Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Hidden Positivity and a New Approach to Numerical Computation of
  Hausdorff Dimension: Higher Order Methods   In [14], the authors developed a new approach to the computation of the
Hausdorff dimension of the invariant set of an iterated function system or IFS.
In this paper, we extend this approach to incorporate high order approximation
methods. We again rely on the fact that we can associate to the IFS a
parametrized family of positive, linear, Perron-Frobenius operators $L_s$, an
idea known in varying degrees of generality for many years. Although $L_s$ is
not compact in the setting we consider, it possesses a strictly positive $C^m$
eigenfunction $v_s$ with eigenvalue $R(L_s)$ for arbitrary $m$ and all other
points $z$ in the spectrum of $L_s$ satisfy $|z| \le b$ for some constant $b <
R(L_s)$. Under appropriate assumptions on the IFS, the Hausdorff dimension of
the invariant set of the IFS is the value $s=s_*$ for which $R(L_s) =1$. This
eigenvalue problem is then approximated by a collocation method at the extended
Chebyshev points of each subinterval using continuous piecewise polynomials of
arbitrary degree $r$. Using an extension of the Perron theory of positive
matrices to matrices that map a cone $K$ to its interior and explicit a priori
bounds on the derivatives of the strictly positive eigenfunction $v_s$, we give
rigorous upper and lower bounds for the Hausdorff dimension $s_*$, and these
bounds converge rapidly to $s_*$ as the mesh size decreases and/or the
polynomial degree increases.

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The categories of this paper are: chao-dyn
5
The categories of this paper are: gr-qc
32
The categories of this paper are: atom-ph
15
The categories of this paper are: nucl-ex
28
The categories of this paper are: alg-geom
20
The categories of this paper are: acc-phys
11
The categories of this paper are: hep-lat
30
The categories of this paper are: q-alg
26
The categories of this paper are: bayes-an
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The categories of this paper are: eess
25
The categories of this paper are: patt-sol
18
The categories of this paper are: econ
31
The categories of this paper are: ao-sci
37
The categories of this paper are: supr-con
12
The categories of this paper are: nucl-th
4
The categories of this paper are: cond-mat