Papers

We study stationary states for actions of countable discrete groups on unital separable $C^*$-algebras. We prove that the stationary state space associated with an inner action of a subgroup of the unitary group that generates the algebra is a Choquet simplex. We also give a locality criterion covering Bernoulli shifts. The simplex structure yields a canonical decomposition of stationary states into tracial and purely nontracial parts. For inner actions, we characterize extreme stationary states by factoriality of their GNS von Neumann algebras. We further show that a stationary state is tracial if and only if its GNS von Neumann algebra is finite, and that it is purely nontracial if and only if this algebra is of type~$\mathrm{III}$. As applications, for $2\leq d\leq\infty$ the stationary state simplex of $C^*(\mathbb{F}_d)$ has a Poulsen face, while for a nontrivial property~$(T)$ group the stationary state simplex of $C^*(\Gamma)$ is not Poulsen. Finally, we give a sufficient spectral-gap condition for $S_{\mu}(A)$ to be a Bauer simplex and construct a family $(A_d,\Gamma_d,\mu_d)$ satisfying this condition.

We study limit theorems for random dynamical systems. We recast the random dynamics via the skew-product of the two-point motion (the Varadhan trick). Using this approach we prove several statistical properties for the system, including quenched central limit theorems with rates, quenched local limit theorem, large deviations, and decay of correlations. Then, in applications we show how to verify the general conditions using spectral properties of appropriate operators. A key application is the study of effectively expanding on average random diffeomorphisms (in any dimension, which are allowed to be dissipative). In particular, this is the first instance of proving local limit theorems in such a broad setting, without imposing restrictions on the dynamics. We provide several new examples.

We introduce a general machinery to study thermodynamic formalism out of equilibrium: The thermodynamics of a topological Markov shift (denoted by $\Sigma^-$) where the potential is given by a random walk on a compact metric space $X$ (and the randomness is driven by a Gibbs process). We introduce the semi-Ruelle operator, which acts on $C(\Sigma^-\times X)$. We construct conformal measures and harmonic functions for the semi-Ruelle operator. We present a few applications: (1) We provide a new proof to the POE variational principle (POE stands for the pressure out of equilibrium which is associated with the process), and we show that maximizing measures in the POE variational principle admit positive entropy out of equilibrium, and satisfy semi-Gibbs estimates. (2) In the setting where the random walk on the fiber $X$ is given by $C^{1+}$ diffeomorphisms (which are allowed to be very dissipative), and it satisfies the open condition of effective expansion on average, we show that the averaged semi-Ruelle operator is quasi-compact when acting on a Sobolev function space. An application includes proving a spectral gap when assuming volume decay of correlations, and proving bounds on the dimension of stationary measure in terms of similarity dimension.

We prove that conjugacy of minimal homeomorphisms of the Cantor space is Borel bireducible with isomorphism of countable graphs, answering the Cantor minimal case of a question of Foreman. We obtain the lower bound by encoding countably based profinite groups. Finite quotient homomorphisms are represented by factor maps between a common family of minimal subshifts. Amalgamation makes the resulting inverse limit independent, up to conjugacy, of the quotient presentation. Conversely, finite-stage factorization recovers the group from any conjugacy of these inverse limits.

In this paper, we develop a linear response theory for a class of stochastic differential equations driven by jump processes. We investigate the response of the system to small time-dependent perturbations from three complementary perspectives: probability density functions, mean exit times, and escape probabilities. By performing perturbation analyses of the corresponding forward and backward Kolmogorov equations, we derive the first-order response equations governing these statistical quantities and establish explicit linear response formulas, which characterize the sensitivity of the evolution of the probability distribution and the escape behavior of the system to variations in its coefficients.

We propose a learning procedure for system identification in interacting particle systems from single-snapshot observations of collective behaviors, unlike existing approaches that rely on observations of trajectories. This setting leads to a fundamentally ill-posed inverse problem, which we solve by using a regularization strategy based on the empirical distribution of observed configurations, drawn from different, unobserved initial conditions. We test our learning procedure on a variety of representative models with steady-state and quasi-stationary patterns, where collective behaviors encode implicit information about the interaction mechanisms, demonstrating that our approach enables stable and accurate recovery of the underlying interaction laws, leading to faithful reproduction of the collective behavior, and in many cases even of the dynamics leading up to it.

We characterize connected Lie groups on which all adapted random walks have only trivial bounded harmonic functions. In the nilpotent case, this answers a question of Furstenberg recorded by Guivarc'h in 1973.

