Entropy, volume growth and SRB measures for Banach space mappings

We consider $C^2$ Fr\'echet differentiable mappings of Banach spaces leaving invariant compactly supported Borel probability measures, and study the relation between entropy and volume growth for a natural notion of volume defined on finite dimensional subspaces. SRB measures are characterized as exactly those measures for which entropy is equal to volume growth on unstable manifolds, equivalently the sum of positive Lyapunov exponents of the map. In addition to numerous difficulties incurred by our infinite-dimensional setting, a crucial aspect to the proof is the technical point that the volume elements induced on unstable manifolds are regular enough to permit distortion control of iterated determinant functions. The results here generalize previously known results for diffeomorphisms of finite dimensional Riemannian manifolds, and are applicable to dynamical systems defined by large classes of dissipative parabolic PDEs.Comment: 51 page

Analysis of a Class of Strange Attractors

This work contains the results from a comprehensive study of a new class of attractors. The attractors in this class are characterized by strong local instability, but they are not uniformly hyperbolic. Rigorous results on their dynamical, geometric and statistical properties are presented.Comment: 103 pages, 11 figure

Dynamic signal tracking in a simple V1 spiking model

This work is part of an effort to understand the neural basis for our visual system's ability, or failure, to accurately track moving visual signals. We consider here a ring model of spiking neurons, intended as a simplified computational model of a single hypercolumn of the primary visual cortex. Signals that consist of edges with time-varying orientations localized in space are considered. Our model is calibrated to produce spontaneous and driven firing rates roughly consistent with experiments, and our two main findings, for which we offer dynamical explanation on the level of neuronal interactions, are the following: (1) We have documented consistent transient overshoots in signal perception following signal switches due to emergent interactions of the E- and I-populations, and (2) for continuously moving signals, we have found that accuracy is considerably lower at reversals of orientation than when continuing in the same direction (as when the signal is a rotating bar). To measure performance, we use two metrics, called fidelity and reliability, to compare signals reconstructed by the system to the ones presented, and to assess trial-to-trial variability. We propose that the same population mechanisms responsible for orientation selectivity also impose constraints on dynamic signal tracking that manifest in perception failures consistent with psychophysical observations.Comment: 27 pages, 4 figure

Absolute continuity of stable foliations for mappings of Banach spaces

We prove the absolute continuity of stable foliations for mappings of Banach spaces satisfying conditions consistent with time-t maps of certain classes of dissipative PDEs. This property is crucial for passing information from submanifolds transversal to the stable foliation to the rest of the phase space; it is also used in proofs of ergodicity. Absolute continuity of stable foliations is well known in finite dimensional hyperbolic theory. On Banach spaces, the absence of nice geometric properties poses some additional difficulties.Comment: 27 page

Nonequilibrium Energy Profiles for a Class of 1-D Models

As a paradigm for heat conduction in 1 dimension, we propose a class of models represented by chains of identical cells, each one of which containing an energy storage device called a "tank". Energy exchange among tanks is mediated by tracer particles, which are injected at characteristic temperatures and rates from heat baths at the two ends of the chain. For stochastic and Hamiltonian models of this type, we develop a theory that allows one to derive rigorously -- under physically natural assumptions -- macroscopic equations for quantities related to heat transport, including mean energy profiles and tracer densities. Concrete examples are treated for illustration, and the validity of the Fourier Law in the present context is discussed.Comment: To appear in Commun. Math. Physic

Correlations in Nonequilibrium Steady States

We present the results of a detailed study of energy correlations at steady state for a 1-D model of coupled energy and matter transport. Our aim is to discover -- via theoretical arguments, conjectures, and numerical simulations -- how spatial covariances scale with system size, their relations to local thermodynamic quantities, and the randomizing effects of heat baths. Among our findings are that short-range covariances respond quadratically to local temperature gradients, and long-range covariances decay linearly with macroscopic distance. These findings are consistent with exact results for the simple exclusion and KMP models.Comment: 30 pages, 20 figure

Dynamics of Periodically-kicked Oscillators

We review some recent results surrounding a general mechanism for producing chaotic behavior in periodically-kicked oscillators. The key geometric ideas are illustrated via a simple linear shear model

Dispersing billiards with moving scatterers

We propose a model of Sinai billiards with moving scatterers, in which the locations and shapes of the scatterers may change by small amounts between collisions. Our main result is the exponential loss of memory of initial data at uniform rates, and our proof consists of a coupling argument for non-stationary compositions of maps similar to classical billiard maps. This can be seen as a prototypical result on the statistical properties of time-dependent dynamical systems.Comment: 39 pages, 3 figures. (Minor improvements/corrections in this version; to appear in Communications in Mathematical Physics.

Local thermal equilibrium for certain stochastic models of heat transport

This paper is about nonequilibrium steady states (NESS) of a class of stochastic models in which particles exchange energy with their "local environments" rather than directly with one another. The physical domain of the system can be a bounded region of $\mathbb R^d$ for any $d \ge 1$. We assume that the temperature at the boundary of the domain is prescribed and is nonconstant, so that the system is forced out of equilibrium. Our main result is local thermal equilibrium in the infinite volume limit. In the Hamiltonian context, this would mean that at any location $x$ in the domain, local marginal distributions of NESS tend to a probability with density $\frac{1}{Z} e^{-\beta (x) H}$, permitting one to define the local temperature at $x$ to be $\beta(x)^{-1}$. We prove also that in the infinite volume limit, the mean energy profile of NESS satisfies Laplace's equation for the prescribed boundary condition. Our method of proof is duality: by reversing the sample paths of particle movements, we convert the problem of studying local marginal energy distributions at $x$ to that of joint hitting distributions of certain random walks starting from $x$, and prove that the walks in question become increasingly independent as system size tends to infinity

Memory loss for time-dependent dynamical systems

This paper discusses the evolution of probability distributions for certain time-dependent dynamical systems. Exponential loss of memory is proved for expanding maps and for one-dimensional piecewise expanding maps with slowly varying parameters.Comment: 10 pages, 1 figur