Double Standard Maps

We investigate the family of double standard maps of the circle onto itself, given by (mod 1), where the parameters a,b are real and 0 ≤ b ≤ 1. Similarly to the well known family of (Arnold) standard maps of the circle, (mod 1), any such map has at most one attracting periodic orbit and the set of parameters (a,b) for which such orbit exists is divided into tongues. However, unlike the classical Arnold tongues that begin at the level b = 0, for double standard maps the tongues begin at higher levels, depending on the tongue. Moreover, the order of the tongues is different. For the standard maps it is governed by the continued fraction expansions of rational numbers; for the double standard maps it is governed by their binary expansions. We investigate closer two families of tongues with different behavior. The first author was partially supported by NSF grant DMS 0456526. The second author was supported by FCT Grant SFRH/BD/18631/2004. CMUP is supported by FCT through POCTI and POSI of Quadro Comunitário de apoio III (2000-2006) with FEDER and national funding.     

Observables and Statistical Maps

This article begins with a review of the framework of fuzzy probability theory. The basic structure is given by the -effect algebra of effects (fuzzy events) and the set of probability measures on a measurable space . An observable is defined, where is the value space of X. It is noted that there exists a one-to-one correspondence between states on and elements of and between observables and -morphisms from to . Various combinations of observables are discussed. These include compositions, products, direct products, and mixtures. Fuzzy stochastic processes are introduced and an application to quantum dynamics is considered. Quantum effects are characterized from among a more general class of effects. An alternative definition of a statistical map is given and it is shown that any statistical map has a unique extension to a statistical operator. Finally, various combinations of statistical maps are discussed and their relationships to the corresponding combinations of observables are derived.     

Planar Maps and Continued Fractions

We present an unexpected connection between two map enumeration problems. The first one consists in counting planar maps with a boundary of prescribed length. The second one consists in counting planar maps with two points at a prescribed distance. We show that, in the general class of maps with controlled face degrees, the solution for both problems is actually encoded into the same quantity, respectively via its power series expansion and its continued fraction expansion. We then use known techniques for tackling the first problem in order to solve the second. This novel viewpoint provides a constructive approach for computing the so-called distance-dependent two-point function of general planar maps. We prove and extend some previously predicted exact formulas, which we identify in terms of particular Schur functions.     

Rational Misiurewicz Maps are Rare

We show that the set of Misiurewicz maps has Lebesgue measure zero in the space of rational functions for any fixed degree d ≥ 2.     

Spectral Conditions for Positive Maps

We provide partial classification of positive linear maps in matrix algebras which is based on a family of spectral conditions. This construction generalizes the celebrated Choi example of a map which is positive but not completely positive. It is shown how the spectral conditions enable one to construct linear maps on tensor products of matrix algebras which are positive but only on a convex subset of separable elements. Such maps provide basic tools to study quantum entanglement in multipartite systems.     

Morrey Potentials and Harmonic Maps

This paper discusses trace estimates for Morrey potentials (i.e., Riesz potential integrals of Morrey functions), leading to a consideration of the C ^∞ smoothness of a class of generalized harmonic maps.     

Semigroup Actions of Expanding Maps

We consider semigroups of Ruelle-expanding maps, parameterized by random walks on the free semigroup, with the aim of examining their complexity and exploring the relation between intrinsic properties of the semigroup action and the thermodynamic formalism of the associated skew-product. In particular, we clarify the connection between the topological entropy of the semigroup action and the growth rate of the periodic points, establish the main properties of the dynamical zeta function of the semigroup action and relate these notions to recent research on annealed and quenched thermodynamic formalism. Meanwhile, we examine how the choice of the random walk in the semigroup unsettles the ergodic properties of the action.     

Conformal Maps and Integrable Hierarchies

We show that conformal maps of simply connected domains with an analytic boundary to a unit disk have an intimate relation to the dispersionless 2D Toda integrable hierarchy. The maps are determined by a particular solution to the hierarchy singled out by the conditions known as “string equations”. The same hierarchy locally solves the 2D inverse potential problem, i.e., reconstruction of the domain out of a set of its harmonic moments. This is the same solution which is known to describe 2D gravity coupled to c= matter. We also introduce a concept of the τ-function for analytic curves. Received: 20 December 1999/ Accepted: 2 March 2000     

Nonuniformly Expanding 1D Maps

This paper attempts to make accessible a body of ideas surrounding the following result: Typical families of (possibly multi-model) 1-dimensional maps passing through ``Misiurewicz points' have invariant densities for positive measure sets of parameters. The research of both authors is partially supported by grant from the NSF     

Maps of intragranular precipitation strengthening

Maps of strengthening by small (semi)coherent precipitates homogeneously dispersed in the grains have been constructed. They can show the most probable dislocation-particle interaction mechanism and the strengthening effect of a given dispersion of particles. As an application, the effect of-Ni₃(Al, Ti) and M₂₃C₆ precipitation in austenitic steels is discussed.     

