In this article, we study an analogue of $tt$-reducibility for points in computable metric spaces. We characterize the notion of the metric $tt$-degree in the context of first-level Borel isomorphism. Then, we study this concept from the perspectives of effective topological dimension theory and of effective fractal dimension theory.

We study the Hausdorff dimension spectrum for Lyapunov exponents for a class of interval maps which includes several non-hyperbolic situations. We also analyze the level sets of points with given lower and upper Lyapunov exponents and, in particular, with zero lower Lyapunov exponent. We prove that the level set of points with zero exponent has full Hausdorff dimension, but carries no topologic...

Let T be the differential field of transseries. We establish some basic properties of the dimension of a definable subset of Tn, also in relation to its codimension in the ambient space Tn. The case of dimension 0 is of special interest, and can be characterized both in topological terms (discreteness) and in terms of the Herwig-Hrushovski-Macpherson notion of co-analyzability.