Conformal trace theorem for Julia sets of quadratic polynomials

Abstract

If $c$ is in the main cardioid of the Mandelbrot set, then the Julia set $J$ of the map $\unicode[STIX]{x1D719}_{c}:z\mapsto z^{2}+c$ is a Jordan curve of Hausdorff dimension $p\in [1,2)$. We provide a full proof of a formula for the Hausdorff measure on $J$ in terms of singular traces announced by the first named author in 1996.

Quantum Differentiability on Quantum Tori

“…Connes introduced the quantised calculus in [7] as an analogue of the algebra of differential forms in a noncommutative setting, and later explored the link with the action functional of Yang-Mills theory [8]. Connes successfully applied quantised calculus in computing the Hausdorff measure of Julia sets and limit sets of Quasi-Fuchsian groups in the plane [9, Chapter 4, Section 3.γ] (for a more recent exposition see [14,12]).…”

Quantum Differentiability on Noncommutative Euclidean Spaces

“…Connes introduced the quantised calculus in [8] as a replacement for the algebra of differential forms for applications in a noncommutative setting, and afterwards this point of view found application to mathematical physics [9]. Connes successfully applied his quantised calculus in providing a formula for the Hausdorff measure of Julia sets and for limit sets of Quasi-Fuchsian groups in the plane [10, Chapter 4, Section 3.γ] (for a more recent exposition see [17,14]).…”

On Some Advances in Dynamics of One Variable Complex Functions

“…In [21], a complete rigorous proof of a formula proves for the Hausdorff measure on the Julia set in terms of singular traces. Moreover, there is reconstructed proofs of some existing results for resolving their purposes.…”