Kaplan–Yorke conjecture

In applied mathematics, the Kaplan–Yorke conjecture concerns the dimension of an attractor, using Lyapunov exponents. By arranging the Lyapunov exponents in order from largest to smallest ${\displaystyle \lambda _{1}\geq \lambda _{2}\geq \dots \geq \lambda _{n}}$, let j be the largest index for which

${\displaystyle \sum _{i=1}^{j}\lambda _{i}\geqslant 0}$

and

${\displaystyle \sum _{i=1}^{j+1}\lambda _{i}<0.}$

Then the conjecture is that the dimension of the attractor is

${\displaystyle D=j+{\frac {\sum _{i=1}^{j}\lambda _{i}}{|\lambda _{j+1}|}}.}$

This idea is used for the definition of the Lyapunov dimension.