scholarly journals On Sinaĭ Billiards on Flat Surfaces with Horns

2021 ◽  
Vol 183 (2) ◽  
Author(s):  
Henk Bruin
Keyword(s):  
Exponential Decay ◽  
Height Function ◽  
Suspension Flow ◽  
Limit Laws ◽  
Suspension Flows ◽  
Flat Surfaces ◽  
Exponential Tails ◽  
Exponential Decay Of Correlations ◽  
Sinai Billiards ◽  
Young Towers

AbstractWe show that certain billiard flows on planar billiard tables with horns can be modeled as suspension flows over Young towers (Ann. Math. 147:585–650, 1998) with exponential tails. This implies exponential decay of correlations for the billiard map. Because the height function of the suspension flow itself is polynomial when the horns are Torricelli-like trumpets, one can derive Limit Laws for the billiard flow, including Stable Limits if the parameter of the Torricelli trumpet is chosen in (1, 2).

scholarly journals Thermodynamics of smooth models of pseudo–Anosov homeomorphisms

2021 ◽  
pp. 1-43
Author(s):  
DOMINIC VECONI
Keyword(s):  
Thermodynamic Formalism ◽  
Equilibrium States ◽  
Maximal Entropy ◽  
Srb Measure ◽  
The Central Limit Theorem ◽  
Exponential Decay Of Correlations ◽  
Surface Homeomorphisms ◽  
Young Towers ◽  
Uniqueness Of Equilibrium ◽  
Measure Of Maximal Entropy

Abstract We develop a thermodynamic formalism for a smooth realization of pseudo-Anosov surface homeomorphisms. In this realization, the singularities of the pseudo-Anosov map are assumed to be fixed, and the trajectories are slowed down so the differential is the identity at these points. Using Young towers, we prove existence and uniqueness of equilibrium states for geometric t-potentials. This family of equilibrium states includes a unique SRB measure and a measure of maximal entropy, the latter of which has exponential decay of correlations and the central limit theorem.

A Central Limit Theorem and Law of the Iterated Logarithm for a Random Field with Exponential Decay of Correlations

2004 ◽  
Vol 56 (1) ◽  
pp. 209-224 ◽  
Author(s):  
Byron Schmuland ◽  
Wei Sun
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Field ◽  
Central Limit ◽  
Exponential Decay ◽  
Gibbs Measures ◽  
Strong Mixing ◽  
Iterated Logarithm ◽  
Exponential Decay Of Correlations ◽  
The Law

AbstractIn [6], Walter Philipp wrote that “… the law of the iterated logarithm holds for any process for which the Borel-Cantelli Lemma, the central limit theorem with a reasonably good remainder and a certain maximal inequality are valid.” Many authors [1], [2], [4], [5], [9] have followed this plan in proving the law of the iterated logarithm for sequences (or fields) of dependent random variables.We carry on this tradition by proving the law of the iterated logarithm for a random field whose correlations satisfy an exponential decay condition like the one obtained by Spohn [8] for certain Gibbs measures. These do not fall into the ϕ-mixing or strong mixing cases established in the literature, but are needed for our investigations [7] into diffusions on configuration space.The proofs are all obtained by patching together standard results from [5], [9] while keeping a careful eye on the correlations.

scholarly journals Statistical Properties of Lorenz-like Flows, Recent Developments and Perspectives

2014 ◽  
Vol 24 (10) ◽  
pp. 1430028 ◽  
Author(s):  
Vitor Araujo ◽  
Stefano Galatolo ◽  
Maria José Pacifico
Keyword(s):  
Exponential Decay ◽  
Chaotic Behavior ◽  
Decay Of Correlations ◽  
Statistical Properties ◽  
Computer Assisted ◽  
Lorenz Attractor ◽  
Lorenz Equations ◽  
Physical Measure ◽  
Exponential Decay Of Correlations ◽  
Statistical Behavior

We comment on the mathematical results about the statistical behavior of Lorenz equations and its attractor, and more generally on the class of singular hyperbolic systems. The mathematical theory of such kind of systems turned out to be surprisingly difficult. It is remarkable that a rigorous proof of the existence of the Lorenz attractor was presented only around the year 2000 with a computer-assisted proof together with an extension of the hyperbolic theory developed to encompass attractors robustly containing equilibria. We present some of the main results on the statistical behavior of such systems. We show that for attractors of three-dimensional flows, robust chaotic behavior is equivalent to the existence of certain hyperbolic structures, known as singular-hyperbolicity. These structures, in turn, are associated with the existence of physical measures: in low dimensions, robust chaotic behavior for flows ensures the existence of a physical measure. We then give more details on recent results on the dynamics of singular-hyperbolic (Lorenz-like) attractors: (1) there exists an invariant foliation whose leaves are forward contracted by the flow (and further properties which are useful to understand the statistical properties of the dynamics); (2) there exists a positive Lyapunov exponent at every orbit; (3) there is a unique physical measure whose support is the whole attractor and which is the equilibrium state with respect to the center-unstable Jacobian; (4) this measure is exact dimensional; (5) the induced measure on a suitable family of cross-sections has exponential decay of correlations for Lipschitz observables with respect to a suitable Poincaré return time map; (6) the hitting time associated to Lorenz-like attractors satisfy a logarithm law; (7) the geometric Lorenz flow satisfies the Almost Sure Invariance Principle (ASIP) and the Central Limit Theorem (CLT); (8) the rate of decay of large deviations for the volume measure on the ergodic basin of a geometric Lorenz attractor is exponential; (9) a class of geometric Lorenz flows exhibits robust exponential decay of correlations; (10) all geometric Lorenz flows are rapidly mixing and their time-1 map satisfies both ASIP and CLT.

scholarly journals Small deformations of Helfrich energy minimising surfaces with applications to biomembranes

2017 ◽  
Vol 27 (08) ◽  
pp. 1547-1586 ◽  
Author(s):  
Charles M. Elliott ◽  
Hans Fritz ◽  
Graham Hobbs
Keyword(s):  
Numerical Solutions ◽  
General Setting ◽  
Height Function ◽  
Lagrange Equations ◽  
Flat Surfaces ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Point Forces ◽  
External Forces ◽  
Small Deformations

In this paper, we introduce a mathematical model for small deformations induced by external forces of closed surfaces that are minimisers of Helfrich-type energies. Our model is suitable for the study of deformations of cell membranes induced by the cytoskeleton. We describe the deformation of the surface as a graph over the undeformed surface. A new Lagrangian and the associated Euler–Lagrange equations for the height function of the graph are derived. This is the natural generalisation of the well-known linearisation in the Monge gauge for initially flat surfaces. We discuss energy perturbations of point constraints and point forces acting on the surface. We establish existence and uniqueness results for weak solutions on spheres and on tori. Algorithms for the computation of numerical solutions in the general setting are provided. We present numerical examples which highlight the behaviour of the surface deformations in different settings at the end of the paper.

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