The Takagi Curve and the $β$-Cantor Function from Mechanical Laws

Limit laws of entrance times for low complexity Cantor minimal systems

This paper is devoted to the study of limit laws of entrance times to cylinder sets for Cantor minimal systems of zero entropy using their representation by means of ordered Bratteli diagrams. We study in detail substitution subshifts and we prove these limit laws are piecewise linear functions. The same kind of results is obtained for classical low complexity systems given by non stationary ordered Bratteli diagrams.

具有类康托尔边缘分布的自动回归过程的极大半稳定极值律

Max-semistable extreme value laws for autoregressive processes with Cantor-like marginals

本文研究边缘分布类似康托尔函数的自动回归过程，证明其边缘分布属于极大半稳定分布的吸引域，并通过引入极值指数得到该过程的极值律，同时揭示其与确定性动力系统极值的联系。

Rare events for Cantor target sets

We study the existence of limiting laws of rare events corresponding to the entrance of the orbits on certain target sets in the phase space. The limiting laws are obtained when the target sets shrink to a Cantor set of zero Lebesgue measure. We consider both the presence and absence of clustering, which is detected by the Extremal Index, which turns out to be very useful to identify the compatibility between the dynamics and the fractal structure of the limiting Cantor set. The computation of the Extremal Index is connected to the box dimension of the intersection between the Cantor set and i…

Takagi型函数与动力系统：SBR测度的光滑性及局部时的存在性与光滑性

Takagi type functions and dynamical systems: the smoothness of the SBR measure and the existence and smoothness of local time

本文研究Takagi型函数的占据测度与局部时，通过嵌入面包师变换动力系统，利用SBR测度与伯努利卷积的联系，证明特定参数下局部时的平方可积性。

Extreme value laws for fractal intensity functions in dynamical systems: Minkowski analysis

Typically, in the dynamical theory of extremal events, the function that gauges the intensity of a phenomenon is assumed to be convex and maximal, or singular, at a single, or at most a finite collection of points in phase--space. In this paper we generalize this situation to fractal landscapes, i.e. intensity functions characterized by an uncountable set of singularities, located on a Cantor set. This reveals the dynamical rôle of classical quantities like the Minkowski dimension and content, whose definition we extend to account for singular continuous invariant measures. We also introduce t…