First return times: multifractal spectra and divergence points
The Poincaré recurrence theorem implies that if \(S:X\to X\) is a measure-preserving map on a probability measure space \((X,\mu)\), with \(X\) a metric space, then the first return time of a point \(x\) to the ball \(B(x,r)\): \[ \tau_{B(x,r)}(x)=\inf \{1\leq k\leq n\mid S^kx\in B(x,r)\} \] is finite for \(\mu\)-almost all points \(x\in X\). In the paper under review is analyzed the quantitative behaviour of the first return times associated to the map \(S:K\to K\), where \(K\) is the selfconformal set associated to a selfconformal IFS \((V, X, (S_{i})_{i=1,\ldots N})\), and \(S\) is the natural self map induced by the shift. More precisely, under some conditions \(S(x)=S_{i}^{-1}(x)\) for \(x\in S_{i}(K)\). It is shown that the first return time exponent \[ {\log \tau_{B(x,r)}(x)}\over{-\log r} \] has the property that for any compact interval \(I\) of \((0,\infty)\), the set of points such that for each \(t\in I\) there exists an arbitrary small \(r>0\) for which the first return time \(\tau_{B(x,r)}(x)\) of \(x\) to the ball \(B(x,r)\) behaves like \(1/r^{t}\) has full Hausdorff dimension on any open set. A consequence of this result is that the multifractal formalism fails comprehensively for the first return time multifractal spectrum.
scientific article; zbMATH DE number 2075935