First return times: multifractal spectra and divergence points

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 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 less than or equal to k less than or equal to n \ S(k)x is an element of B(x,r)},

of a point x to the ball B(x, r) as r tends to 0. For a function phi : (0, infinity) --> R, let A(phi(r)) denote the set of accumulation points of phi(r) as r SE arrow 0. We show that the first return time exponent, logtau(B(x,r))(x)/-logr , has an extremely complicated and surprisingly intricate structure: for any compact subinterval I of (0, infinityo), the set of points x such that for each t is an element of 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.

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for any open set G with G boolean AND K not equal 0. 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.