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First return times: multifractal spectra and divergence points

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  • 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.

    Mathematics Subject Classification: 37D20, 28A80.


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