This article analyzes in detail the statistical and measure-theoretical properties of the nonuniform stationary measure, referred to as the w-invariant measure, associated with the spatial length distribution of the integral manifolds of the unstable invariant foliation in two-dimensional differentiable area-preserving systems. The analysis is developed starting from a sequence of analytical approximations for the associated density. These approximations are related to the properties of the Jacobian matrix of the nth iteration of a Poincaré map. The w-invariant measure plays a fundamental role in the study of transport phenomena in laminar-chaotic fluid-mixing systems, for which it furnishes the asymptotic invariant distribution of intermaterial contact length between two fluids. The w-invariant measure turns out to be singular and exhibits multifractal features. Its associated density displays local self-similarity in an ε neighborhood of hyperbolic periodic points. The cancellation exponent of the signed measure associated with the w measure by attaching at each point the direction of the field of the asymptotic unstable eigenvectors is also analyzed. The only case for which the w-invariant measure is absolutely continuous is given by the conjugation of hyperbolic toral automorphisms with a linear automorphism. The connections with the statistical properties, and in particular with the stretching dynamics, are addressed in detail. ©1999 The American Physical Society.

Measure-theoretical properties of the unstable foliation of two-dimensional differentiable area-preserving systems / Adrover, Alessandra; Giona, Massimiliano. - In: PHYSICAL REVIEW E. - ISSN 1063-651X. - 60:1(1999), pp. 347-362.

Measure-theoretical properties of the unstable foliation of two-dimensional differentiable area-preserving systems

ADROVER, Alessandra;GIONA, Massimiliano
1999

Abstract

This article analyzes in detail the statistical and measure-theoretical properties of the nonuniform stationary measure, referred to as the w-invariant measure, associated with the spatial length distribution of the integral manifolds of the unstable invariant foliation in two-dimensional differentiable area-preserving systems. The analysis is developed starting from a sequence of analytical approximations for the associated density. These approximations are related to the properties of the Jacobian matrix of the nth iteration of a Poincaré map. The w-invariant measure plays a fundamental role in the study of transport phenomena in laminar-chaotic fluid-mixing systems, for which it furnishes the asymptotic invariant distribution of intermaterial contact length between two fluids. The w-invariant measure turns out to be singular and exhibits multifractal features. Its associated density displays local self-similarity in an ε neighborhood of hyperbolic periodic points. The cancellation exponent of the signed measure associated with the w measure by attaching at each point the direction of the field of the asymptotic unstable eigenvectors is also analyzed. The only case for which the w-invariant measure is absolutely continuous is given by the conjugation of hyperbolic toral automorphisms with a linear automorphism. The connections with the statistical properties, and in particular with the stretching dynamics, are addressed in detail. ©1999 The American Physical Society.
1999
01 Pubblicazione su rivista::01a Articolo in rivista
Measure-theoretical properties of the unstable foliation of two-dimensional differentiable area-preserving systems / Adrover, Alessandra; Giona, Massimiliano. - In: PHYSICAL REVIEW E. - ISSN 1063-651X. - 60:1(1999), pp. 347-362.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/247666
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