Papers

In the late 1980s, Baulieu and Grossman demonstrated that the supersymmetric formulation of Langevin stochastic differential equations (SDEs), proposed earlier by Parisi and Sourlas, belongs to the family of Witten-type topological field theories (TFTs). From a certain angle, this finding may appear puzzling: TFTs have no local degrees of freedom, whereas SDEs do exhibit local fluctuations. In this paper, we address this apparent contradiction in the context of the supersymmetric theory of stochastic dynamics (STS), a generalization of the Parisi-Sourlas-Baulieu-Grossman approach to SDEs of arbitrary form. We further discuss how, within STS, dynamical chaos can be identified with the spontaneous breakdown of the corresponding topological supersymmetry, while 1/f noise can be understood as a consequence of the Goldstone theorem. The butterfly effect, in turn, calls for an effective field-theoretic description that may itself possess a hidden topological structure, as we speculate on the basis of the Ginzburg-Landau approach and insights from AdS/CFT duality.

Satellite gravimetry has become essential in many areas of earth science. However, the resolution of satellite gravitational models remains low at scales of a few hundreds km and no gravity recovery methods can take full advantages of unprecedented high accuracy of satellite tracking measurements. We first provide a unified theoretical framework of parameter estimation in differential equations for satellite gravimetry and then briefly review the mathematical methods to compute the gravity field of the Earth from satellite tracking. We focus on the collocation method, Kaula linear perturbations, two-point boundary value problems and orbit-energy-based methods. The numerical integration method is also included in this review, though it has been proved to be mathematically incorrect and physically not permitted. The reason is that it has become the standard method to routinely produce global gravitational models from satellite tracking data, which have been widely applied in many different areas of earth science. Because it is not clear how the incorrect foundation would affect gravity products from satellite tracking, we do not review any applications of these products. We then present a measurement-based perturbation theory to estimate the gravity field of the Earth, which can fully utilize both precise satellite orbits of arbitrary length and unprecedented high accuracy of satellite and inter-satellite tracking. The method is theoretically free of modeling errors, is capable of extracting any small forces from satellite and inter-satellite tracking data and provides a guarantee for high-precision and high-resolution global gravity models. Finally, we assume a reference gravity model and derive local solutions to the Newton's nonlinear governing differential equations of satellite motion for scattered tracking data that can still be important in some applications.

The consensus of continuous-time multi-agent systems with unbounded and heterogeneous constant delays is investigated by combining frequency-domain analysis and algebraic graph theory. Several types of signed Laplacians are constructed to characterize consensusability under delays. The core results are established based on the defined delay-embedded signed Laplacian, where a small-delay link creates a cooperative interaction and a possibly unbounded large-delay link creates an antagonistic interaction between the agents. The dividing line between small and large delays is given by $\tau_{ij}=\pi/2\lambda_{\max}(L_0)$, where $\lambda_{\max}(L_0)$ refers to the maximum eigenvalue of the conventional graph Laplacian. It is proved that the consensusability is preserved if the delay-embedded signed Laplacian is positive semi-definite with a simple zero eigenvalue. Moreover, we derived some consensus conditions in terms of the extended effective resistance which measures the overall coupling between two sets of agents. The obtained results provide new insights into the mechanism of delayed consensus from the interplay between the small-delay-induced cooperativeness and large-delay-induced antagonism in the underlying network topology.

We study the structure of the Birkhoff center and the recurrent behavior of differentially positive systems whose linearizations along trajectories preserve a homogeneous cone field on a homogeneous space. Such cone fields arise naturally from general relativity and Lie theory. We establish an order-structural dichotomy for the Birkhoff center: every connected component of the Birkhoff center, as well as the support of any invariant measure, is either strongly ordered or unordered. This yields a comprehensive characterization of recurrent dynamics for differentially positive systems on homogeneous spaces, interpreted through the lens of the underlying order relation.

