Dimension Theory of some non-Markovian repellers Part II: Dynamically defined function graphs

Dimension Theory of some non-Markovian repellers Part I: A gentle introduction

Michael Barnsley introduced a family of fractals sets which are repellers of piecewise affine systems. The study of these fractals was motivated by certain problems that arose in fractal image compression but the results we obtained can be applied for the computation of the Hausdorff dimension of the graph of some functions, like generalized Takagi functions and fractal interpolation functions. In this paper we introduce this class of fractals and present the tools in the one-dimensional dynamics and nonconformal fractal theory that are needed to investigate them. This is the first part in a s…

动力学定义的 Cantor 集的 Hausdorff 维数新界

New bounds for the Hausdorff dimension of a dynamically defined Cantor set

本文利用加性热力学形式，为光滑流形上分段扩张映射定义的非共形双曲排斥子推导出 Hausdorff 维数与计盒维数的新界。

Warning signs for non-Markovian bifurcations: colour blindness and scaling laws

Warning signs for tipping points (or critical transitions) have been very actively studied. Although the theory has been applied successfully in models and in experiments for many complex systems such as for tipping in climate systems, there are ongoing debates, when warning signs can be extracted from data. In this work, we shed light on this debate by considering different types of underlying noise. Thereby, we significantly advance the general theory of warning signs for nonlinear stochastic dynamics. A key scenario deals with stochastic systems approaching a bifurcation point dynamically u…