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Loewner evolution of hedgehogs and 2-conformal measures of circle maps | | Kingshook Biswas
; | | Date: |
2 Mar 2016 | | Abstract: | Let $f$ be a germ of holomorphic diffeomorphism with an irrationally
indifferent fixed point at the origin in $C$ (i.e. $f(0) = 0, f’(0) = e^{2pi
i alpha}, alpha in R - Q$). Perez-Marco showed the existence of a unique
continuous monotone one-parameter family of nontrivial invariant full continua
containing the fixed point called Siegel compacta, and gave a correspondence
between germs and families $(g_t)$ of circle maps obtained by conformally
mapping the complement of these compacts to the complement of the unit disk.
The family of circle maps $(g_t)$ is the orbit of a locally-defined semigroup
$(Phi_t)$ on the space of analytic circle maps which we show has a
well-defined infinitesimal generator $X$. The explicit form of $X$ is obtained
by using the Loewner equation associated to the family of hulls $(K_t)$. We
show that the Loewner measures $(mu_t)$ driving the equation are 2-conformal
measures on the circle for the circle maps $(g_t)$. | | Source: | arXiv, 1603.0830 | | Services: | Forum | Review | PDF | Favorites | |
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