A scuba diver holding constant depth balances on an unstable equilibrium: the gas carried in the suit and buoyancy compensator compresses with depth, so the buoyant force falls as the diver sinks and rises as the diver ascends. We represent the diver as a proportional-derivative controller that regulates this compressible-buoyancy saddle after a finite reaction delay, and we derive the governing delay differential equation from the vertical force balance and the isothermal gas law, reducing it to a damping ratio, two control gains, and a dimensionless delay. The characteristic spectrum, obtained by pseudospectral collocation of the semigroup generator and checked against a direct Newton solution of the characteristic equation, locates the Hopf boundary that separates stable hovering from sustained porpoising; for the baseline diver the critical reaction delay is 3.36 s and the onset period is 28.8 s. The bifurcation is supercritical, and because the saturating force is the quadratic hydrodynamic drag, the limit-cycle amplitude grows in proportion to the delay excess rather than as its square root. The safe-operating envelope shows that runaway ascent is triggered by saturation of the compensator, not by loss of linear stability, so a stable and an unstable diver can share the same escape threshold. As onset is approached, the lag-one autocorrelation and variance rise while the fitted recovery rate falls and matches the spectral abscissa, giving an eigenvalue-exact early warning of the transition.

An admissible configuration $\Lambda\subset\Hh^m\simeq\Rr^{4m}$ defines a compact intersection of quadrics $Z_{\Hh}(\Lambda)$ and a free diagonal $\Sp(1)$ quotient $N_{\Hh}(\Lambda)$. We realize this quotient as the leaf space of a real dilation action on an open subset of quaternionic projective space. The total space is a quaternionic moment-angle polyhedral product. Every quotient is $2$-connected; when there are no indispensable coordinates, it is $3$-connected and its fourth integral cohomology is generated by the Euler class $e$ of the associated oriented rank-four bundle. We compute $p_1(TN_{\Hh})=2(n-2)e$ and prove that no LVMQ manifold admits a quaternionic Kähler metric. An explicit infinite family consists of products of spheres admitting both homogeneous positive Einstein metrics and quaternionic toric structures in the sense of Gentili–Gori–Sarfatti. We also describe the trace foliations of the real action in the Poincaré domain. For the separate rank-one distribution $X\Hh$, an invertible quaternionic-linear field is integrable only in the real scalar case. A nonlinear field with positive real scalar derivative at the origin and involutive $X\Hh$ has exactly the standard quaternionic Hopf foliation on every sufficiently small centered sphere.

Pathological neuronal synchrony provides one practical motivation for studying sparse desynchronisation control in networked oscillatory systems, particularly in applications where actuation and communication are resource constrained, as in deep brain stimulation. Motivated by this challenge, we study how to stabilise desynchronisation in coupled oscillatory dynamical systems without continuously updated control, even when the oscillators' phase is unavailable. We develop a general event-triggered control framework for stabilising the desynchronised state of coupled limit-cycle oscillatory dynamical systems. Our analysis establishes a unified theoretical result showing that desynchronisation can be achieved by various feedback controllers that act sparsely and depend only on an order parameter observable. The proposed controllers admit a gradient-descent interpretation and stabilise the desynchronisation state of general phase-reduced oscillator networks. We further prove the controlled systems under event-triggered mechanisms possess a strictly positive lower bound of the inter-event dwell-time, excluding Zeno behaviour. To address the practical unavailability of exact phase reductions, we introduce a pseudo-phase construction that yields a computable order parameter from state measurements alone. Numerical studies on representative oscillator networks demonstrate the effectiveness and robustness of the proposed framework.

The empirical transfer function estimate (ETFE) is a widely used method for system identification of linear time-invariant (LTI) systems in engineering disciplines such as acoustics, audio engineering, seismography, and tomography. However, ETFE suffers from numerical limitations when the excitation signal is band-limited or vanishes at certain frequencies, which is a common physical constraint of the excitation in practice. In such cases, division in the frequency domain becomes heavily ill-conditioned, and small measurement disturbances or numerical inaccuracies can degrade the solution. This paper presents L-ETFE, a generalization of ETFE based on Laguerre-Fourier expansions that addresses these limitations. After a suitable transformation, the method can yield a well-conditioned circulant problem even when the original ETFE system is ill-conditioned. Solving this problem via classical ETFE yields the discrete Laguerre-Fourier coefficients of the system's transfer function. The desired impulse response (IR) of the system-to-be-identified can then be recovered by a subsequent transformation pipeline. We derive novel and efficient algorithms for performing these transformations and analyse the conditioning of the transformed problem, explicitly characterizing its dependence on the input and a parameter used in the Laguerre-Fourier expansion. We evaluate the method on two simulated discrete-time LTI systems of varying complexity. The experiments demonstrate accurate IR recovery for spectral-zero and band-limited excitation, where standard ETFE fails.

