Growth of positive harmonic functions and Kleinian group limit sets of zero planar measure and hausdorf dimension two In celebration of Nico Kuiper's sixtieth birthday

1981 ◽  
pp. 127-144 ◽  
Author(s):  
Dennis Sullivan
Keyword(s):  
Harmonic Functions ◽  
Kleinian Group ◽  
Sixtieth Birthday ◽  
Limit Sets ◽  
Group Limit ◽  
Positive Harmonic Functions

scholarly journals Free vs. locally free Kleinian groups

2019 ◽  
Vol 2019 (746) ◽  
pp. 149-170
Author(s):  
Pekka Pankka ◽  
Juan Souto
Keyword(s):  
Hausdorff Dimension ◽  
Kleinian Group ◽  
Cantor Set ◽  
Cantor Sets ◽  
Kleinian Groups ◽  
The Other ◽  
Limit Set ◽  
Limit Sets ◽  
Other Hand

Abstract We prove that Kleinian groups whose limit sets are Cantor sets of Hausdorff dimension < 1 are free. On the other hand we construct for any ε > 0 an example of a non-free purely hyperbolic Kleinian group whose limit set is a Cantor set of Hausdorff dimension < 1 + ε.

A note on positive harmonic functions

1948 ◽  
Vol 44 (2) ◽  
pp. 289-291 ◽  
Author(s):  
S. Verblunsky
Keyword(s):  
Harmonic Function ◽  
Harmonic Functions ◽  
Duke Math ◽  
Negative Number ◽  
Image Position ◽  
American Math ◽  
Positive Harmonic Functions

If H(ξ, η) is a harmonic function which is defined and positive in η > 0, then there is a non-negative number D and a bounded non-decreasing function G(x) such that(For a proof, see Loomis and Widder, Duke Math. J. 9 (1942), 643–5.) If we writewhere λ > 1, then the equationdefines a harmonic function h which is positive in υ > 0. Hence there is a non-negative number d and a bounded non-decreasing function g(x) such thatThe problem of finding the connexion between the functions G(x) and g(x) has been mentioned by Loomis (Trans. American Math. Soc. 53 (1943), 239–50, 244).

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