Results 1  10
of
246
The pinwheel tilings of the plane
, 1994
"... Abstract. We discuss two new results on tilings of the plane. In the first, we give sufficient conditions for the tilings associated with an inflation rule to be uniquely ergodic under translations, the conditions holding for the pinwheel inflation rule. In the second result we prove there are match ..."
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Cited by 62 (14 self)
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Abstract. We discuss two new results on tilings of the plane. In the first, we give sufficient conditions for the tilings associated with an inflation rule to be uniquely ergodic under translations, the conditions holding for the pinwheel inflation rule. In the second result we prove there are matching rules for the pinwheel inflation rule, making the system the first known to have complete rotational symmetry. We consider tilings of the Euclidean plane, E2, by (orientationpreserving) congruent copies of a fixed finite set of prototiles. Prototiles are topological disks in the plane satisfying some mild restrictions on their shapes, as detailed below. Congruent copies of prototiles are called tiles, and a tiling is simply an unordered collection of tiles whose union is the plane and in which each pair of tiles has disjoint interiors. We are concerned here with two constructions associated with a fixed finite set S = {Pj} of prototiles, the most important of which is the set X(S) of all tilings by tiles from S. In particular, we are interested in understanding the purest cases, in which all the tilings in X(S) are “essentially the same”; we will define this precisely further on. Two examples are exhibited in Figures 1 and 2 on the next page, both with two prototiles; in Figure 1, S = SK produces only a checkerboardlike tiling (and all congruences), and in Figure 2, S = SP produces the wellknown tilings of Penrose [3, 4, 6]. Tilings like those of Penrose are not usually invariant under any congruence of the plane (other than the identity), so to analyze their symmetries we introduce some elementary ergodic theory and another basic construction which can sometimes be associated with a prototile set S, the set XF (S) of tilings defined by an “inflation function ” F. An “inflation rule ” for S, if it exists, consists of a dilation DF of E2 by some factor λF < 1 and a finite set {Cjk} of congruences of E2, such that for each Pj ∈ S we have
Billiards and Teichmüller curves on Hilbert modular surfaces
, 2003
"... This paper exhibits an infinite collection of algebraic curves isometrically embedded in the moduli space of Riemann surfaces of genus two. These Teichmüller curves lie on Hilbert modular surfaces parameterizing Abelian varieties with real multiplication. Explicit examples, constructed from Lshaped ..."
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Cited by 54 (7 self)
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This paper exhibits an infinite collection of algebraic curves isometrically embedded in the moduli space of Riemann surfaces of genus two. These Teichmüller curves lie on Hilbert modular surfaces parameterizing Abelian varieties with real multiplication. Explicit examples, constructed from Lshaped polygons, give billiard tables with optimal dynamical properties.
Convection enhanced diffusion for periodic flows
"... We study the in uence of convection by periodic or cellular ows on the effective diffusivity of a passive scalar transported by the fluid when the molecular diffusivity is small. The flows are generated by twodimensional, steady, divergencefree, periodic velocity fields. ..."
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Cited by 53 (5 self)
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We study the in uence of convection by periodic or cellular ows on the effective diffusivity of a passive scalar transported by the fluid when the molecular diffusivity is small. The flows are generated by twodimensional, steady, divergencefree, periodic velocity fields.
Equipartition of the Eigenfunctions of Quantized Ergodic Maps on the Torus
, 1997
"... We give a simple proof of the equipartition of the eigenfunctions of a class of quantized ergodic areapreserving maps on the torus. Examples are the irrational translations, the skew translations, the hyperbolic automorphisms and some of their perturbations. ..."
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Cited by 49 (0 self)
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We give a simple proof of the equipartition of the eigenfunctions of a class of quantized ergodic areapreserving maps on the torus. Examples are the irrational translations, the skew translations, the hyperbolic automorphisms and some of their perturbations.