Motivated by Keller's W-shaped maps and its variants, we introduce a general class of families of expanding maps such that the perturbed maps have a shrinking almost invariant neighborhood about a fixed point of the limit map. The unique absolutely continuous invariant measures (ACIM) of the perturbed maps can converge to limit measures of various types. A local quantity is identified which determines if the limit is absolutely continuous, singular, or a non-trivial convex combination of these. Furthermore, for the case of a nontrivial convex combination, we prove that the dynamics of the perturbed system, when viewed on an appropriate slow time scale, converges to a jump Markov process, a convergence that extends to the diffusion coefficients for observables of bounded variation. Compared to analogous results on expanding maps with metastable behavior, a new feature of our setting is the emergence of a localized state of the Markov process which corresponds to the shrinking almost invariant interval about the fixed point. Our approach provides a general framework which, in particular, accommodates, to the best of our knowledge, all previously studied families that limit to Keller's W-shaped map.

We investigate the persistence of most probable paths through the Onsager–Machlup functional for multidimensional stochastic differential equations driven by fractional Brownian motion with time-dependent diffusion coefficients and Hurst parameter $H\in(1/4,1)$. Under suitable structural and variational conditions, deterministic trajectories remain most probable paths for sufficiently small noise in both the fixed-endpoint transition problem and the free-endpoint evolution problem, whereas sufficiently large noise destroys their local minimality. More generally, when exact persistence does not hold, global most probable paths converge to the corresponding trajectories of the noise-free system in both the uniform and Hölder topologies at the rate $O(\epsilon)$. We further analyze the second variation along periodic deterministic trajectories over time intervals of length $NT$. For $H>1/2$, positive definiteness, and hence local minimality, is lost on sufficiently long intervals. For $H\in(1/4,1/2]$, long-time positive definiteness holds for the fixed-endpoint problem, but this conclusion does not directly extend to the free-endpoint setting. We also establish the persistence of KAM tori in nearly integrable Hamiltonian systems in the sense of most probable evolution paths. Finally, a two-dimensional numerical example illustrates the persistence of deterministic trajectories under small noise and their pronounced deviation under large noise.

We prove that a homeomorphism of the two-torus in a Dehn twist homotopy class is a topological extension of an irrational circle rotation if and only if its vertical rotation set reduces to a single irrational number and the homeomorphism exhibits uniformly bounded rotational deviations in the vertical direction. This result has been previously obtained by the second author under an additional hypothesis about the geometry of the non-wandering set. We show that this hypothesis can be entirely dropped: the large-scale shear intrinsic to a Dehn twist eliminates the obstructions coming from the wandering set, which cannot be removed in the homotopy class of the identity. Finally, we prove that the factor is unique up to a circle rotation and is locally constant on the wandering set. Consequently, every connected component of the wandering set is contained in a single fiber, is itself a wandering domain and is either inessential or annular.

For the Lozi map $L_{a,b}$, we consider parameter pairs for which the fixed point $X$ in the first quadrant has no homoclinic points and the period-two orbit $\{P,P'\}$ is attracting. For such parameters, let $\ell$ denote the set of accumulation points of the unstable manifold $W_X^u$ that do not belong to $W_X^u$. We completely classify the forward asymptotic behavior of points in the phase space. The forward orbit of every point in the plane either converges to $X$, to the other fixed point $Y$ in the third quadrant, or to $\ell$, or it escapes to infinity. The global phase space is organized by the stable manifolds of the fixed points: $W_Y^s$ separates the basin of $\ell$ from the region of escaping orbits, while $W_X^s$ is the exceptional set of points whose orbits converge to $X$. In particular, if $\mathcal{A}_1$ denotes the component of $\mathbb{R}^2 \setminus W_Y^s$ containing $X$, then $\mathcal{A}_1 \setminus W_X^s$ is precisely the basin of attraction of $\ell$.

In this paper, we consider the classical time-optimal elevator problem in the presence of higher-order state constraints. The main contribution is a constructive indirect solution framework based on a Pontryagin Maximum Principle specifically formulated for higher-order constrained systems. First, we derive specialized optimality conditions and analyze the resulting structure of extremal trajectories. Second, we show that the infinite-dimensional optimal control problem can be transformed into a finite-dimensional system of nonlinear algebraic equations involving switching times, boundary-contact times, and multiplier parameters. This provides a computationally efficient procedure for calculating candidate optimal trajectories using standard nonlinear equation solvers. Third, we characterize the geometry of boundary contacts and explain the emergence of contact-chattering phenomena in higher-order constrained systems. In particular, we show that normal extremals cannot evolve along the constraint boundary for a positive amount of time, although sequences of boundary contacts may accumulate indefinitely. The proposed approach is demonstrated on a fourth-order elevator model, and the computed solutions are independently verified using a direct IPOPT-based optimization method.

