Conjecture
Let \( C_N \) be the Carleman matrix of \( (x+s)^p-s \) (truncated to N rows and columns), \( s>0 \), \( p>0 \) real.
Then the set of eigenvalues of \( C_N \) converges to the set \( \{p^k:k\ge 0\} \) for \( N\to\infty \) in the sense that there exist an enumeration \( v_{N,k} \) of the Eigenvalues of \( C_N \) such that \( \lim_{N\to\infty} v_{N,k} = p^k \) for each \( k \).
Discussion
This is about the function \( f(x)=x^p \) shifted by \( s \).
The fixed point 0 is a singularity for \( f \) (for non-natural \( p \)), so \( f \) has to be developed at the different point \( s \).
In the particular case \( s=1 \) we have the fixed point at 0 and the first derivative is \( p \). So the Carleman matrix is triangular and we can solve it exactly, getting \( f^{\circ t}(x)=x^{p^t} \).
The conjecture is again about the independence of the matrix function method with respect to the development point.
\( f \) can even be developed at the fixed point 0 in the particular case \( p\in\mathbb{N} \). However in this case \( f'(0)=0 \) except \( p=1 \) and regular iteration can not be applied, which makes sense as \( x^{p^t} \) can for most t not be developed at 0.
Let \( C_N \) be the Carleman matrix of \( (x+s)^p-s \) (truncated to N rows and columns), \( s>0 \), \( p>0 \) real.
Then the set of eigenvalues of \( C_N \) converges to the set \( \{p^k:k\ge 0\} \) for \( N\to\infty \) in the sense that there exist an enumeration \( v_{N,k} \) of the Eigenvalues of \( C_N \) such that \( \lim_{N\to\infty} v_{N,k} = p^k \) for each \( k \).
Discussion
This is about the function \( f(x)=x^p \) shifted by \( s \).
The fixed point 0 is a singularity for \( f \) (for non-natural \( p \)), so \( f \) has to be developed at the different point \( s \).
In the particular case \( s=1 \) we have the fixed point at 0 and the first derivative is \( p \). So the Carleman matrix is triangular and we can solve it exactly, getting \( f^{\circ t}(x)=x^{p^t} \).
The conjecture is again about the independence of the matrix function method with respect to the development point.
\( f \) can even be developed at the fixed point 0 in the particular case \( p\in\mathbb{N} \). However in this case \( f'(0)=0 \) except \( p=1 \) and regular iteration can not be applied, which makes sense as \( x^{p^t} \) can for most t not be developed at 0.