arXiv subjects

Andrew D Burbanks

Publications and source records attributed to Andrew D Burbanks.

3 recordsLinked to original sources

arXiv2021

Rigorous computer-assisted bounds on the period doubling renormalisation fixed point and eigenfunctions in maps with critical point of degree 4

We gain tight rigorous bounds on the renormalisation fixed point for period doubling in families of unimodal maps with degree $4$ critical point. We use a contraction mapping argument to bound essential eigenfunctions and eigenvalues for the linearisation of the operator and for the operator controlling the scaling of added noise. Multi-precision arithmetic with rigorous directed rounding is used to bound operations in a space of analytic functions yielding tight bounds on power series coefficients and universal constants to over $320$ significant figures.

math.DS↗

arXiv2021

Rigorous computer-assisted bounds on renormalisation fixed point functions, eigenfunctions, and universal constants

We gain tight rigorous bounds on the renormalisation fixed point function for period doubling in families of unimodal maps with degree 2 critical point. By writing the relevant eigenproblems in a modified nonlinear form, we use these bounds, together with a contraction mapping argument, to gain tight bounds on the essential eigenvalues and eigenfunctions of the linearised renormalisation operator at the fixed point and also those of the operator encoding the universal scaling of added uncorrelated noise. We gain bounds on the corresponding power series coefficients and universal constants accurate to over 400 significant figures, confirming and (in the case of noise) extending the accuracy of previous numerical estimates, by using multi-precision interval arithmetic with rigorous directed rounding to implement operations on a space of analytic functions.

math.DS↗

arXiv2021

Rigorous bounds on the Hausdorff dimension of Feigenbaum attractors

We calculate rigorous bounds on the Hausdorff dimension of the attractor at the accumulation of the period-doubling cascade for families of maps with quadratic, cubic, and quartic critical point. To do this, we express the attractors as the limit sets of appropriate Iterated Function Systems constructed using rigorous bounds on the corresponding renormalisation fixed point functions. We use interval arithmetic with rigorous directed rounding modes to show that the respective dimensions lie in subintervals of the intervals (0.5370,0.5392), (0.6040,0.6091), and (0.6395,0.6474).

math.DS↗