This paper studies periodic inventory oscillations in delayed supply-chain networks of identical warehouses arranged in a ring. After reducing the inventory–replenishment system to a second-order delay equation, we formulate the periodic problem as an equivariant compact perturbation of the identity, with symmetries generated by the ring structure, the odd nonlinearity, and time translations. Using equivariant degree theory, we characterize critical parameter pairs, compute the associated crossing numbers, and construct a local bifurcation invariant whose nonvanishing guarantees nonconstant periodic solutions and provides information about their spatial and spatio-temporal symmetry types. By restricting the two-parameter problem to an admissible transversal curve, we further establish a global continuation alternative for the bifurcating branches. Examples with \(D_7\)- and \(D_8\)-symmetry illustrate the computation of critical parameters, bifurcation invariants, and admissible orbit types. Finally, Floquet theory is applied to small-amplitude periodic branches, yielding a leading-order stability criterion for nonresonant perturbation modes. The results provide a unified framework for describing the existence, symmetry, global continuation, and leading-order stability of periodic inventory oscillations in delayed ring supply-chain networks.

We develop an abstract Banach principle in vector lattices and apply it to obtain lattice-theoretic versions of the individual and maximal ergodic theorems, without recourse to any measure representation. Considering a sequence of bounded operators with values in a Dedekind $\sigma$-complete vector lattice endowed with a locally solid topology satisfying the $\sigma$-Lebesgue property, we prove that the set of points at which the sequence is order convergent is a closed subspace and coincides with the whole space whenever convergence holds on a dense subset. Investigating positive, power-bounded, mean ergodic operators on order continuous Banach lattices, we construct a topology on the universal completion induced by a strictly positive order continuous functional and prove that the Cesàro means converge in order in the universal completion and, in particular, uo-converge in the original lattice. Moreover, we introduce the notion of a superinvariant pair and derive a lattice-theoretic Hopf inequality together with a weak type estimate via band projections, which yields an abstract maximal ergodic theorem. A spectral-theoretic version of the theorem follows from classical Perron–Frobenius theory. Finally, we specialise the abstract framework to the model space $L^0(\Omega)$ and revisit the classical Banach principle, the Hopf–Dunford–Schwartz theorem and Doob's martingale convergence theorem.

Most models assess whether a newly introduced pathogen strain can grow when rare by testing it against a steady endemic background. Resident strains, however, can have more than one long-term behavior under the same parameters. They may settle to a steady state or continue through recurring outbreaks, leaving different fractions of hosts with different infection histories. We study this possibility in a stylized immune-history model. For two resident strains, we identify a parameter point where a stable endemic equilibrium and a stable outbreak cycle are both locally attracting. Numerical continuation maps a Hopf branch and a branch of folds of periodic orbits that bound a candidate bistability region, organized locally by a generalized Hopf (Bautin) point. We then embed the resident model exactly in a 20-state, three-strain system. When strain 3 infectiousness and total removal are common across four host-history classes, rare growth is determined by a weighted sum of the corresponding uninfected host fractions. The equilibrium and cycle yield different weighted sums, so one fixed strain 3 parameter set grows near one resident attractor but declines near the other. A second parameter set gives the reverse ordering, and both patterns persist over open regions of invader parameter space. Sixty full-model simulations with progressively smaller inocula reproduce these linear predictions across cycle phases and numerical solvers. Thus, knowing which resident strains are present may not suffice to predict initial invasion; the realized resident attractor can also matter. These results concern local growth from rarity, not eventual establishment, long-term coexistence, or post-invasion dynamics.

