seminar: Asymptotical behavior of piecewise contractions
speaker: Arnaldo Nogueira (Institut de Mathematiques de Marseille)abstract: (See also http:/arxiv.orgabs1310.5784)
Poincar'e first return maps induced by some Cherry flows on transverse intervals are, up to topological conjugacy, piecewise contractions. These maps also appear in dynamical systems describing the time evolution of manufacturing process adopting some decision-making policies. An injective map \(f:[0,1)to [0,1)\) is a {it piecewise contraction of \(n\) intervals}, if there exists a partition of the interval \([0,1)\) into intervals mbox{\(I_1\), ldots, \(I_n\)} such that for every \(i\), the restriction \(fvert_{I_i}\) is \(kappa\)-Lipschitz for \(0\). In the talk we will consider piecewise contractions defined by a finite family of contractions. Let \(phi_i:[0,1]to (0,1)\), \(1le ile n\), be \(C^2\)-diffeomorphisms with \(sup_{xin (0,1)} vert Dphi_i(x)vert<1\) whose images \(phi_1([0,1]), ldots , phi_n([0,1])\) are pairwise disjoint. Let \(0 I_1,ldots , I_n\) be a partition of the interval \([0,1)\) into subintervals \(I_i\) having interior \((x_{i-1},x_i)\), where \(x_0=0\) and \(x_n=1\). Let \(f_{x_1,ldots,x_{n-1}}\) be the map given by \(xmapsto phi_i(x)\) if \(xin I_i\) for every \(i\). Among other results we will show that for Lebesgue almost every \((x_1,ldots,x_{n-1})\), the piecewise contraction \(f_{x_1,ldots,x_{n-1}}\) is asymptotically periodic. {it The talk is based in a joint work with B. Pires and R. Rosales (see in Arxiv).}
timetable:
Mon 11 Nov, 14:30 - 15:30, Sala Conferenze Centro De Giorgi
<< Go back