Embedded continued fractals and their Hausdorff dimension (Q1114796)

Let \(\alpha\) be irrational and \(g\) be a given monotone decreasing function. A construction is given which attaches to a number \(k\) a polygon \(Q_k\). Under certain hypotheses on \(\alpha\) and \(g\) a subsequence of \((Q_k)\) converges to a continuous limit curve \(Q\). If \(\alpha^2+\alpha N=1\), where \(N\) is an even number, then the curve \(Q\) is embedded for special choices of \(g\) and its Hausdorff dimension can be computed. The main tool is the continued fraction expansion of \(\alpha\) which fact obviously is the reason why the curve \(Q\) is called a ``continued fractal''.