A switched server system semiconjugate to a minimal interval exchange

Switched server systems are mathematical models of manufacturing, traffic and queueing systems that have being studied since the early 1990s. In particular, it is known that typically the dynamics of such systems is asymptotically periodic: each orbit of the system converges to one of its finitely many limit cycles. In this article, we provide an explicit example of a switched server system with exotic behaviour: each orbit of the system converges to the same Cantor attractor. To accomplish this goal, we bring together recent advances in the understanding of the topological dynamics of piecewise contractions and interval exchange transformations (IETs) with flips. The ultimate result is a switched server system whose Poincaré map is semiconjugate to a minimal and uniquely ergodic IET with flips.

[1]  J. Yorke,et al.  Period Three Implies Chaos , 1975 .

[2]  M. Keane Interval exchange transformations , 1975 .

[3]  A. Katok Interval exchange transformations and some special flows are not mixing , 1980 .

[4]  M. Rees An alternative approach to the ergodic theory of measured foliations on surfaces , 1981, Ergodic Theory and Dynamical Systems.

[5]  H. Masur Interval Exchange Transformations and Measured Foliations , 1982 .

[6]  W. Veech Gauss measures for transformations on the space of interval exchange maps , 1982 .

[7]  M. Boshernitzan A condition for minimal interval exchange maps to be uniquely ergodic , 1985 .

[8]  S. Kerckhoff Simplicial systems for interval exchange maps and measured foliations , 1985, Ergodic Theory and Dynamical Systems.

[9]  C. Gutierrez Smoothing continuous flows on two-manifolds and recurrences , 1986, Ergodic Theory and Dynamical Systems.

[10]  A. Nogueira Almost all interval exchange transformations with flips are nonergodic , 1989, Ergodic Theory and Dynamical Systems.

[11]  P. Ramadge,et al.  Periodicity and chaos from switched flow systems: contrasting examples of discretely controlled continuous systems , 1993, IEEE Trans. Autom. Control..

[12]  Ricardo Camelier,et al.  Affine interval exchange transformations with wandering intervals , 1997, Ergodic Theory and Dynamical Systems.

[13]  J. Zukas Introduction to the Modern Theory of Dynamical Systems , 1998 .

[14]  Andrey V. Savkin,et al.  Hybrid dynamical systems: Stability and chaos , 2001 .

[15]  M. Cobo Piece-wise affine maps conjugate to interval exchanges , 2002, Ergodic Theory and Dynamical Systems.

[16]  L. Bunimovich,et al.  Switched flow systems: pseudo billiard dynamics , 2004, math/0408241.

[17]  E. Salinelli,et al.  Modelli dinamici discreti , 2009 .

[18]  B. Pires,et al.  Orbit structure of interval exchange transformations with flip , 2011, 1104.2015.

[19]  B. Pires,et al.  Asymptotically periodic piecewise contractions of the interval , 2013, 1310.5784.

[20]  Invariant measures for piecewise continuous maps , 2016, 1603.02542.

[21]  B. Pires,et al.  Topological dynamics of piecewise $\unicode[STIX]{x1D706}$ -affine maps , 2016, Ergodic Theory and Dynamical Systems.

[22]  Topological dynamics of piecewise λ-affine maps , 2018 .

[23]  B. Pires Symbolic dynamics of piecewise contractions , 2018, Nonlinearity.