Invariant measure under the trace map for binary quasiperiodic lattices

Dynamical systems have attracted much interest since the discovery of chaotic phenomena I I ]. Many applications to physical systems have appeared [2 ]. The definition of dynamical systems is the following [3]: Let M be a smooth manifold, I a measure on M defined by a continuous positive density, /,: M-M a one-parameter group of measure-preserving diffeomorphisms. Then the collection (M, p, /,) is called a classical dynamical system (CDS). The parameter t canbe either a real number (teR) or an integer (tez). If tez., /, is the discrete group generated by a measure-preserving diffeomorphism o=or. Then the system is merely denoted by (M, p, d) and called an automorphism. When a CDS consists of an invariant (i.e., a conserved quantit-v), it is called a Hamilton dynamical system (HDS). The HDS is defined as follows: Let {Pt, Pz, ..., Pni 4r, Qz, '.., Q,)=(p, q) be a coordinate system in the phase space of R2' and H(p, q) a smooth function (the Hamiltonian). The equations of motion are given by PH YS!CS LETTERS A