DinAmicI: Another Internet Seminar

Il più antico sistema astronomico che prevede una Terra in moto è quello attribuito al pitagorico Filolao, del V secolo a.C., nel quale la Terra e una misteriosa e invisibile "Antiterra" ruoterebbero entrambe, insieme alla Luna, ai pianeti e al Sole, intorno a un misterioso e invisibile "Fuoco centrale”. Nel seminario si sostiene che le stranezze del sistema (quasi completamente privo di relazione con dati osservativi) non siano dovute a Filolao, ma al fraintendimento del suo pensiero da parte di Aristotele e all'adesione quasi unanime all'interpretazione aristotelica degli studiosi dei successivi ventitré secoli.

In this talk, we introduce higher-order expansions in the mixing local limit theorems for Birkhoff sums. We will discuss the general results under technical assumptions, and illustrate them by different examples (e.g. subshifts of finite type, Young towers, Sinai billiards, and random matrix products), including situations of unbounded observables with integrability order arbitrarily close to the optimal moment condition required in the i.i.d. setting. This is joint work with Françoise Pène.

Double rotations are the simplest subclass of interval translation mappings. A double rotation is of finite type if its attractor is an interval and of infinite type if it is a Cantor set. It is easy to see that the restriction of a double rotation of finite type to its attractor is simply a rotation. It is known due to Suzuki – Ito – Aihara and Bruin – Clark that double rotations of infinite type are defined by a subset of zero measure in the parameter set. We introduce a new renormalization procedure on double rotations, which is reminiscent of the classical Rauzy induction. Using this renormalization, we prove that the set of parameters which induce infinite type double rotations has Hausdorff dimension strictly smaller than 3. Moreover, we construct a natural invariant measure supported on these parameters and show that, with respect to this measure, almost all double rotations are uniquely ergodic. In my talk I plan to outline this proof that is based on the recent result by Fougeron for simplicial systems. I also hope to discuss briefly some challenging open questions and further research plans related to double rotations. The talk is based on joint work with Charles Fougeron, Pascal Hubert and Sasha Skripchenko.

We prove that skew systems with a sufficiently expanding base have “approximate” statistical properties similar to random ergodic Markov chains. For example, they exhibit approximate exponential decay of correlations, meaning that the exponential rate is observed modulo a controlled error. The fiber maps are only assumed to be Lipschitz regular and to depend on the base in a way that guarantees diffusive behaviour on the vertical component. The assumptions do not imply an hyperbolic picture and one cannot rely on the spectral properties of the transfer operators involved. The approximate nature of the result is the inevitable price one pays for having so mild assumptions on the dynamics on the vertical component. The error in the approximation is shown to go to zero when the expansion of the base tends to infinity.

After a very brief review of basics on elliptic curves and their families, we shall consider "sections" of such  families, and especially their "torsion values".  For instance, what can be said of the complex numbers b for which (2, \sqrt{2(2-b)}) is torsion on the Legendre curve y^2=x(x-1)(x-b)? In particular, we shall recall results of "Manin-Mumford type" and focus to illustrate some applications to elliptical billiards. Finally, if time allows we shall frame these issues as special cases of a general question in arithmetic dynamics, which can be treated with different methods, depending on the context. (Most results refer to work with Pietro Corvaja and David Masser.)

In this talk I will consider the following question: if one picks a piecewise affine map of the circle “at random”, what dynamical behaviour are we likely to observe? The case of standard circle diffeomorphisms has been studied by Herman in the 80s; in this emblematic case the problem can be reduced to theorems close to KAM theory. For the piecewise affine case, we put forward a geometric approach, inspired by methods from both Teichmüller theory and hyperbolic geometry.

We are interested in the limit behaviour of Birkhoff sums over an infinite sigma-finite measure space. If the observable is integrable then — by a classical theorem by Aaronson — there exists no sequence of real numbers such that the Birkhoff sum normed by this sequence converges almost surely to 1. Under strong mixing conditions on the underlying system we prove a generalized strong law of large numbers for integrable observables using a truncated sum adding a suitable number of terms depending on the point of evaluation. For f not integrable we give conditions on f such that the Birkhoff sum normed by a sequence of real numbers converges almost surely to 1. Joint work with Claudio Bonanno.

The dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by the Euler equations. Recently, Tao launched a programme to address the global existence problem for the Euler and Navier Stokes equations based on the concept of universality. Inspired by this proposal, we show that the stationary Euler equations exhibit several universality features, In the sense that, any non-autonomous flow on a compact manifold can be extended to a smooth stationary solution of the Euler equations on some Riemannian manifold of possibly higher dimension. These results can be viewed as lending support to the intuition that solutions to the Euler equations can be extremely complicated in nature. A key point in the proof is looking at the h-principle in contact geometry through a contact mirror, unveiled by Sullivan, Etnyre and Ghrist more than two decades ago. We end up this talk addressing an apparently different question: What kind of physics might be non-computational? Using the former universality result, we can establish the Turing completeness of the steady Euler flows, i.e., there exist solutions that encode a universal Turing machine and, in particular, these solutions have undecidable trajectories. But, in view of the increase of dimension yielded by our proof, the question is: can this be done in dimension 3? We will prove the existence of Turing complete fluid flows on a 3-dimensional geometric domain. Our novel strategy uses the computational power of symbolic dynamics and the contact mirror again. This talk is based on joint work with Robert Cardona and Fran Presas (arXiv:1911.01963 and arXiv:2012.12828).

When one considers manifolds with boundary, billiard dynamics are the natural analogue of standard geodesic dynamics. Namely, instead of having geodesics escape at the boundary, we force them back into the manifold using the reflection law. In other dynamical settings, similar constructions are possible: In 2006, B. Khesin and S. Tabachnikov initiated the study of billiards in the semiriemannian setting, studying the integrability of various tables. In recent years we have also seen the appearance of several billiard setups of symplectic nature. In this talk I will discuss recent work with L. Dahinden in which we look at billiards in subriemannian geometry. I will sketch how the reflection law arises naturally both from the control-theoretical and symplectic perspectives, how the reflection is problematic at tangency points between the distribution and the boundary of the table, and I will introduce some concrete examples. My ultimate goal will be to pose several intriguing open questions.

The symbolic action of a substitution on the binary expansion generates naturally an interval map. In case of erasing substitutions, the map is typically Baire-1 and not Darboux. As a model case, we consider what is arguably the simplest erasing substitution, mapping 0s to the empty word and 1s to 0 or 1 depending on the parity. The corresponding interval map is shown to have fractal properties (bounds are given on the Hausdorff dimension of the fibers) and to display rich dynamical behavior, including Devaney chaos, uniform distributional chaos of type 1, infinite topological entropy as well as the presence of cycles attracting in a finite time every rational.

I will present some decay of correlations results on skew products which are locally accessible. The results rely on the study of a twisted transfer operator and could be generalized to many other situations. I will also present numerical counterparts to such results.

In this talk I shall discuss Birkhoff-Poritsky conjecture for centrally-symmetric C^2-smooth convex planar billiards. We assume that the domain A between the invariant curve of 4-periodic orbits and the boundary of the phase cylinder is foliated by C^0-invariant curves. Under this assumption we prove that the billiard curve is an ellipse. Other versions of Birkhoff-Poritsky conjecture follow from this result. For the original Birkhoff-Poritsky formulation we show that if a neighborhood of the boundary of billiard domain has a C^1-smooth foliation by convex caustics of rotation numbers in the interval (0; 1/4] then the boundary curve is an ellipse. The main ingredients of the proof are: (1) the non-standard generating function for convex billiards; (2) the remarkable structure of the invariant curve consisting of 4-periodic orbits; and (3) the integral-geometry approach initiated by the author for rigidity results of circular billiards. Surprisingly, we establish a Hopf-type rigidity for billiards in the ellipse. Based on joint work with Andrey E. Mironov (Novosibirsk).

Inspired by the problem of modeling the so called anthropogenic forcing in climatology, e.g. the effects of the emissions of greenhouse gases in the atmosphere, we introduce a novel type of random perturbation for the classical Lorenz flow and prove its stochastic stability. The perturbation acts on the system in an impulsive way, hence is not of diffusive type. Namely, given a cross-section M for the unperturbed flow, each time the trajectory of the system crosses M the phase velocity field is changed with a new one sampled at random from a suitable neighborhood of the unperturbed one. The resulting random evolution is therefore described by a piecewise deterministic Markov process. The proof of the stochastic stability for the unperturbed flow is then carried on working either in the framework of the Random Dynamical Systems or in that of semi-Markov processes. Joint work with Sandro Vaienti.

