The fractal dimensions of the spectrum of Sturm Hamiltonian

We also show the following results: s⁎(V) and s⁎(V) are Lipschitz continuous on any bounded interval of [24,∞); the limits s⁎(V)lnV and s⁎(V)lnV exist as V tends to infinity, and the limits are constants only depending on α; s⁎(V)=1 if and only if limsupn→∞(a1⋯an)1/n=∞, which can be compared with the fact: s⁎(V)=1 if and only if liminfn→∞(a1⋯an)1/n=∞ (Liu and Wen, 2004) [13].