Metalinsulator transition for the almost Mathieu operator
, 1999
"... We prove that for Diophantine ω and almost every θ, the almost Mathieu operator, (Hω,λ,θΨ)(n) = Ψ(n+1)+Ψ(n −1)+λcos2π(ωn+θ)Ψ(n), exhibits localization for λ> 2 and purely absolutely continuous spectrum for λ < 2. This completes the proof of (a correct version of) the AubryAndré conjecture. ..."
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Cited by 47 (5 self)
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We prove that for Diophantine ω and almost every θ, the almost Mathieu operator, (Hω,λ,θΨ)(n) = Ψ(n+1)+Ψ(n −1)+λcos2π(ωn+θ)Ψ(n), exhibits localization for λ> 2 and purely absolutely continuous spectrum for λ < 2. This completes the proof of (a correct version of) the AubryAndré conjecture.
The Riemann Zeros and Eigenvalue Asymptotics
 SIAM Rev
, 1999
"... Comparison between formulae for the counting functions of the heights t n of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the t n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many feat ..."
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Cited by 42 (5 self)
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Comparison between formulae for the counting functions of the heights t n of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the t n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many features of H cl are provided by the analogy; for example, the "Riemann dynamics" should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers. Statistics of the t n have a similar structure to those of the semiclassical En ; in particular, they display randommatrix universality at short range, and nonuniversal behaviour over longer ranges. Very refined features of the statistics of the t n can be computed accurately from formulae with quantum analogues. The RiemannSiegel formula for the zeta function is described in detail. Its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations. We speculate that the Riemann dynamics is related to the trajectories generated by the classical hamiltonian H cl = XP. Key words. spectral asymptotics, number theory AMS subject classifications. 11M26, 11M06, 35P20, 35Q40, 41A60, 81Q10, 81Q50 PII. S0036144598347497 1.
Expansive Subdynamics
, 1997
"... . This paper provides a framework for studying the dynamics of commuting homeomorphisms. Let ff be a continuous action of Z d on an infinite compact metric space. For each subspace V of R d we introduce a notion of expansiveness for ff along V , and show that there are nonexpansive subspaces in ever ..."
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Cited by 39 (8 self)
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. This paper provides a framework for studying the dynamics of commuting homeomorphisms. Let ff be a continuous action of Z d on an infinite compact metric space. For each subspace V of R d we introduce a notion of expansiveness for ff along V , and show that there are nonexpansive subspaces in every dimension d \Gamma 1. For each k d the set E k (ff) of expansive k dimensional subspaces is open in the Grassman manifold of all kdimensional subspaces of R d . Various dynamicalproperties of ff are constant, or vary nicely, within a connected component of E k (ff), but change abruptly when passing from one expansive component to another. We give several examples of this sort of "phase transition," including the topological and measuretheoretic directional entropies studied by Milnor, zeta functions, and dimension groups. For d = 2 we show that, except for one unresolved case, every open set of directions whose complement is nonempty can arise as an E 1 (ff). The unr...
High Temperature Expansions and Dynamical Systems
 Comm. Math. Phys
, 1996
"... We develop a resummed hightemperature expansion for lattice spin systems with long range interactions, in models where the free energy is not, in general, analytic. We establish uniqueness of the Gibbs state and exponential decay of the correlation functions. Then, we apply this expansion to the Pe ..."
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Cited by 39 (2 self)
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We develop a resummed hightemperature expansion for lattice spin systems with long range interactions, in models where the free energy is not, in general, analytic. We establish uniqueness of the Gibbs state and exponential decay of the correlation functions. Then, we apply this expansion to the PerronFrobenius operator of weakly coupled map lattices. 1 Introduction. The theory of Gibbs states was originally developed for the mathematical analysis of equilibrium statistical mechanics. An interesting application of the theory was found by Sinai, Ruelle and Bowen in the 70's [41, 42, 38, 1] who applied it to the ergodic theory of uniformly hyperbolic dynamical systems. While this so called thermodynamic formalism has been very successful in ergodic theory, the Gibbs states that describe the statistics of such dynamical systems are quite simple from the point of view of statistical mechanics: they describe one dimensional spin systems with spins taking values in a finite set and intera...