Fix a positive integer N ≥ 2. For a real number x ∈ [0, 1] and a digit i ∈ {0, 1,..., N − 1}, let Π i (x, n) denote the frequency of the digit i among the first nN-adic digits of x. It is well-known that for a typical (in the sense of Baire) x ∈ [0, 1], the sequence of digit frequencies diverges as n →∞. In this paper we show that for any regular linear transformation T there exists a residual set of points x ∈ [0,1] such that the T -averaged version of the sequence (Π i (x, n)) n also diverges significantly.

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<p style='text-indent:20px;'>In this article, we consider a counting problem for orbits of hyperbolic rational maps on the Riemann sphere, where constraints are placed on the multipliers of orbits. Using arguments from work of Dolgopyat, we consider varying and potentially shrinking intervals, and obtain a result which resembles a local central limit theorem for the logarithm of the absolute value of the multiplier and an equidistribution theorem for the holonomies.</p>

Fix a positive integer N ≥ 2. For a real number x ∈ [0, 1] and a digit i ∈ {0, 1, ..., N − 1}, let Πi(x, n) denote the frequency of the digit i among the first n N -adic digits of x. It is well-known that for a typical (in the sense of Baire) x ∈ [0, 1], the frequencies diverge as n → ∞. In this paper we provide a substantial strengthening of this result. Namely, we show that for a typical x ∈ [0, 1] any regular linear average of the sequence (Πi(x, n))n also diverges spectacularly.