By Juan M. Delgado Sanchez, Tomas Dominguez Benavides

This quantity contains a suite of articles via best researchers in mathematical research. It presents the reader with an intensive assessment of the present-day examine in numerous parts of mathematical research (complex variable, harmonic research, genuine research and sensible research) that holds nice promise for present and destiny advancements. those evaluate articles are hugely necessary if you are looking to know about those issues, as many effects scattered within the literature are mirrored throughout the many separate papers featured herein.

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Gillespie, Bilinear Hilbert transform on measure spaces, J. Fourier Anal. and Appl. 11 (2005), 459–470. 5. O. Blasco and F. Villarroya, Transference of bilinear multipliers on Lorentz spaces, Illinois J. Math. 47(4) (2005), 1327–1343. 6. R. R. Coifman and Y. Meyer, Fourier Analysis of multilinear convolution, Calder´ on theorem and analysis of Lipschitz curves, Euclidean Harmonic Analysis (Proc. Sem. Univ. , Md) Lecture Notes in Math. 779 (1979), 104–122. 7. R. R. Coifman and W. Weiss, Transference Methods in Analysis, Regional Conf.

2 was based on the method of majorant series, that requires finding a convergent series whose coefficients are greater than the coefficients of the formal linearization. A different proof is in the spirit of the so-called Kolmogorov–Arnold–Moser (or KAM) method (see [15]). Unfortunately, both proofs (as well as the proofs of the next two theorems) are well beyond the scope of this survey. A bit of terminology is now useful: if f ∈ End (C, 0) is elliptic, we shall say that the origin is a Siegel point if f is holomorphically linearizable; otherwise, it is a Cremer point.

3 (Ecalle, 1981 [10,11]; Voronin, 1981 [36]). Let f , g ∈ End (C, 0) be two holomorphic local dynamical systems tangent to the identity. Then f and g are holomorphically locally conjugated if and only if they have the same multiplicity, the same index and the same sectorial invariant. Furthermore, for any r ≥ 1, β ∈ C and µ ∈ Mr there exists f ∈ End (C, 0) tangent to the identity with multiplicity r + 1, index β and sectorial invariant µ. 2. In particular, holomorphic local dynamical systems tangent to the identity give examples of local dynamical systems that are topologically conjugated without being neither holomorphically nor formally conjugated and of local dynamical systems that are formally conjugated without being holomorphically conjugated.