Mathematics

A classical result in thermodynamic formalism is that for uniformly hyperbolic systems, every Hölder continuous potential has a unique equilibrium state. One proof of this fact is due to Rufus Bowen and uses the fact that such systems satisfy expansivity and specification properties. In these notes, we survey recent progress that uses generalizations of these properties to extend Bowen's arguments beyond uniform hyperbolicity, including applications to partially hyperbolic systems and geodesic flows beyond negative curvature. We include a new criterion for uniqueness of equilibrium states for partially hyperbolic systems with 1-dimensional center.

This chapter presents a dynamical systems point of view of the study of systems with delays. The focus is on how advanced tools from bifurcation theory, as implemented for example in the package DDE-BIFTOOL, can be applied to the study of delay differential equations (DDEs) arising in applications, including those that feature state-dependent delays. We discuss the present capabilities of the most recent release of DDE-BIFTOOL. They include the numerical continuation of steady states, periodic orbits and their bifurcations of codimension one, as well as the detection of certain bifurcations of codimension two and the calculation of their normal forms. Two longer case studies, of a conceptual DDE model for the El Ni{ñ}o phenomenon and of a prototypical scalar DDE with two state-dependent feedback terms, demonstrate what kind of insights can be obtained in this way.

Given two positive real numbers M and m and an integer n>2 , it is well known that we can find a family of solutions of the (n+1) -body problem where the body with mass M stays put at the origin and the other n bodies, all with the same mass m , move on the x - y plane following ellipses with eccentri\-city e . It is expected that this geometrical family that depends on e , has some bifurcations that produce solutions where the body in the center moves on the z -axis instead of staying put in the origin. By doing an analytic continuation of a periodic numerical solution of the 4 -body problem --the one displayed on the video this http URL --we surprisingly discovered that the origin of this periodic solution is not part of the geometrical family of elliptical solutions parametrized by the eccentricity e . It comes from a not so geometrical but easier to describe family. Having noticed this new family, the authors find an exact formula for the bifurcation point in this new family and use it to show the existence of a non-planar periodic solution for any pair of masses M , m , and any integer n . As a particular example, we find a solution where three bodies with mass 3 move around a body with mass 7 that moves up and down.

The Maslov index is a powerful tool for assessing the stability of solitary waves. Although it is difficult to calculate in general, a framework for doing so was recently established for singularly perturbed systems. In this paper, we apply this framework to standing wave solutions of a three-component activator-inhibitor model. These standing waves are known to become unstable as parameters vary. Our goal is to see how this established stability criterion manifests itself in the Maslov index calculation. In so doing, we obtain new insight into the mechanism for instability. We further suggest how this mechanism might be used to reveal new instabilities in singularly perturbed models.

We introduce a multi-parameter three-dimensional system of ordinary differential equations that exhibits dynamics on three distinct timescales. Our system is an extension of both a prototypical example introduced by Krupa et al. [14] and a canonical form suggested by Letson et al. [18]; in the three-timescale context, it admits a one-dimensional S-shaped supercritical manifold that is embedded into a two-dimensional S-shaped critical manifold in a symmetric fashion. We apply geometric singular perturbation theory to explore the dependence of the geometry of our system on its parameters. Then, we study the implications of that geometry for both the local and global dynamics. Our focus is on mixed-mode oscillations (MMOs) and their bifurcations; in particular, we uncover a geometric mechanism that encodes the transition from MMOs with single epochs of small-amplitude oscillations (SAOs) to those with double epochs of SAOs, and we show that the latter are more robust than in the two-timescale context. We demonstrate our results for the Koper model from chemical kinetics [13], which represents one particular realisation of our prototypical system. Finally, we illustrate how some of our results can be extended to more general systems with similar geometric properties, such as to a three-dimensional reduction of the Hodgkin-Huxley equations derived by Rubin and Wechselberger [20].

A long standing open problem in the theory of hyperfinite equivalence relations asks if the orbit equivalence relation generated by a Borel action of a countable amenable group is hyperfinite. In this paper we prove that this question always has a positive answer when the acting group is polycyclic, and we obtain a positive answer for all free actions of a large class of solvable groups including the Baumslag--Solitar group BS(1,2) and the lamplighter group. This marks the first time that a group of exponential volume-growth has been verified to have this property. In obtaining this result we introduce a new tool for studying Borel equivalence relations by extending Gromov's notion of asymptotic dimension to the Borel setting. We show that countable Borel equivalence relations of finite Borel asymptotic dimension are hyperfinite, and more generally we prove under a mild compatibility assumption that increasing unions of such equivalence relations are hyperfinite. As part of our main theorem, we prove for a large class of solvable groups that all of their free Borel actions have finite Borel asymptotic dimension (and finite dynamic asymptotic dimension in the case of a continuous action on a zero-dimensional space). We also provide applications to Borel chromatic numbers, Borel and continuous Folner tilings, topological dynamics, and C ∗ -algebras.

In this paper we provide a concrete construction of Furstenberg entropy values of τ -boundaries of the group Z[ 1 p 1 ,…, 1 p l ]⋊{ p n 1 1 ⋯ p n l l : n i ∈Z} by choosing an appropriate random walk τ . We show that the boundary entropy spectrum can be realized as the subsum-set for any given finite sequence of positive numbers.

We study the box dimensions of sets invariant under the toral endomorphism (x,y)↦(mx mod 1,ny mod 1) for integers n>m≥2 . The basic examples of such sets are Bedford-McMullen carpets and, more generally, invariant sets are modelled by subshifts on the associated symbolic space. When this subshift is topologically mixing and sofic the situation is well-understood by results of Kenyon and Peres. Moreover, other work of Kenyon and Peres shows that the Hausdorff dimension is generally given by a variational principle. Therefore, our work is focused on the box dimensions in the case where the underlying shift is not topologically mixing and sofic. We establish straightforward upper and lower bounds for the box dimensions in terms of entropy which hold for all subshifts and show that the upper bound is the correct value for coded subshifts whose entropy can be realised by words which can be freely concatenated, which includes many well-known families such as β -shifts, (generalised) S -gap shifts, and transitive sofic shifts. We also provide examples of transitive coded subshifts where the general upper bound fails and the box dimension is actually given by the general lower bound. In the non-transitive sofic setting, we provide a formula for the box dimensions which is often intermediate between the general lower and upper bounds.

It is well-known that billiards in polygons cannot be chaotic (hyperbolic). Particularly Kolmogorov-Sinai entropy of any polygonal billiard is zero. We consider physical polygonal billiards where a moving particle is a hard disc rather than a point (mathematical) particle and show that typical physical polygonal billiard is hyperbolic at least on a subset of positive measure and therefore has a positive Kolmogorov- Sinai entropy for any positive radius of the moving particle (provided that the particle is not so big that it cannot move within a polygon). This happens because a typical physical polygonal billiard is equivalent to a mathematical (point particle) semi-dispersing billiard. We also conjecture that in fact typical physical billiard in polygon is ergodic under the same conditions.

The moduli space of algebraic foliations on P2 of a fixed degree and with a center singularity has many irreducible components. We find a basis of the Brieskorn module defined for a rational function and prove that set of pull-back foliations forms an irreducible component of the moduli space. The main tools are the Picard-Lefschetz theory of a rational function in two variables, period integrals, and the Brieskorn module.