Recently, there has been considerable interest in the study of non-deterministic dynamical systems. To analyze the chaotic behavior of such systems from a measure-theoretic viewpoint, it is desirable to consider ergodicity. However, the classical definition of ergodicity involves invariant sets, whose definition is not unique for non-deterministic dynamical systems. Thus, we are led to the question of which invariance yields an interesting definition of ergodicity. Here, we propose a definition based on the strong backward invariance and show that analogs of classical results hold. We also consider implications of the Birkhoff ergodic theorem for systems without uniqueness of orbits.

Polynomial skew products in $\mathbb{C}^2$ are maps of the form $F(z,w)=(P(z),Q(z,w))$, where $P$ and $Q$ are polynomials. Their local dynamics have been widely investigated. In this paper, we study the global dynamics inside Fatou components of some skew products. We consider all the inverse images in a Fatou component of a given point and use the Kobayashi metric to measure the distance between points. In the cases we consider, there are always arbitrarily large Kobayashi balls in the complement of these inverse sets.

We develop a perturbative KAM reducibility theory for one-frequency $\mathrm{SL}(2,\mathbb{R})$ cocycles based on an admissible Fourier length $\ell$. The regularity relevant to the iteration is measured by positive adapted Fourier width rather than ordinary smoothness in the Euclidean length $|n|$. The same length governs Fourier decay, truncation and resonance scales, and the arithmetic condition controlling the small divisors. This framework contains the classical analytic and Gevrey settings, while non-monotone choices of $\ell$ allow classical nowhere differentiable Weierstrass-type perturbations and continuous perturbations outside every positive Hölder class. As spectral applications, we obtain purely absolutely continuous spectrum for every phase and $1/2$-Hölder continuity of the integrated density of states for the associated quasiperiodic Schrödinger operators. The Aubry dual has pure point spectrum for Lebesgue almost every dual phase, with eigenfunctions exponentially localized in the metric induced by $\ell$. We also construct nowhere differentiable quasiperiodic potentials with purely absolutely continuous Cantor spectrum.

We develop a framework for offline full-window branch discovery, optionally followed by online continuation with an ensemble Kalman filter (EnKF). Three mechanisms drive the branch search: adjoint path-kernel (APK) differentiation balances kernel differentiation and correction-stabilized path perturbation, shifting the optimization from exploration to exploitation; an optimized Gaussian initial law broadens the search over initial-state basins; and loss-weighted mixing across independent runs recombines successful path components. We may then select an interior state using a local loss and continue online with an EnKF. In 40-dimensional Lorenz-96 experiments, the mean offline path RMSE of APK is 4.3 times smaller than that of population weak-$\mathrm{4D\text{-}Var}_x$. The resulting APK-EnKF method has a mean online RMSE 64 times smaller than that of ordinary EnKF.

We study the dynamics of collisions and near-collisions in the system of three point vortices in the plane. The dynamics is known to be completely integrable and can be reduced to the study of Hamiltonian systems on two-dimensional symplectic leaves. This still allows for interesting behavior of the solutions, various aspects of which have been investigated in a large body of work by many authors. Our focus will be on collisions and the possibility of their regularization through perturbations of suitable parameters of the system. We will use the reduced coordinate given by $\zeta=(z_2-z_1)/(z_3-z_1)$, where $z_1,z_2,z_3$ are the complex numbers representing the positions of the vortices. This choice seems very natural, although we have not come across it in the existing literature on the topic. Some of our results concerning the near-collision behavior appear to be new, and we also prove inequalities between critical energy values that we have not found in the literature. The calculations we do to analyze the collisions and near-collisions also reproduce various classical results that were previously obtained by other authors using different coordinates, and parts of the paper therefore have an expository character.

This paper is a sequel to the authors' paper on rigidity of higher dimensional contact Anosov flows~[GRH]. At the time the authors were unaware of much earlier work of Hamenstädt~[Ham] devoted to the same problem. The methods of~[GRH] and~[Ham] are overlapping but not entirely the same. In this paper we strengthen some the results of Hamenstädt, by further bootstrapping the regularity of the conjugacy of 2-pinched contact Anosov flows. We also introduce a notion of symplecto-rigidity for symplectomorphisms and prove that some Anosov diffeomorphisms are symplecto-rigid. We further combine symplecto-rigidity with various rigidity techniques to establish a number of smooth rigidity results of Anosov symplectomorphisms. Some new phenomena are present in the realm of Anosov symplectomorphism. For example, while the well-known de la Llave example on the 4-torus demonstrates absence of rigidity in the space of smooth Anosov diffeomorphisms, however, we prove that it is rigid in the space of Anosov sympelctomorphisms.

