Mathematics – Dynamical Systems
Scientific paper
2012-04-09
Mathematics
Dynamical Systems
13 pages, 1 figure
Scientific paper
For a map $S: X\to X$ and an open set $H\subset X$ we define $J_H(S)$ to be the set of points in $X$ whose $S$-orbit avoids $H$. We say that a hole $H_0$ is supercritical if (i) for any hole $H$ such that $\bar{H_0}\subset H$ the set $J_H(S)$ is either empty or contains only fixed points of $S$; (ii) for any hole $H$ such that $\barH\subset H_0$ the Hausdorff dimension of $J_H(S)$ is positive. The purpose of this note is to show that for any $\beta\in(1,2]$ there exists a continuum of supercritical holes for the $\beta$-transformation which are naturally parametrized by Sturmian words. We also prove that any cyclic Pisot automorphism of a torus has a supercritical hole (subject to a standard number-theoretic conjecture). Furthermore, we show that a supercritical hole can be arbitrarily large. Our methods are symbolic in nature.
No associations
LandOfFree
Supercritical holes does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Supercritical holes, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Supercritical holes will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-652180