Hausdorff dimension of canonical height spectra

Source abstract

We study the Hausdorff dimension of the closure of the canonical height values associated to finite compositions of two rational maps. For rational maps with common periodic critical points, a nonzero local--global height discrepancy gives a nonempty set of starting points with positive-dimensional spectra. For polynomials, equality of the normalized logarithms of the leading coefficients at every place forces dimension zero; a single-place discrepancy gives positive dimension under boundedness at the other places. We construct rational cubic pairs with spectra of dimension $\log2/\log3$. The spectrum dimension gives a lower bound for the growth exponent of the number of distinct orbit points of bounded height. At the degree exponent, positive Hausdorff measure gives matching growth bounds for multiplicatively dependent degrees. For independent degrees, a restriction of the covering intervals and a Diophantine rotation estimate give a lower bound with logarithmic loss of exponent strictly less than $1/2$. Quadratic monomial examples show that the spectrum dimension can depend on the starting point even when both individual orbits are infinite.