July  2004, 10(3): 635-656. doi: 10.3934/dcds.2004.10.635

First return times: multifractal spectra and divergence points

1. 

Department of Mathematics, University of St. Andrews, St. Andrews, Fife KY16 9SS, Scotland, United Kingdom

Received  December 2002 Revised  June 2003 Published  January 2004

We provide a detailed study of the quantitative behavior of first return times of points to small neighborhoods of themselves. Let $K$ be a self-conformal set (satisfying a certain separation condition) and let $S:K\to K$ be the natural self-map induced by the shift. We study the quantitative behavior of the first return time,

$ \tau_{B(x,r)}(x)=$inf{$1\le k \le n | S^k x \in B(x,r)},$

of a point $x$ to the ball $B(x,r)$ as $r$ tends to $0$. For a function $\varphi:(0,\infty)\to\mathbb R$, let A$(\varphi(r))$ denote the set of accumulation points of $\varphi(r)$ as $r\to 0$. We show that the first return time exponent, $\frac{\log\tau_{B(x,r)}(x)} {-\log r}$, has an extremely complicated and surprisingly intricate structure: for any compact subinterval $I$ of $(0,\infty)$, the set of points $x$ such that for each $t\in I$ there exists arbitrarily small $r>0$ for which the first return time $\tau_{B(x,r)}(x)$ of $x$ to the neighborhood $B(x,r)$ behaves like $1/r^t$, has full Hausdorff dimension on any open set, i. e.

dim$(G\cap ${$x\in K| $A ($\frac {\log\tau_{B(x,r)}(x)}{-\log r}) =I$}) $=$dim $K$

for any open set $G$ with $G\cap K$≠$\emptyset$. As a consequence we deduce that the so-called multifractal formalism fails comprehensively for the first return time multifractal spectrum. Another application of our results concerns the construction of a certain class of Darboux functions.

Citation: Lars Olsen. First return times: multifractal spectra and divergence points. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 635-656. doi: 10.3934/dcds.2004.10.635
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