Consider the action of a rational function on the Berkovich projective line over an algebraically closed field complete with respect to a nontrivial non-archimedean absolute value. We say a connected component of the fixed locus is a "non-attracting subtree" if it contains no attracting points. We define a "degree of a non-attracting subtree," in terms of the local degrees of the rational function at its points. We show that the degrees of non-attracting subtrees sum up to the degree of the rational function. This counting result is obtained by studying the potential of a discrete measure charging each non-attracting subtree with mass equal to its degree. We prove that as we iterate the rational function these measures equidistribute to the equilibrium measure. Under certain conditions, notably in the case of polynomials, this reduces to the equidistribution of repelling points. We also show that repelling type II points become negligible under iteration when the Lyapunov exponent is positive. These results give a partial and a complete answer to questions of Favre and Rivera-Letelier.

We investigate proper actions of arbitrary semigroups on separable locally compact metric spaces, where point orbits are allowed to escape to infinity. An action is pointwise uniquely ergodic when every compact orbit closure supports exactly one invariant probability measure and non-compact orbit closures support none. The associated ergodic map therefore assigns the selected probability measure to non-escaping points and the zero subprobability to escaping ones. Under the hypothesis that compact orbit closures admit at least one invariant measure, we establish that the weak* continuity of this ergodic map together with a vanishing at infinity condition is equivalent to the mean ergodicity of the Koopman representation on the space of continuous functions vanishing at infinity. In consequence, every such function and every finite signed measure split uniquely into invariant components and limits of coboundaries. The corresponding projections are obtained by integration against the ergodic map. Because this operator-theoretic characterization avoids explicit averaging schemes, it remains applicable even to semigroups without Følner sequences. When restricted to countable, discrete, bicancellative, and left amenable semigroups, these properties are shown to be equivalent to the uniform convergence of Følner averages and the weak-star continuity of their dual limits, extending classical results for group actions on compact spaces. Furthermore, we identify the space of ergodic measures with a compactified ergodic quotient, prove that the invariant measure simplex is Bauer, and show that these structural properties descend through proper factor maps. The theoretical framework is complemented by dynamical examples, including a continuously pointwise ergodic subshift that exhibits discontinuous entropy along the ergodic map.

We study a two–dimensional discrete nonlinear Schrödinger equation with saturable nonlinearity on the lattice $\mathbb Z^2$. Using a variational approach based on the Nehari manifold, we establish the existence of nontrivial periodic ground states on finite lattices and establish the existence of exponentially localized ground states in $\ell^2(\mathbb Z^2)$. A principal result is the rigorous passage from periodic to localized states: we show that, up to lattice translations, periodic ground states converge strongly in $\ell^2(\mathbb Z^2)$ to a localized ground state as the lattice periods tend to infinity. The analysis combines variational methods, spectral properties of the discrete Laplacian, and concentration–compactness techniques adapted to the two–dimensional discrete setting. We further derive qualitative properties of the resulting solutions, including positivity and exponential localization, and establish a conditional orbital stability result within the Grillakis–Shatah–Strauss framework. Numerical computations illustrate the theoretical results and confirm the predicted convergence and localization behavior.

We show that every $C^2$ immersion of the circle $\mathbb{S}^1$ into itself has low emergence.

We consider simple autoregressive processes of type AR(1), whose stationary distribution is supported on a subset of the unit interval and is an affine transformation of a Bernoulli convolution. A new structural representation of the stationary distribution as a product of a power function with a log-periodic function near the origin is given, which gives structural insight to the stationary distribution on the whole unit interval by using a characteristic functional equation. This enables to prove that the stationary distribution of the AR(1)-process belongs to the domain of geometric partial attraction of a max-semistable law. We further prove uniform convergence of the distribution function of normalized maxima of the AR(1)-process to a certain power of the max-semistable law in the spirit of a merge theorem and point out connections to deterministic and random dynamical systems.

