We obtain a generalized law of the iterated logarithm for a class of dependent processes with superdiffusive behaviour. Our results apply in particular to the Lorentz gas with infinite horizon.

Building upon previous works by Young, Chernov-Zhang and Bruin-Melbourne-Terhesiu, we present a general scheme to improve bounds on the statistical properties (in particular, decay of correlations, and rates in the almost sure invariant principle) for a class of non-uniformly hyperbolic dynamical systems. Specifically, for systems with polynomial, yet summable mixing rates, our method removes logarithmic factors of earlier arguments, resulting in essentially optimal bounds. Applications include Wojtkowski's system of two falling balls, dispersing billiards with flat points and Bunimovich's flower-shaped billiard tables.

Az isteni gondviselés nem csak a hős jutalmazásáról szól, de a különféle bűnök elkövetése utáni büntetésről is. A kisebb vagy súlyosabb bűnöknek számos formája ismeretes a népmesékben. A mesemondók saját tapasztalataikat vetik össze a bibliai bűnökkel, és mindenkor figyelembe veszik az istenítéletet. A büntetés, az ítélet történelmi koronként és mesemondói értelmezéstől függően változik a narratívákban.

We prove limit laws for infinite horizon planar periodic Lorentz gases when, as time n tends to infinity, the scatterer size $$\rho $$ may also tend to zero simultaneously at a sufficiently slow pace. In particular we obtain a non-standard Central Limit Theorem as well as a Local Limit Theorem for the displacement function. To the best of our knowledge, these are the first results on an intermediate case between the two well-studied regimes with superdiffusive $$\sqrt{n\log n}$$ scaling (i) for fixed infinite horizon configurations—letting first $$n\rightarrow \infty $$ and then $$\rho \rightarrow 0$$ —studied e.g. by Szász and Varjú (J Stat Phys 129(1):59–80, 2007) and (ii) Boltzmann–Grad type situations—letting first $$\rho \rightarrow 0$$ and then $$n\rightarrow \infty $$ —studied by Marklof and Tóth (Commun Math Phys 347(3):933–981, 2016) .

We introduce and study a general framework for modeling the evolution of crack networks. The evolution steps are triggered by exponential clocks corresponding to local micro-events, and thus reflect the state of the pattern. In an appropriate simultaneous limit of pattern domain tending to infinity and time step tending to zero, a continuous time model, specifically a system of ODE is derived that describes the dynamics of averaged quantities. In comparison with the previous, discrete time model, studied recently by two of the present three authors, this approach has several advantages. In particular, the emergence of non-physical solutions characteristic to the discrete time model is ruled out in the relevant nonlinear version of the new model. We also comment on the possibilities of studying further types of pattern formation phenomena based on the introduced general framework.

Az Isten fia kifejezés ritkán fordul elő a népmesékben, a mesemondók inkább az istenküldötte emberről beszélnek. Az istenfiúság elvont fogalom számukra, bár sejtik, hogy a Messiást vagy egy prófétát kell érteni rajta. A narratívákban megkülönböztetjük az istenküldötte embert a segítő lénytől. Egyfelől másként definiálják magukat, másfelől a hőssel folytatott diskurzusokban is mindkettő másként viselkedik.

Enthesitis is considered a hallmark manifestation of spondyloarthritis including axial spondyloarthritis and psoriatic arthritis. Detection of enthesitis might be challenging in both diagnostic and classification processes. In this debate, we discuss the controversy on the role of imaging in the detection of enthesitis including the relevance for treatment decisions in spondyloarthritis.

We prove limit laws for infinite horizon planar periodic Lorentz gases when, as time n tends to infinity, the scatterer size ρ may also tend to zero simultaneously at a sufficiently slow pace. In particular we obtain a non-standard Central Limit Theorem as well as a Local Limit Theorem for the displacement function. To the best of our knowledge, these are the first results on an intermediate case between the two well-studied regimes with superdiffusive √(nlog n) scaling (i) for fixed infinite horizon configurations – letting first n→∞ and then ρ→ 0 – studied e.g. by Szász & Varjú (2007) and (ii) Boltzmann-Grad type situations – letting first ρ→ 0 and then n →∞ – studied by Marklof & Tóth (2016).

We prove limit laws for infinite horizon planar periodic Lorentz gases when, as time n tends to infinity, the scatterer size ρ may also tend to zero simultaneously at a sufficiently slow pace. In particular we obtain a non-standard Central Limit Theorem as well as a Local Limit Theorem for the displacement function. To the best of our knowledge, these are the first results on an intermediate case between the two well-studied regimes with superdiffusive √ n log n scaling (i) for fixed infinite horizon configurations – letting first n→ ∞ and then ρ→ 0 – studied e.g. by Szász & Varjú (2007) and (ii) Boltzmann-Grad type situations – letting first ρ→ 0 and then n→ ∞ – studied by Marklof & Tóth (2016). Mathematics Subject Classification (2010): Primary: 37D50, Secondary 37A60, 60F05, 60F17, 82C05, 82C40

A Kárpát-medencei mesekorpusz ismeretében elmondhatjuk, hogy a Két testvér vagy a Két egyforma barát története a Bibliában és a Koránban is több egymást követő epizódból egységessé szőtt példázatos elbeszélés, mely az évszázadok alatt különböző forrásokból került át a mesehagyományba. A különböző lokális és vallási felekezetekhez tartozó elbeszélők jól érezték: magát a történetet további epizódokra lehet szétszálazni, hogy az egyes szövegek példázatos jellege mind az erénygyakorlás módozatai, mind a diskurzusok élettel telítettsége szempontjából a hallgatóság számára könnyebben befogadhatóvá váljon.