Place one grain at every nonsink vertex of the wired $n\times n$ square, and let $L(n)$ be the order of this operation in the sandpile group. Thus $L(n)$ is the least positive $q$ for which $q$ uniform grain layers form an integral combination of toppling moves. We prove that, for every $n\ge1$, \[ \nu_2(L(n))= \begin{cases} 2,&n=1,\\ 1,&n\ge2\text{ even},\\ \nu_2(n+1)+2,&n\ge3\text{ odd}. \end{cases} \] For even squares, this follows from the domino–sandpile results of Florescu, Morar, Perkinson, Salter, and Xu, completed by a short parity observation. For odd squares, a unimodular cyclic basis identifies the folded cokernel with a quotient by two shifted Chebyshev polynomials and sends the all-ones class to $1$. Its order is determined by the constant part of this polynomial ideal, not just by a determinant. Two normalized Euclidean remainders reduce to consecutive Fibonacci polynomials over $\mathbb F_2$, giving the exact valuation.

We prove exact dimensionality of ergodic stationary measures for random $C^1$ diffeomorphisms in the single negative Lyapunov scale setting. Let $\nu$ be a Borel probability measure on $\mathrm{Diff}^1(M)$ satisfying a logarithmic $C^1$ moment condition, and let $\mu$ be a $\nu$-stationary ergodic probability measure. If $\lambda_{\mathrm{top}} = \lambda_{\mathrm{bot}} = \lambda<0,$ then $\mu$ is exact dimensional and $ \mathrm{dim}(\mu)={h_\mu^{\mathrm{F}}(\nu)}/{(-\lambda)}.$ No discreteness assumption is imposed on the driving measure.

We compare eight variants of a three-variable pollution-based tourism model obtained by assigning the same discrete delay to different terms. The delay may enter three slots: the visitor factor in the deterrence term, pollution perception, or visitation-driven growth and reinvestment. We call a placement feasible when every nonnegative initial history produces a solution that exists for all future time and remains nonnegative. For every $\tau>0$, feasibility holds exactly when the visitor factor in the deterrence term is current, which occurs in four of the eight placements. In three of these four feasible placements, the positive equilibrium is locally asymptotically stable for every delay. In the fourth – the perception-lag model, with delayed pollution perception but current visitation – a conjugate pair of characteristic roots crosses the imaginary axis at a finite critical delay for every set of positive parameters. The equilibrium then loses stability and does not restabilise as the delay increases. A center-manifold reduction and numerical parameter sweep show that both supercritical and subcritical Hopf bifurcations can occur. The visitor-delayed placement, closest to earlier delayed-tourism models, can undergo a local Hopf bifurcation under additional conditions, but it does not preserve nonnegativity for all initial histories.

We construct a faithful functor from the category of cubespaces, arising in structured ergodic theory and nilspace theory, to the category of (condensed) simplicial sets. This functor translates the weak structure theorem of a nilspace to a Postnikov tower of the corresponding simplicial set. In particular, nilspaces are mapped to Kan complexes and the structure groups of the nilspace correspond to the simplicial homotopy groups of the Kan complex. The functor is constructed via pullback along a special cosimplicial cubical set, whose combinatorial properties may be of independent interest.

Originating in ergodic structure theory, nilspace theory is concerned with the study of cubical spaces. A first key result in this theory, the weak structure theorem, is the representation of a concrete fibrant compact cubespace as a tower of its canonical truncations, exhibiting each step over the previous one as a principal bundle by a compact abelian group. We present a categorical proof of the weak structure theorem by systematically interpreting the relevant concepts in an appropriate context.

In this paper, we count the number of (almost) extremally stretched closed geodesics for a pair of (non-conjugate) Fuchsian representations of a closed surface group, and show that it has subexponential growth. We then deduce that, as a discrete subgroup of $\mathsf{PO}(2, 1) \times \mathsf{PO}(2, 1)$, the growth indicator of the product representation vanishes on the boundary of the Benoist limit cone. We also prove that for a general Zariski dense Borel Anosov subgroup of $\mathsf{PO}(2, 1) \times \mathsf{PO}(2, 1)$, its growth indicator vanishes on at least one boundary component of the Benoist limit cone, but not necessarily on both. One may view our first result as a sharpening of Thurston's result that there is a unique geodesic lamination $\lambda$ such that every measured lamination maximizing the ratio of lengths with respect to the two representations has support contained in $\lambda$. We hope this will be a starting point for a more general study of extremal entropy.