We prove for the N-body problem the existence of hyperbolic motions for any prescribed limit shape and any given initial configuration of the bodies. The energy level h>0 of the motion can also be chosen arbitrarily. Our approach is based on the construction of a global viscosity solutions for the Hamilton-Jacobi equation H(x,du(x))=h. Our hyperbolic motion is in fact a calibrating curve of the viscosity solution. The presented results can also be viewed as a new application of Marchal’s theorem, whose main use in recent literature has been to prove the existence of periodic orbits. Joint work with Ezequiel Maderna.

We study the spectrum of transfer operators associated to various dynamical systems. Our aim is to obtain precise information on discrete spectrum. To this end we propose a unitary approach. We consider various settings where new information can be obtained following different branches along the proposed path. These settings include affine expanding Markov maps, uniformly expanding Markov maps, non-uniformly expanding maps, hyperbolic diffeomorphisms. We believe this to be the germ of a general theory. Joint work with O. Butterley and N. Kiamari.

Defining entropy on noncompact metric spaces is a tricky business, since there are several natural and nonequivalent generalizations of the usual notions of entropy for continuous maps on compact spaces. By defining entropy for transcendental maps on the complex plane as the sup over the entropy restricted to compact forward invariant subsets, we prove that with this definition the entropy of such functions is infinite. The proof relies on covering results which are distinctive to holomorphic maps.

In this talk I will present a family of one dimensional systems with random additive noise such that, as the noise size increases, the Lyapunov exponent of the stationary measure transitions from positive to negative. This phenomena is known in literature as Noise Induced Order, and was first observed in a model of the Belosouv-Zhabotinsky reaction and its existence was proven only recently by Galatolo-Monge-Nisoli. In the talk I will show how this phenomena is strictly connected with non-uniform hyperbolicity and the coexistence of regions of expansion and contraction in phase space; the result is attained through a result on the continuity of the Lyapunov exponent of the stationary measure with respect to the size of the noise.

We consider the free motion of a point particle inside a circular billiard with periodically moving boundary, with the assumption that the collisions of the particle with the boundary are elastic so that the energy of the particle is not preserved. It is known that if the motion of the boundary is regular enough then the energy is bounded due to the existence of invariant curves. We show that it is nevertheless possible that the motion of the particle is chaotic, also under regularity assumptions for the moving boundary. More precisely, we show that there exists a class of functions describing the motion of the boundary for which the billiard map admits invariant probability measures with positive metric entropy. The proof relies on variational techniques based on Aubry-Mather theory. Joint work with Claudio Bonanno.

It is well known that the integral of an observable is preserved by induction. We are interested here in extensions of this result to moments of order 2 and 3. We have two natural candidates for the second and third order moments: the classical asymptotic variance (given by the Green-Kubo formula) and an analogous quantity of the third order. This question arises from the proof of CLT. In some cases, the asymptotic variance in the CLT can be expressed on the one hand in terms of the classical Green-Kubo formula and on the other hand in terms of the Green-Kubo formula for the induced system. Under general assumptions (involving transfer operators), we prove that the asymptotic variance is preserved by induction and that the natural third order quantity is preserved up to an error term. This is joint work with Damien Thomine.

We consider families of skew-product maps, representing systems evolving in discrete time in which, at each time step, one of a number of transformations is chosen according to an i.i.d process and applied. We extend the notion of matching for such dynamical systems and we show that, for a certain family of piecewise affine random maps of the interval, the property of random matching implies that any invariant density is piecewise constant. We give an application by introducing a one-parameter family of random maps generating signed binary expansions of numbers. This family has random matching for Lebesgue almost every parameter, producing matching intervals that are related to the ones obtained for the Nakada continued fraction transformations. We use this property to study the expansions with minimal weight. Joint with K. Dajani, and C. Kalle.

