Hidden positivity and a new approach to numerical computation of Hausdorff dimension: higher order methods

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