By Nevanlinna Paatero
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E ∩ {−N, . . , N}) 2N + 1 N →+∞ d Z (E) := lim sup #(E ∩ {−N, . . , N}) . N →+∞ 2N + 1 d Z (E) := lim inf For Δ ∈ R, Δ = 0 we say that a function f ∈ O(S) is strongly recurrent modulo Δ, if for each compact set K ⊂ S, and for each > 0, the set of k ∈ Z such that maxK |f (z) − f (z + i(kΔ))| < is of positive upper density in Z. The following particular case of a theorem of Walter H. Gottschalk and Gustav A. Hedlund [8] allows us to pass between continuous and discrete dynamics. Theorem 2 (Inheritance Theorem) Let f ∈ O(S) and Δ ∈ R, Δ = 0.
Mat. Pura Appl. 51(1), 107–138 (1960) 24. : Weighted norm inequalities for the conjugate function and Hilbert transform. Trans. Am. Math. Soc. 176, 227–251 (1973) 25. : Some consequences of the Hunt–Muckenhoupt–Wheeden theorem. Mat. Issled. 47, 64–70 (1978) (Russian) 26. : Banach Algebras with Symbol and Singular Integral Operators. Birkhäuser, Basel (1987) 27. : Representation Theorems in Hardy Spaces. Cambridge University Press, Cambridge (2009) 28. : Composition operators. Can. J. Math. 20, 442–449 (1968) 29.
Finally, we would like to reiterate that very little is known about the (essential) spectrum of T (a) : H p (T) → H p (T), 1 < p < ∞ for a general a ∈ L∞ (T). Acknowledgements The first author was supported by PROMEP (México) via “Proyecto de Redes” and by CONACYT grant 102800. References 1. : Toeplitz operators with discontinuous symbols: phenomena beyond piecewise continuity. Oper. Theory, Adv. Appl. 90, 55–118 (1996) 2. : On the composition of Muckenhoupt weights and inner functions. J. Lond.