Partly with Jacques Fejoz, Richard Montgomery, Stefan Fleischer and Manuel Quaschner. The past and future of scattering particle systems is partly determined by their asymptotic velocity, that is, the Cesàro limit of the velocity. That this exists for bounded interactions and all initial conditions, is part of a statement sometimes called ‘asymptotic completeness’. The same statement does not apply to individual initial conditions in celestial mechanics. However, at least for up to four particles, nonexistence of asymptotic velocity is a measure zero phenomenon. We explain some new ideas connected with the proof (Poincaré section techniques for wandering sets, non-deterministic particle systems, and walks on a poset of set partitions).

A square-tiled surface (STS) is a branched cover of the standard square torus with branching over exactly one point. They are concrete examples of translation surfaces which are an important class of singular flat metrics on 2-manifolds with applications in Teichmüller theory and polygonal billiards. In this talk we will consider a randomizing model for STSs based on permutation pairs and use it to compute the genus distribution. We also study holonomy vectors (Euclidean displacement vectors between cone points) on a random STS. Holonomy vectors of translation surfaces provide coordinates on the space of translation surfaces and their enumeration up to a fixed length has been studied by various authors such as Eskin and Masur. In this talk, we obtain finer information about the set of holonomy vectors, Hol(S), of a random STS. In particular, we will see how often Hol(S) contains the set of primitive integer vectors and find how often these sets are exactly equal.

We develop a quenched thermodynamic formalism for random dynamical systems generated by countably branched, piecewise-monotone mappings of the interval that satisfy a random covering condition. Joint with J. Atnip, G. Froyland and C. Gonzalez-Tokman

Equidistribution of horocycles on hyperbolic surfaces has been used to dynamically answer several probabilistic questions about number-theoretical objects. In this talk we focus on horocycle lifts, i.e. curves on higher-dimensional manifolds whose projection to the hyperbolic surface is a classical horocycle, and their behaviour under the action of the geodesic flow. It is known that when such horocycle lifts are `generic’, then their push forward via the geodesic flow becomes equidistributed in the ambient manifold. We consider certain ‘non-generic’ (i.e. rational) horocycle lifts, in which case the equidistribution takes place on a sub-manifold. We then use this fact to study the tail distribution of quadratic Weyl sums when one of their arguments is random and the other is rational. In this case we obtain random variables with heavy tails, all of which only possess moments of order less than 4. Depending on the rational argument, we establish the exact tail decay, which can be described with the help of the Dedekind \psi-function. Joint work with Tariq Osman.

In a project with P. Bálint, J. De Simoi and V. Kaloshin, we have been studying the inverse problem for a class of open dispersing billiards obtained by removing from the plane a finite number of smooth strictly convex scatterers satisfying a non-eclipse condition. The dynamics of such billiards is hyperbolic (Axiom A), and there is a natural labeling of periodic orbits. We show that it is generically possible, in the analytic category and for billiard tables with two (partial) axial symmetries, to determine completely the geometry of those billiards from the purely dynamical data encoded in their Marked Length Spectrum (lengths of periodic orbits + marking). An important step is the obtention of asymptotic estimates for the Lyapunov exponents of certain periodic points accumulating a reference periodic point, which turn out to be useful in the study of other rigidity problems. In particular, I will explain the results obtained in a joint work with J. De Simoi, K. Vinhage and Y. Yang on the question of entropy rigidity for 3-dimensional Anosov flows and dispersing billiards.

We establish general central limit theorems for an action of a group on a hyperbolic space with respect to counting for the word length in the group. In 2013, Chas, Li, and Maskit produced numerical experiments on random closed geodesics on a hyperbolic pair of pants. Namely, they drew uniformly at random conjugacy classes of a given word length, and considered the hyperbolic length of the corresponding closed geodesic on the pair of pants. Their experiments lead to the conjecture that the length of these closed geodesics satisfies a central limit theorem, and we proved this conjecture in 2018. In our new work, we remove the assumptions of properness and smoothness of the space, or cocompactness of the action, thus proving a general central limit theorem for group actions on hyperbolic spaces. We will see how our techniques replace the classical thermodynamic formalism and allow us to provide new applications, including to lengths of geodesics in geometrically finite manifolds and to intersection numbers with submanifolds. Joint work with I. Gekhtman and S. Taylor.