A Family of Semitoric Systems with Four Focus–Focus Singularities and Two Double Pinched Tori

Abstract

AbstractWe construct a one-parameter family $$F_t=(J, H_t)_{0 \le t \le 1}$$ F t = ( J , H t ) 0 ≤ t ≤ 1 of integrable systems on a compact 4-dimensional symplectic manifold $$(M, \omega )$$ ( M , ω ) that changes smoothly from a toric system $$F_0$$ F 0 with eight elliptic–elliptic singular points via toric type systems to a semitoric system $$F_t$$ F t for $$ t^-< t < t^+$$ t - < t < t + . These semitoric systems $$F_t$$ F t have precisely four elliptic–elliptic and four focus–focus singular points. Moreover, at $$t= \frac{1}{2}$$ t = 1 2 , the system has precisely two focus–focus fibres each of which contains exactly two focus–focus points, giving these fibres the shape of double pinched tori. We exemplarily parametrise one of these fibres explicitly.