We investigate the dynamics and spectral stability of traveling kink and antikink solutions in a dissipative sine–Gordon equation with two distinct fourth–order regularization mechanisms: a mixed space–time (inertial) term and a purely spatial (elliptic) term. The model includes damping, bias forcing, and higher–order dissipative effects, and is motivated by refined descriptions of fluxon dynamics in long Josephson junctions. Using a collective–coordinate reduction, we derive a Melnikov–type condition for speed selection, yielding explicit predictions for asymptotic propagation speeds, which are validated by direct numerical simulations of the full partial differential equation. Spectral stability is analyzed using Evans function techniques adapted to the singular slow–fast structure induced by the regularization. By formulating the linearized problem on a consistent–splitting domain, we show that no additional point spectrum bifurcates from the origin. Near the edges of the essential spectrum, a square–root transformation is used to resolve branch singularities and establish analyticity in a lifted spectral variable. Numerical Evans function computations near $\lambda=0$ and near the essential spectrum edges confirm the analytical results, indicating absence of unstable eigenvalues for both kink and antikink solutions.

We construct symbolic dynamics for non-uniformly hyperbolic flows, in any dimension, possibly with fixed points. More precisely, for each $\chi>0$, we code a set which has full measure for every $\chi$-hyperbolic invariant probability measure that gives zero mass to the set of fixed points. As a main application, we prove that a three dimensional $C^\infty$ flow with positive topological entropy on a closed manifold has finitely many ergodic measures of maximal entropy. For flows in any dimension, we also provide applications to the number of periodic orbits and to the Bernoulli property for equilibrium states of Hölder continuous potentials. The main technical result of this paper is a new method for handling singularities of vector fields by modifying the Riemannian metric. This technique is analogous to a blowup, which allows many results for nonsingular vector fields to be directly applied to vector fields with singularities.

Environmental variability is a defining feature of natural ecosystems, yet many theories of ecological stability assume static environments. Here, we develop an analytical framework for local stability in complex ecological communities subjected to periodic environmental forcing. We show that, in slowly varying environments and under a large-system approximation, the maximal Lyapunov exponent can be approximated by the time-averaged rightmost spectral edge of the instantaneous interaction matrix, yielding explicit stability criteria for large ecological communities. Within the random-matrix interaction ensembles considered here, our minimal theoretical framework predicts a topology-dependent ordering of local perturbation stability under periodic temporal modulation in high-dimensional ecological networks, which is supported by numerical simulations. Beyond the adiabatic regime, rapid environmental oscillations dynamically stabilize otherwise unstable ecosystems, revealing a high-frequency rescue effect that is absent from static theories. These results extend ecological stability theory beyond autonomous systems and provide a general framework for understanding resilience in fluctuating environments.

The aim of this article is two-fold. First, we obtain a normal form for automorphisms of the escaping sets of Hénon maps in a neighborhood of the attracting fixed point at infinity. Building on this local description, we characterize the global automorphisms of $\mathbb{C}^2$ that preserve the escaping sets. The analytic structure of the escaping sets, as established by Hubbard–Oberste-Vorth (\cite{HOV}), plays a crucial role in the proof of these results. In a similar spirit, we investigate the analytic structure of the super-level sets of the Green's functions associated with Hénon maps and give a description of the automorphisms of these domains.

Takens' time-delay embedding theorem provides conditions under which delay-coordinate maps, formed using uniformly-sampled time series of trajectories evolving on attractors of dynamical systems, can faithfully represent the dynamics of the original system. Nonlinear systems can be highly sensitive, and Takens' theorem does not provide guarantees about the stability of time-delay embeddings. In the linear setting, statements about the stability of time-delay embeddings are more tractable and have been proven for delay-coordinate maps with evenly-spaced delays. In many experimental applications, however, time series data may be non-uniformly-sampled, especially in systems with multiple timescales or when using event-based rather than time-based sampling techniques. In this paper, we extend the theorems for the stable linear Takens' embeddings to the setting where the delay-coordinate maps involve unevenly-spaced delays. We pose a conjecture about the rank of generalized Vandermonde matrices that capture the temporal structure of time-delay embeddings. We prove that, provided the conjecture holds, existing theorems about stable linear Takens' embeddings readily extend to unevenly-sampled settings, and the quality of the embedding converges to the same asymptotic bounds when using a large number of delays in the delay-coordinate map, a result which is supported by numerical simulations.

