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We study the decimation to a sublattice of half the sites, of theone-dimensional Dyson-Ising ferromagnet with slowly decaying long-range pairinteractions of the form $\frac{1}{{|i-j|}^{\alpha}}$, in the phase transitionregion (1< $\alpha \leq$ 2, and low temperature). We prove non-Gibbsianness ofthe decimated measure at low enough temperatures by exhibiting a point ofessential discontinuity for the finite-volume conditional probabilities ofdecimated Gibbs measures. Thus result complements previous work provingconservation of Gibbsianness for fastly decaying potentials ($\alpha$ > 2) andprovides an example of a "standard" non-Gibbsian result in one dimension, inthe vein of similar resuts in higher dimensions for short-range models. We alsodiscuss how these measures could fit within a generalized (almost vs. weak)Gibbsian framework. Moreover we comment on the possibility of similar resultsfor some other transformations.

Decimation of the Dyson-Ising Ferromagnet