The doubling property of conformal measures of infinite iterated function systems
A Borel probability measure \(m\) on a compact metric space \((Y,\rho)\) satisfies the doubling property provided there exists a constant \(C>0\) and \(\tau>1\) such that \(m(B(x,\tau r))\leq Cm(B(x,r))\) for \(m\)-almost all \(x\in Y\) and all \(r\leq 1\). (This property holds for some \(\tau>1\) if and only if it holds for \(\tau=2\), and this is known as doubling property). Measures with the doubling property naturally arise in the studies of conformal mappings or finite conformal iterated function systems. The authors investigate conformal measures induced by an infinite conformal iterated function system. They find some easily verifiable conditions under which the conformal measure satisfies (or does not satisfy) the doubling property. A particular attention is given to systems generated by continued fractions with restricted entries.
scientific article; zbMATH DE number 1967960