We establish an asymptotic formula for the number of integer solutions to the Markoff-Hurwitz equation \[ x_{ 1}^{ 2}+x_{ 2}^{ 2}+\ldots+x_{ n}^{ 2}=ax_{ 1}x_{ 2}\ldots x_{ n}+k. \] When $n\geq4$ the previous best result is by Baragar (1998) that gives an exponential rate of growth with exponent $\beta$ that is not in general an integer when $n\geq 4$. We give a new interpretation of this exponent of growth in terms of the unique parameter for which there exists a certain conformal measure on projective space.
Categories: math.NT math.DS math.GR