Maps of Matrices and Portrait Maps of the Density Operators of Composite and Noncomposite Systems

We obtain a new inequality for arbitrary Hermitian matrices. We describe particular linear maps called the matrix portrait of arbitrary N × N matrices. The maps are obtained as analogs of partial tracing of density matrices of multipartite qudit systems. The structure of the maps is inspired by “portrait” map of the probability vectors corresponding to the action on the vectors by stochastic matrices containing either unity or zero matrix elements. We obtain new entropic inequalities for arbitrary qudit states including a single qudit and discuss entangled single qudit state. We consider in detail the examples of N = 3 and 4. Also we point out the possible use of entangled states of systems without subsystems (e.g., a single qudit) as a resource for quantum computations.     

Boundary from Bulk Integrability in Three Dimensions: 3D Reflection Maps from Tetrahedron Maps

We derive the macroscopic laws that govern the evolution of the density of particles in the exclusion process on the Sierpinski gasket in the presence of a variable speed boundary. We obtain, at the hydrodynamics level, the heat equation evolving on the Sierpinski gasket with either Dirichlet or Neumann boundary conditions, depending on whether the reservoirs are fast or slow. For a particular strength of the boundary dynamics we obtain linear Robin boundary conditions. As for the fluctuations, we prove that, when starting from the stationary measure, namely the product Bernoulli measure in the equilibrium setting, they are governed by Ornstein-Uhlenbeck processes with the respective boundary conditions.

     

Ergodic Jacobi Matrices and Conformal Maps

We study structural properties of the Lyapunov exponent γ and the density of states k for ergodic (or just invariant) Jacobi matrices in a general framework. In this analysis, a central role is played by the function w = − γ + iπ k as a conformal map between certain domains. This idea goes back to Marchenko and Ostrovskii, who used this device in their analysis of the periodic problem.     

Ergodic Theory of Generic Continuous Maps

We study the ergodic properties of generic continuous dynamical systems on compact manifolds. As a main result we prove that generic homeomorphisms have convergent Birkhoff averages under continuous observables at Lebesgue almost every point. In spite of this, when the underlying manifold has dimension greater than one, generic homeomorphisms have no physical measures—a somewhat strange result which stands in sharp contrast to current trends in generic differentiable dynamics. Similar results hold for generic continuous maps. To further explore the mysterious behaviour of C ⁰ generic dynamics, we also study the ergodic properties of continuous maps which are conjugated to expanding circle maps. In this context, generic maps have divergent Birkhoff averages along orbits starting from Lebesgue almost every point.     

Melnikov Potential for Exact Symplectic Maps

The splitting of separatrices of hyperbolic fixed points for exact symplectic maps of n degrees of freedom is considered. The non-degenerate critical points of a real-valued function (called the Melnikov potential) are associated to transverse homoclinic orbits and an asymptotic expression for the symplectic area between homoclinic orbits is given. Moreover, if the unperturbed invariant manifolds are completely doubled, it is shown that there exist, in general, at least $4$ primary homoclinic orbits (4n in antisymmetric maps). Both lower bounds are optimal. Two examples are presented: a 2n-dimensional central standard-like map and the Hamiltonian map associated to a magnetized spherical pendulum. Several topics are studied about these examples: existence of splitting, explicit computations of Melnikov potentials, transverse homoclinic orbits, exponentially small splitting, etc. Received: 6 June 1996 / Accepted: 16 April 1997     

Large Deviations for Non-Uniformly Expanding Maps

We obtain large deviation bounds for non-uniformly expanding maps with non-flat singularities or criticalities and for partially hyperbolic non-uniformly expanding attracting sets. That is, given a continuous function we consider its space average with respect to a physical measure and compare this with the time averages along orbits of the map, showing that the Lebesgue measure of the set of points whose time averages stay away from the space average tends to zero exponentially fast with the number of iterates involved. As easy by-products we deduce escape rates from subsets of the basins of physical measures for these types of maps. The rates of decay are naturally related to the metric entropy and pressure function of the system with respect to a family of equilibrium states. 2000 Mathematics Subject Classification: 37D25, 37A50, 37B40, 37C40     

Sequential Gibbs Measures and Factor Maps

We define the notion of sequential Gibbs measures, inspired by on the classical notion of Gibbs measures and recent examples from the study of non-uniform hyperbolic dynamics. Extending previous results of Kempton and Pollicott (Factors of Gibbs measures for full shifts, entropy of hidden Markov processes and connections to dynamical systems, Cambridge University Press, Cambridge, 2011) and Chazottes and Ugalde (On the preservation of Gibbsianness under symbol amalgamation, entropy of hidden Markov processes and connections to dynamical systems, Cambridge University Press, Cambridge, 2011), we show that the images of one block factor maps of a sequential Gibbs measure are also a sequential Gibbs measure, with the same sequence of Gibbs times. We obtain some estimates on the regularity of the potential of the image measure at almost every point.     

Constrained Wave Equations and Wave Maps

In this paper we establish that wave maps can be obtained by a penalization method if the initial data is well prepared. When the data is not well prepared, we prove that the solution of the penalized equation converges weakly to the solution of the system of coupled equations obtained in [11] by a multi-scale formal analysis. In particular, the interaction between the rapid normal oscillations and the tangential motions creates a new term in the limit system whose well-posedness is proved by using the Nash-Moser Implicit Function Theorem. The first author is funded in part by NSF DMS 0203485.The second author is funded in part by NSF DMS 0101969.