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Flatness-induced phase transition in Lyapunov spectrum for unimodal maps

Hold Date 2016-10-13 16:00~2016-10-13 19:00

Place Seminar Room W1-C-615, West Zone 1, Ito campus, Kyushu University

Object person  

Speaker Hiroki TAKAHASI (Keio University)

For a class of Misiurewicz $S$-unimodal maps with flat critical points, we study the occurrence of phase transitions, i.e., the lack of analyticity in their Lyapunov spectra. We show the Lyapunov spectrum is not real-analytic if the map behaves near its critical point $x=c$ like $|x-c|^{|x-c|^{-p}}$, $p¥geq1$. By the result of Benedicks $¥&$ Misiurewicz, the absolutely continuous invariant measure of this map is an infinite measure. Therefore, our result reveals a contrast to Nakaishi's theorem for interval maps with parabolic fixed points which asserts that the Lyapunov spectrum is real-analytic if and only if the absolutely continuous invariant measure is an infinite measure.