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3x+1 inverse orbit generating functions almost always have natural boundaries

Jason P. Bell, Jeffrey C. Lagarias (2015)

Acta Arithmetica

The 3x+k function T k ( n ) sends n to (3n+k)/2, resp. n/2, according as n is odd, resp. even, where k ≡ ±1 (mod 6). The map T k ( · ) sends integers to integers; for m ≥1 let n → m mean that m is in the forward orbit of n under iteration of T k ( · ) . We consider the generating functions f k , m ( z ) = n > 0 , n m z n , which are holomorphic in the unit disk. We give sufficient conditions on (k,m) for the functions f k , m ( z ) to have the unit circle |z|=1 as a natural boundary to analytic continuation. For the 3x+1 function these conditions hold for all m...


Е.М. Никишин (1980)

Matematiceskij sbornik

A characterization of Fuchsian groups acting on complex hyperbolic spaces

Xi Fu, Liulan Li, Xiantao Wang (2012)

Czechoslovak Mathematical Journal

Let G 𝐒𝐔 ( 2 , 1 ) be a non-elementary complex hyperbolic Kleinian group. If G preserves a complex line, then G is -Fuchsian; if G preserves a Lagrangian plane, then G is -Fuchsian; G is Fuchsian if G is either -Fuchsian or -Fuchsian. In this paper, we prove that if the traces of all elements in G are real, then G is Fuchsian. This is an analogous result of Theorem V.G. 18 of B. Maskit, Kleinian Groups, Springer-Verlag, Berlin, 1988, in the setting of complex hyperbolic isometric groups. As an application...

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