We study the spatial $(n+1)$-body problem formed by one heavy central mass together with $n$ equal masses placed on a single orbit of a polyhedral rotation group $H\in\{T,O,I\}$, so that $n=|H|\in\{12,24,60\}$. Imposing the symmetry $q_{L}=L\,q_{I}$ for $L\in H$ reduces the problem to a single $2\pi$-periodic reference curve, with reduced action $A_{H}=A_{0}+\varepsilon A_{1}$, in which $\varepsilon$ is the inverse central mass and $A_{0}$ is the Kepler action. At $\varepsilon=0$ the critical set contains, as one connected component, the five-dimensional manifold of Kepler ellipses of minimal period $2\pi$, which we prove to be a nondegenerate critical manifold. A Lyapunov–Schmidt reduction along this manifold turns the continuation problem into the search for nondegenerate critical points of an explicit function $\Phi(e,\psi)$ of the eccentricity $e$ and the spatial orientation $\psi$, a nondegeneracy we verify by a computer-assisted proof. We thereby obtain, for each of the three groups, families of periodic solutions of the $(n+1)$-body problem with $n+1\in\{13,25,61\}$ bodies, bifurcating from Kepler ellipses and carrying the full tetrahedral, octahedral, or icosahedral symmetry.

We give a class of local diffeomorphisms which are homotopic to toral hyperbolic endomorphisms, but which are not topologically semi-conjugate, within the homotopic class of the identity, to any Anosov local diffeomorphism. In particular, we show that for a non-invertible partially hyperbolic $C^1$-smooth local diffeomorphism with expanding directions on the $2$-torus, if it is semi-conjugate to an Anosov local diffeomorphism, then it is also an Anosov local diffeomorphism.

We study high-dimensional conditioned dynamics obtained from an arbitrary number of independent copies of a mixing Markov chain. The dynamics is conditioned to avoid a family of holes whose stationary measure vanishes as the dimension grows, and we ask whether the resulting process admits a quasi-stationary measure close to the stationary product measure. Our problem is motivated by conditioning on holes with complicated geometry in high dimension, such as sets arising naturally from large deviations of suitable observables. Our main tool is the ANOVA decomposition, which separates functions according to their dependence on different subsets of coordinates. This allows us to exploit the product structure of the dynamics and obtain contraction estimates that are uniform in the dimension. Combined with a Keller Liverani perturbation argument, these estimates yield the existence of quasi-stationary densities converging to the stationary density as the size of the hole vanishes. Under stronger one-step mixing and regularization assumptions, we obtain sharper convergence in a Sobolev norm, with a square-root dependence on the measure of the hole. The improvement relies on the increasingly strong contraction of higher-order ANOVA components, thereby overcoming the usual loss of control with dimension. Finally, we apply our results to additive-noise Markov chains on the circle with smooth, uniformly positive transition densities

We study the Hopf decomposition of subgroup actions of the Teichmüller modular group on the Thurston boundary with respect to the Thurston measure class. We identify the conservative part with the big horospherical limit set modulo null sets. For a basepoint with trivial stabilizer in the subgroup, the ideal boundary of the associated Dirichlet polyhedron is wandering, and its subgroup translates cover the dissipative part modulo null sets. The dissipative part also agrees with the set of Dirichlet points modulo null sets. The description using Dirichlet polyhedra relies on a separation theorem for extremal length: every level set of an extremal length ratio at distinct points of Teichmüller space has measure zero. Kaimanovich's Radon–Nikodym criterion characterizes the two parts by the divergence and convergence, respectively, of a series of extremal length ratios. For the Torelli group of a closed surface of genus at least two, we use radial limits of the period map to prove that its conical limit set has measure zero. Together with the conservativity established by Choi, Gekhtman, Yang, and Zheng, our geometric characterization implies that its big horospherical limit set has full measure and that the ideal boundary of every Dirichlet polyhedron has measure zero.

We present computer-assisted proofs of partial hyperbolicity, existence of a blender and robust transitivity for diffeomorphisms on closed manifolds. These proofs are implemented for a family of derived-from-Anosov systems on the 3-torus.

We study the dynamics of self-consistent transfer operators associated with mean-field coupled systems whose unit dynamics is given by an interval map with finitely many singularities of cusp type. We show that, when the coupling strength is sufficiently weak, such an operator has a unique fixed point in a suitable cone of the associated Banach space, and that this fixed point attracts the orbit of every density in the cone at an exponential rate in the corresponding norm.