arXiv subjects

J. J. P. Veerman

Publications and source records attributed to J. J. P. Veerman.

39 records · Page 3Linked to original sources

arXiv1997

Rigidity properties of locally scaling fractals

Local scaling of a set means that in a neighborhood of a point the structure of the set can be mapped into a finer scale structure of the set. These scaling transformations are compact sets of locally affine (that is: with uniformly $α$-Hölder continuous derivatives) contractions. In this setting, without any assumption on the spacing of these contractions such as the open set condition, we show that the measure of the set is an upper semi-continuous of the scaling transformation in the $C^0$-topology. With a restriction on the 'non-conformality' (see below) the Hausdorff dimension is lower semi-continous function in the $C^{1}$-topology. We include some examples to show that neither of these notions is continuous.

math.DS↗

arXiv1992

Scalings in circle maps III

Circle maps with a flat spot are studied which are differentiable, even on the boundary of the flat spot. Estimates on the Lebesgue measure and the Hausdorff dimension of the non-wandering set are obtained. Also, a sharp transition is found from degenerate geometry similar to what was found earlier for non-differentiable maps with a flat spot to bounded geometry as in critical maps without a flat spot.

math.DS↗

arXiv1991

Dynamics of certain non-conformal degree two maps on the plane

In this paper we consider maps on the plane which are similar to quadratic maps in that they are degree 2 branched covers of the plane. In fact, consider for $α$ fixed, maps $f_c$ which have the following form (in polar coordinates): $$f_c(r\,e^{iθ})\;=\;r^{2α}\,e^{2iθ}\,+\,c$$ When $α=1$, these maps are quadratic ($z \maps z^2 + c$), and their dynamics and bifurcation theory are to some degree understood. When $α$ is different from one, the dynamics is no longer conformal. In particular, the dynamics is not completely determined by the orbit of the critical point. Nevertheless, for many values of the parameter c, the dynamics has strong similarities to that of the quadratic family. For other parameter values the dynamics is dominated by 2 dimensional behavior: saddles and the like. The objects of study are Julia sets, filled-in Julia sets and the connectedness locus. These are defined in analogy to the conformal case. The main drive in this study is to see to what extent the results in the conformal case generalize to that of maps which are topologically like quadratic maps (and when $α$ is close to one, close to being quadratic).

math.DS↗