# Map Lattices Coupled by Collisions

@article{Keller2008MapLC,
title={Map Lattices Coupled by Collisions},
author={Gerhard Keller and Carlangelo Liverani},
journal={Communications in Mathematical Physics},
year={2008},
volume={291},
pages={591-597}
}
• Published 21 November 2008
• Mathematics, Physics
• Communications in Mathematical Physics
We introduce a new coupled map lattice model in which the weak interaction takes place via rare “collisions”. By “collision” we mean a strong (possibly discontinuous) change in the system. For such models we prove uniqueness of the SRB measure and exponential space-time decay of correlations.
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#### References

SHOWING 1-10 OF 12 REFERENCES
Uniqueness of the SRB Measure for Piecewise Expanding Weakly Coupled Map Lattices in Any Dimension
• Mathematics
• 2006
We prove the existence of a unique SRB measure for a wide range of multidimensional weakly coupled map lattices. These include piecewise expanding maps with diffusive coupling.
A spectral gap for a one-dimensional lattice of coupled piecewise expanding interval maps
• Physics
• 2005
We study one-dimensional lattices of weakly coupled piecewise expand- ing interval maps as dynamical systems. Since neither the local maps need to have full branches nor the coupling map needs to beExpand
Limit theorems for coupled interval maps
• Mathematics
• 2006
We prove a local limit theorem for Lipschitz continuous observables on a weakly coupled lattice of piecewise expanding mixing interval maps. The core of the paper is a proof that the spectral radiiExpand
Ergodic systems ofn balls in a billiard table
• Mathematics
• 1992
We consider the motion ofn balls in billiard tables of a special form and we prove that the resulting dynamical systems are ergodic on a constant energy surface; in fact, they enjoy theK-property.Expand
Stability of statistical properties in two-dimensional piecewise hyperbolic maps
• Mathematics
• 2006
We investigate the statistical properties of a piecewise smooth dynamical system by studying directly the action of the transfer operator on appropriate spaces of distributions. We accomplish suchExpand
Heat conduction and Fourier's law in a class of many particle dispersing billiards
• Physics
• 2008
We consider the motion of many confined billiard balls in interaction and discuss their transport and chaotic properties. In spite of the absence of mass transport, due to confinement, energyExpand
Heat conduction and Fourier's law by consecutive local mixing and thermalization.
• Physics, Medicine
• Physical review letters
• 2008
We present a first-principles study of heat conduction in a class of models which exhibit a new multistep local thermalization mechanism which gives rise to Fourier's law. Local thermalization in ourExpand
Good Banach spaces for piecewise hyperbolic maps via interpolation
• Mathematics
• 2007
We introduce a weak transversality condition for piecewise C 1+α and piecewise hyperbolic maps which admit a C 1+α stable distribution. We show bounds on the essential spectral radius of theExpand
Lectures from the school-forum (CML 2004) held in Paris
• Lecture Notes in Physics
• 2004
Bismarckstr . 1 1 2 , 91052 Erlangen, Germany E-mail address: keller@mi.uni-erlangen
• Bismarckstr . 1 1 2 , 91052 Erlangen, Germany E-mail address: keller@mi.uni-erlangen