Pure Maths Colloquium: Catherine Bruce

In 2012 Hochman and Shmerkin proved that, given Borel probability measures on \([0,1]\) invariant under multiplication by 2 and 3 respectively, the Hausdorff dimension of the orthogonal projection of the product of these measures is equal to the maximum possible value in every direction except the horizontal and vertical directions. Their result holds beyond multiplication by 2,3 to natural numbers m,n which are multiplicatively independent. We discuss a generalisation of this theorem to include random cascade measures on subsets of \([0,1]\) invariant under multiplication by multiplicatively independent \(m\),\(n\). We will define random cascade measures in a heuristic way, as a natural randomisation of invariant measures on symbolic space. The theorem of Hochman and Shmerkin fully resolved a conjecture of Furstenberg originating in the late 1960s concerning sumsets of these invariant sets. We apply our main result to present a random version of this conjecture which holds for products of percolations on \(\times m, \times n\)-invariant sets.