resp U1 I get the following possible results (consistent with the assumtions of eigensystem-decomposition)
a1) U0 = W0^-1 * D0 * W0
or
a2) W0 * U0 = D0 * W0
b1) U1 = W1^-1 * D1 * W1
or
b2) W1 * U1 = D1 * W1
and rearranging the fixpoint-shift
c1) C = X0^-1 * D0 * X0 = X1^-1 * D1 * X1
or in vector-invariance-notation:
c2a) X0*C = D0 * X0
c2b) X1*C = D1 * X1
With fixpoint t0(=-1/4) I show the first 3 invariant vectors X0[0..2,0..inf]
Code:
´
1 -1/4 1/16 -1/64 1/256 -1/1024 ...
0 1 -1/2 3/16 -1/16 5/256 ...
0 -4 3 -3/2 5/8 -15/64 ...
such that
X0[0,] * C~ = 1 *X0[0,]
X0[1,] * C~ = 1/2*X0[1,]
X0[2,] * C~ = 1/4*X0[2,]
and with fixpoint t1(=1/4) I show the first 3 invariant vectors X1[0..2,0..inf]
Code:
´
1 1/4 1/16 1/64 1/256 1/1024 ...
0 1 1/2 3/16 1/16 5/256 ...
0 -4/3 1/3 1/2 7/24 25/192 ...
such that
X1[0,] * C~ = 1 *X1[0,]
X1[1,] * C~ = 3/2*X1[1,]
X1[2,] * C~ = 9/4*X1[2,]
which -when continued- proves, that both assumtions for eigenvalues of C lead to possible solutions for the eigenvectors.
The schroeder-functions
)
)
and their inverses are nasty; to determine their powerseries-coefficients we need for all except
divergent summation of high order as long as we don't have analytical expressions, so not only the result of the schroeder-functions are depending on truncation, but even their coefficents themselves are more or less good approximations. Only for
we have coefficients, based on evaluation of convergent series...
