ACTA issues

Markoff constants for Cantor sets

Vilmos Totik

Acta Sci. Math. (Szeged) 60:3-4(1995), 715-734

Abstract. It is proven, that for Cantor type sets the Markoff property (i.e. the fact that the Markoff constants are of polynomial growth) is equivalent to the Hölder continuity property of the associated Green function, which in turn is equivalent to the fact that the length of the generating intervals at level $n$ is not smaller than $\alpha ^n$ for some $\alpha >0$. It is also shown that for regular Cantor sets the fastest possible increase of the Markoff constants is subexponential.

AMS Subject Classification (1991): 26B10, 31A15

Received October 19, 1994, and in revised form March 21, 1995. (Registered under 5661/2009.)