Let $X$ be a planar vector field with an equilibrium $p$ at which the Jacobian $J=DX(p)$ is nilpotent of rank one, and let $q_0$ span $\ker J$. We prove that the two coefficients $a$ and $b$ of the Bogdanov–Takens (BT) normal form are the directional derivatives, along $q_0$, of the two invariants of the Jacobian: $a=-\frac{1}{2}\langle\nabla\det DX(p),q_0\rangle$, $b=\langle\nabla\operatorname{tr}DX(p),q_0\rangle$. The identity is invariant under changes of phase-space coordinates and equivariant under the rescaling of $q_0$, and the resulting formula requires neither generalized eigenvectors nor the second-order multilinear form. It yields a coordinate-free reading of the BT nondegeneracy conditions in terms of the kernel line field of the projection of the equilibrium manifold onto parameter space, the transformation rule $(a,b)\mapsto(h^2a,hb)$ under orbital equivalence, and the fact that $a=0$ whenever the vector field factors through a function vanishing at $p$. We then prove an obstruction of a combinatorial nature. For a Kolmogorov system $\dot{x}=xA/g_1$, $\dot{y}=yB/g_2$ and an equilibrium $p$ in the torus $(\mathbb{C}^*)^2$ at which the Jacobian is nilpotent and nonzero, the order $m=\operatorname{ord}f$ in the Takens normal form is bounded by the mixed volume of the Newton polytopes of $A$ and $B$. In particular, if $\operatorname{MV}(\operatorname{Newt}A,\operatorname{Newt}B)\leq2$ then $a\neq0$ unless the equilibrium fails to be isolated, and the nilpotent singularities of saddle, focus and elliptic type are unreachable: only cusp-type singularities occur at isolated equilibria, while their exact codimension is not controlled by the mixed-volume bound. The class of systems with cross-product cubic terms, for which the mixed volume equals $2$, is treated in detail, and two classical Bazykin models are shown to be instances.

We study mixing at a double exponential rate on analytic observables and ask how much of a map is remembered by its rate of mixing. For finite Blaschke products of the circle with a fixed point in the unit disk, and for the free semigroup actions they generate, we give a complete classification: the rate of mixing (no mixing, exponential, or double exponential) is determined by the multiplier of the generators at that fixed point, the invariant measure being the harmonic measure with a pole there. In the double exponential regime, the exponent equals $\log p$, where $p$ is the minimal local degree of the generators at the fixed point, and we show that this value is sharp. Consequently, the rate is not rigid: it is not stable under $C^1$-perturbations and does not imply $C^1$-conjugacy to affine models. Rigidity holds when the exponent is maximal for the degree: a map of degree $q$ whose exponent attains $\log q$ is Möbius conjugate to an affine model.

The dynamic of a rational map $f:\widehat{\mathbb C}\to \widehat{\mathbb C}$ is determined by the forward orbits of its critical points. Such a map is called postcritically finite if every critical point has finite forward orbit, or equivalently, if every critical point eventually maps into a periodic cycle. These orbits can be encoded in a finite directed graph called a dynamic portrait. In this work, we classify which abstract bicritical dynamic portraits of degree $d\geq2$ with at least 4 postcritical points are realizable exclusively by rational Thurston maps.

