24 May 1999 Ž . Physics Letters A 256 1999 31–38 Continuous formulation of global invariant properties of 2D time-periodic chaotic flows M. Giona, A. Adrover Dipartimento di Ingegneria Chimica, UniÕersita di Roma ‘‘La Sapienza’’, Õia Eudossiana 18, 00184 Rome, Italy ´ Received 21 January 1999; accepted 10 March 1999 Communicated by A.R. Bishop Abstract This article analyzes the continuum-mechanical representation of the global invariant geometric properties of 2D time-periodic Hamiltonian systems and chaotic flows. An application of this analysis concerns the evolution in time of the invariant measure associated with the space-filling properties of the invariant unstable foliation related in laminar chaotic flows to the pointwise intermaterial contact-area density between fluid elements. q 1999 Elsevier Science B.V. All rights reserved. PACS: 05.45 qb; 42.52 qj; 47.53 qn; 83.50 Ws Keywords: Dynamical systems; Area-preserving diffeomorphism; Hyperbolic behavior 1. Introduction There are two basic conceptual frameworks in the analysis of dynamical systems. The first is based on the local analysis of the dynamics in the neighbor- hood of some characteristic fixedrperiodic points and on the bifurcation structure in the parameter wx space as it regards their stability 1 . The second approach is global in nature and focuses on the invariant statistical and geometrical properties ap- pearing in the dynamic evolution of ensembles of initial conditions in the phase space. It thus encom- wx w x passes ergodic theory 2 and hyperbolic theory 3,4 . wx Initially pioneered by Anosov 5 in the 1960s and further developed by Oseledec, Pesin, Mane, Bowen, ˜´ Ž w x. Sinai and Ruelle see 6,7 , hyperbolic theory pro- vides the formal conceptual apparatus required to describe the invariant geometric properties of a wide class of dynamical systems of physical interest, thus connecting geometry with the statistical description Ž of the dynamics in particular with stretching dynam- ics expressed e.g. by the short-time Liapunov expo- . nents . w x Ž Recent studies 8–10 confirm the conjecture re- ferred to by some authors as Ruelle’s chaotic hy- w x. pothesis 11 that 2D area-preserving chaotic diffeo- Ž . morphisms F , x s F x possess hyperbolic nq1 n w x properties within a chaotic region C 12 . This im- plies that within an invariant chaotic submanifold Ž . C < F C , the tangent space T C at x g C, decom- x Ž u . poses into the direct sum of an unstable E invari- x Ž s . ant vector subspace and a stable E invariant x vector subspace, T C s E u [ E s , which are invari- x x x ) Ž. Ž. < ant under the differential F x s EF y rE y of ysx 0375-9601r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. Ž . PII: S0375-9601 99 00191-7