We survey applications of the theory of hyperbolic (and to a lesser extent non hyperbolic) billiards to some fundamental problems of statistical physics and their mathematically rigorous derivations in the framework of classical Hamiltonian systems.

We consider billiards obtained by removing three strictly convex obstacles satisfying the non-eclipse condition on the plane. The restriction of the dynamics to the set of non-escaping orbits is conjugated to a subshift on three symbols that provides a natural labeling of all periodic orbits. We study the following inverse problem: does the Marked Length Spectrum (i.e., the set of lengths of periodic orbits together with their labeling), determine the geometry of the billiard table? We show that from the Marked Length Spectrum it is possible to recover the curvature at periodic points of period two, as well as the Lyapunov exponent of each periodic orbit.

We prove sharp results on polynomial decay of correlations for nonuniformly hyperbolic flows. Applications include intermittent solenoidal flows and various Lorentz gas models including the infinite horizon Lorentz gas.

We prove exponential correlation decay in dispersing billiard flows on the 2-torus assuming finite horizon and lack of corner points. With applications aimed at describing heat conduction, the highly singular initial measures are concentrated here on 1-dimensional submanifolds (given by standard pairs) and the observables are supposed to satisfy a generalized Hölder continuity property. The result is based on the exponential correlation decay bound of Baladi et al. (Invent Math, 211:39–117, 2018. https://doi.org/10.1007/s00222-017-0745-1 ) obtained for Hölder continuous observables in these billiards. The model dependence of the bounds is also discussed.

For geometric Lorenz attractors (including the classical Lorenz attractor) we obtain a greatly simplified proof of the central limit theorem which applies also to the more general class of codimension two singular hyperbolic attractors. We also obtain the functional central limit theorem and moment estimates, as well as iterated versions of these results. A consequence is deterministic homogenisation (convergence to a stochastic differential equation) for fast-slow dynamical systems whenever the fast dynamics is singularly hyperbolic of codimension two.

We consider billiards obtained by removing three strictly convex obstacles satisfying the non-eclipse condition on the plane. The restriction of the dynamics to the set of non-escaping orbits is conjugated to a subshift on three symbols that provides a natural labeling of all periodic orbits. We study the following inverse problem: does the Marked Length Spectrum (i.e., the set of lengths of periodic orbits together with their labeling), determine the geometry of the billiard table? We show that from the Marked Length Spectrum it is possible to recover the curvature at periodic points of period two, as well as the Lyapunov exponent of each periodic orbit.

We study a class of globally coupled maps in the continuum limit, where the individual maps are expanding maps of the circle. The circle maps in question are such that the uncoupled system admits a unique absolutely continuous invariant measure, which is furthermore mixing. Interaction arises in the form of diffusive coupling, which involves a function that is discontinuous on the circle. We show that for sufficiently small coupling strength the coupled map system admits a unique absolutely continuous invariant distribution, which depends on the coupling strength ε. Furthermore, the invariant density exponentially attracts all initial distributions considered in our framework. We also show that the dependence of the invariant density on the coupling strength ε is Lipschitz continuous in the BV norm. When the coupling is sufficiently strong, the limit behavior of the system is more complex. We prove that a wide class of initial measures approach a point mass with support moving chaotically on the circle. This can be interpreted as synchronization in a chaotic state.

We analyse the process of energy exchanges generated by the elastic collisions between a point-particle, confined to a two-dimensional cell with convex boundaries, and a ‘piston’, i.e. a line-segment, which moves back and forth along a one-dimensional interval partially intersecting the cell. This model can be considered as the elementary building block of a spatially extended high-dimensional billiard modeling heat transport in a class of hybrid materials exhibiting the kinetics of gases and spatial structure of solids. Using heuristic arguments and numerical analysis, we argue that, in a regime of rare interactions, the billiard process converges to a Markov jump process for the energy exchanges and obtain the expression of its generator.

Dispersing billiards with cusps are deterministic dynamical systems with a mild degree of chaos, exhibiting "intermittent" behavior that alternates between regular and chaotic patterns. They are characterized by decay of correlations of order 1/n and a central limit theorem with a non-classical scaling factor of root n log n. As for the growth of the pth moments of the appropriately normalized Birkhoff sums, it follows from the results of [Electron. J. Probab. 28 (2014), pp. 30] that these converge to the moments of the limit normal distribution only for p < 2 and diverge for p > 2. Here we focus on the critical case p = 2 and prove a doubling effect: the second moments converge, but their limit is twice the second moment of the limit normal distribution.

Globally coupled doubling maps are studied in this paper. In this setting and for finitely many sites, two distinct bifurcation values of the coupling strength have been identified in the literature, corresponding to the emergence of contracting directions (Koiller and Young in Nonlinearity, 23(5):1121, 2010) and, specifically for $$N=3$$ sites, to the loss of ergodicity (Fernandez in Journal of Statistical Physics, 154(4):999–1029, 2014). On the one hand, we reconsider these results and provide an interpretation of the observed dynamical phenomena in terms of the synchronization of the sites. On the other hand, we initiate a new point of view which focuses on the evolution of distributions and allows to incorporate the investigation of a continuum of sites. In particular, we observe phenomena that is analogous to the limit states of the contracting regime of $$N=3$$ sites.