The identification and reconstruction of the boundaries separating basins of attraction in multistable, multidimensional dynamical systems presents a fundamental challenge in computational dynamics. These structures govern transition pathways and other important large timescale behavior, yet they remain typically under-sampled since their neighborhood does not get routinely visited during direct simulations. Traditional computational approaches face computational limitations in high-dimensional systems and require a priori knowledge of the dynamical system and its equations. Simplistic sampling methods such as random or uniform sampling of the phase space typically fail to quantitatively approximate separatrices and their structure altogether. We introduce and implement a framework that combines supervised classification with generative modeling to address this challenge. Our approach first trains neural network classifiers on uniformly or randomly sampled initial conditions labeled by their corresponding basins of attraction in the system of interest. Using uncertainty metrics of the trained classifier to quantify decision boundaries, the method then identifies these high uncertainty regions and boundaries of the classifier as preliminary approximate separatrices. Subsequently, score-based generative models are trained specifically on samples from high-uncertainty regions, ultimately generating densities of samples consistent with the empirical density of samples on or close to the manifold that constitutes the separatrix between basins in the sampled region. This approach leverages the complementary strengths of (a) discriminative models for global phase space partitioning and (b) generative models for detailed geometric sampling, resulting in a systematic, iterative, data-driven framework that produces empirically consistent reconstructions of (approximate) separatrix manifolds.

We introduce a class of nonautonomous nonlinear oscillator equations of mixed Liénard type that arises from a parametric deformation of the commutative factorization procedure applied to second-order ordinary differential equations with periodic solutions. Their solutions, in particular the isochronous waveforms, are obtained in closed form through a Riccati reduction scheme for the power-law choice of the factorization function, $\phi(x)=kx^{q}$, $k\in \mathbb{R}$, and $q\in\mathbb{N}$, corresponding to the so-called modified Emden oscillators, and do not depend on the arbitrary deformation parameter $A_1$. The variational structure of the equation is characterised by a Lagrangian supplemented with a generalised Rayleigh dissipation function that contains a non-standard cubic term in $\dot{x}$. We also show that multiplying the equation of motion by the Jacobi multiplier $M(x)=x^{-2A_1}$ a position-dependent-mass (PDM) form is obtained with related friction and restoring force.

Everybody learns in school that an out-of-equilibrium system coupled to a heat bath at a fixed temperature evolves to thermodynamic equilibrium as time goes on, and the free energy will only decrease on its way there. At least since Holley's 1971 work, mathematicians know this too, in the modest context of classical Ising Glauber dynamics. But does the free energy also asymptotically decrease to the free energy of an equilibrium state? A typical school kid would say "yes, of course", but since the free energy is only lower semicontinuous, this question is less straightforward than it initially seems. To the best of our knowledge, apart from the uniqueness regime, where one can make use of classical functional inequalities, this question has not previously been resolved rigorously. We prove the asymptotically complete dissipation of the free energy for the classical Ising Glauber dynamics by introducing a coarse modified log-Sobolev inequality, which holds at every positive temperature, in particular in the phase-coexistence regime.

For a negatively curved surface, magnetic (or twisted geodesic) flows with nonpositive ”magnetic curvature” have a large-scale geometry on the universal cover that retains some features associated with geodesic flows: magnetic boundary and visibility, possibly magnetically flat strips. This has dynamical consequences: orbit-equivalence to the geodesic flow, density of periodic points, transitivity, entropy-expansivity, existence of equilibrium states.

In this article, we give a characterization of when two pseudo-Anosov flows obtained via gluings of pieces of pseudo-Anosov flows are orbit equivalent. As an application of this work, and the description of pseudo-Anosov flows in Seifert pieces due to Barbot and Fenley, we prove a “virtual” version of the Finiteness Conjecture for transitive pseudo-Anosov flows on graph-manifolds.

We describe a mechanism of affine symmetry emergence in the maximal density regime of the prototypical model of self-organized criticality when the inverse square of the mesh of the underlying lattice is much larger than the number of random perturbation points distributed according to a prescribed probability measure supported in the interior of the ambient convex domain. Moreover, an appropriate scaling limit of the toppling function (aka odometer), which counts the number of operations per site, is a solution to a non-linear partial differential equation well known in the context of optimal transport and differential geometry, making it possible to accurately estimate the deviation of the density from its maximal value in any macroscopic window. The mechanism for the affine symmetry emergence is due to the novel empirical fact, supported in addition by inductive arguments that have recently being upgraded to a rigorous proof, that the scaling limit of the toppling function is the unique concave solution of the Monge-Ampère equation with Dirichlet boundary condition on the convex domain with the potential given by the probability measure used above as the infinite-perturbation profile.