On a problem of A. Weil

A topological invariant of the geodesic laminations on a modular surface is constructed. The invariant has a continuous part (the tail of a continued fraction) and a combinatorial part (the singularity data). It is shown, that the invariant is complete, i.e. the geodesic lamination can be recovered from the invariant. The continuous part of the invariant has geometric meaning of a slope of lamination on the surface.Comment: to appear Beitr\"age zur Algebra und Geometri

Chaos in the Gauge/Gravity Correspondence

We study the motion of a string in the background of the Schwarzschild black hole in AdS_5 by applying the standard arsenal of dynamical systems. Our description of the phase space includes: the power spectrum, the largest Lyapunov exponent, Poincare sections and basins of attractions. We find convincing evidence that the motion is chaotic. We discuss the implications of some of the quantities associated with chaotic systems for aspects of the gauge/gravity correspondence. In particular, we suggest some potential relevance for the information loss paradox.Comment: 29 pages, 11 figure

Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors

We consider two dimensional maps preserving a foliation which is uniformly contracting and a one dimensional associated quotient map having exponential convergence to equilibrium (iterates of Lebesgue measure converge exponentially fast to physical measure). We prove that these maps have exponential decay of correlations over a large class of observables. We use this result to deduce exponential decay of correlations for the Poincare maps of a large class of singular hyperbolic flows. From this we deduce logarithm laws for these flows.Comment: 39 pages; 03 figures; proof of Theorem 1 corrected; many typos corrected; improvements on the statements and comments suggested by a referee. Keywords: singular flows, singular-hyperbolic attractor, exponential decay of correlations, exact dimensionality, logarithm la

Reinforcement learning or active inference?

This paper questions the need for reinforcement learning or control theory when optimising behaviour. We show that it is fairly simple to teach an agent complicated and adaptive behaviours using a free-energy formulation of perception. In this formulation, agents adjust their internal states and sampling of the environment to minimize their free-energy. Such agents learn causal structure in the environment and sample it in an adaptive and self-supervised fashion. This results in behavioural policies that reproduce those optimised by reinforcement learning and dynamic programming. Critically, we do not need to invoke the notion of reward, value or utility. We illustrate these points by solving a benchmark problem in dynamic programming; namely the mountain-car problem, using active perception or inference under the free-energy principle. The ensuing proof-of-concept may be important because the free-energy formulation furnishes a unified account of both action and perception and may speak to a reappraisal of the role of dopamine in the brain

Multiscale Systems, Homogenization, and Rough Paths:VAR75 2016: Probability and Analysis in Interacting Physical Systems

In recent years, substantial progress was made towards understanding convergence of fast-slow deterministic systems to stochastic differential equations. In contrast to more classical approaches, the assumptions on the fast flow are very mild. We survey the origins of this theory and then revisit and improve the analysis of Kelly-Melbourne [Ann. Probab. Volume 44, Number 1 (2016), 479-520], taking into account recent progress in $p$-variation and c\`adl\`ag rough path theory.Comment: 27 pages. Minor corrections. To appear in Proceedings of the Conference in Honor of the 75th Birthday of S.R.S. Varadha

The Conley Conjecture and Beyond

This is (mainly) a survey of recent results on the problem of the existence of infinitely many periodic orbits for Hamiltonian diffeomorphisms and Reeb flows. We focus on the Conley conjecture, proved for a broad class of closed symplectic manifolds, asserting that under some natural conditions on the manifold every Hamiltonian diffeomorphism has infinitely many (simple) periodic orbits. We discuss in detail the established cases of the conjecture and related results including an analog of the conjecture for Reeb flows, the cases where the conjecture is known to fail, the question of the generic existence of infinitely many periodic orbits, and local geometrical conditions that force the existence of infinitely many periodic orbits. We also show how a recently established variant of the Conley conjecture for Reeb flows can be applied to prove the existence of infinitely many periodic orbits of a low-energy charge in a non-vanishing magnetic field on a surface other than a sphere.Comment: 34 pages, 1 figur