Papers

We establish dimension bounds and an exact dimension formula for equilibrium measures of degree $d$ holomorphic endomorphisms of $\mathbb P^2=\mathbb P^2(\mathbb C)$ satisfying suitable expansion and domination conditions. Write $\lambda_1>\lambda_2>0$ for their Lyapunov exponents. If two inverse histories ending at the same point determine different fast directions, we prove that ${\mathrm dim}_H \mu_F>\log d/\lambda_1+\log d/\lambda_2$ when $\lambda_2>\log d$, and that ${\mathrm dim}_H \mu_F=2\log d/\lambda_2$ when $\lambda_2\ge2\log d$. Applying these results to an explicit quadratic family, we disprove the Binder–DeMarco conjecture on a non-empty open set of holomorphic endomorphisms. The proof adapts the projection growth strategy of Li–Pan–Tong–Xu to the holomorphic setting, combining strong leaf geometry and the Ledrappier–Young theory for endomorphisms with a multidimensional extension of Wu's restricted sum estimate.

Subset currents on free groups, on surface groups and, more generally, on word-hyperbolic groups, were introduced by Kapovich and Nagnibeda, and further studied by Sasaki. Subset currents extend ordinary geodesic currents and provide measure-theoretic generalizations of conjugacy classes of infinite quasiconvex subgroups. Their supports can be viewed as “branching laminations". For free groups and surface groups, we construct non-rational subset currents with genuinely branching supports that carry no other subset currents except scalar multiples. For every nonabelian finite rank free group and every closed hyperbolic surface group, we show that such a current can be chosen so that its support is uncountable, is disjoint from the locus of two-point boundary subsets, and consists entirely of Cantor subsets of the boundary. More generally, we prove that every nonempty metrizable Choquet simplex occurs as a compact convex base for the cone of currents carried by a minimal genuinely branching subset lamination. The same universality phenomenon extends, apart from the usual rational-current terminology, to every non-elementary torsion-free hyperbolic group. We also study finite-pattern complexity. We prove that if $R_L(n)$ counts the allowed round patterns of radius $n$ in a minimal subset lamination $L$, then $R_L(n)=O(n)$ forces the carried-current cone to be finite-dimensional, while infinite-dimensionality forces $R_L(n)/n\to\infty$; this universal threshold is sharp. We show that the coarse growth type of $R_L$ is independent of the chosen free basis and, more generally, of the chosen finite marked graph. We also show that a refined switch system recovers the exact dimension of the carried-current cone.

We study the free-energy gradient flow associated with the depth-$N$ deep linear network on the space of invertible real $d\times d$ matrices. The loss observes only the diagonal, \[ E_d(W)=\frac12\sum_{i=1}^d(W_{ii}-1)^2, \] and the regularizer is the Boltzmann entropy of the balanced factorization fiber computed by Menon and Yu. This is a natural higher-dimensional version of a matrix-completion problem posed by Menon as a test of whether entropy selects among a noncompact family of minimizers. For diagonal completion, every width $d\geq2$, depth $N>2$, and inverse temperature $\beta>0$, we prove that the free energy is unbounded below and has exactly $2^d$ full-rank critical points, one in each diagonal sign chamber. Every critical point is diagonal and hyperbolic. Its unstable dimension is $\binom d2$ and its stable dimension is $d(d+1)/2$. We also prove that a full-rank trajectory cannot converge to a finite rank-deficient matrix. Consequently, almost every full-rank initial condition has an unbounded forward orbit. Thus finite-temperature fiber entropy does not provide an equilibrium selection principle for diagonal completion.

We develop an algebraic-dynamical correspondence for type-I oscillatory blow-up solutions of autonomous ODEs with asymptotically quasi-homogeneous vector fields. Periodic solutions of the balance law arising from asymptotic expansions of blow-ups are related to periodic orbits on the horizon for the desingularized vector field. Unlike the stationary case, the correspondence involves a positive periodic scaling function and a nontrivial time reparametrization. We further establish the correspondence of linearized structures: distinguished phase and scaling directions generate invariant subbundles, while the remaining characteristic multipliers are identified through associated quotient bundles. These results yield a criterion for the existence of type-I periodic blow-up solutions based on periodic solutions of the balance law and their characteristic multipliers, without explicitly constructing compactifications or periodic orbits at infinity. The theory also clarifies the difference between stability information in the balance law and in dynamics at infinity. Two examples illustrate the correspondence.

Fix a strip width \(\rho>0\), a Diophantine frequency \(\alpha\), an integer \(N\ge2\), and \(\lambda>1\). For the family \[ V_{\varepsilon,\delta}(x) =2\lambda\cos(2\pi N x)+\varepsilon\delta(x), \] we prove that there is an open prevalent set \(\mathcal P_{\rm strat}\) of real-analytic strip-\(\rho\) perturbations such that every \(\delta\in\mathcal P_{\rm strat}\) admits \(\varepsilon_0(\delta)>0\) for which \[ \begin{gathered} \operatorname{Leb}\left\{ E\in\Sigma(V_{\varepsilon,\delta},\alpha): \omega(\alpha,E;V_{\varepsilon,\delta})=s \right\}>0, 0<\varepsilon<\varepsilon_0(\delta), \qquad s=1,\ldots,N. \end{gathered} \] The proof relates acceleration to the total multiplicity of real zeros of analytic overlap coefficients for first-return cocycles. Under quantitative control of the return products, Poisson integration and Kac normalization identify acceleration with half this multiplicity. A positive analytic dual state gives the finite-volume spectral gaps and Schur curvature needed at the upper spectral edge. Schur variations separate the perturbed maxima, and parameter exclusion preserves \(s\) pairs of simple zeros on a positive-measure spectral set.

Identifying kinetic parameters and conserved quantities that support multistationarity and multistability remains challenging in chemical reaction network analysis. Structural theory provides existence and exclusion results, but does not always yield explicit, interpretable descriptions of the corresponding parameter regimes. We present a computational and machine-learning framework for discovering and characterizing parameter regimes with numerical evidence of multistationarity and multistability in mass-action systems. We refer to these numerically detected behaviors as approximate multistationarity and approximate multistability. The framework combines Latin hypercube sampling, adaptive sampling, and interpretable surrogate modeling. Logistic-regression and neural-network surrogates are trained separately on accumulated detection labels and independently guide later sampling toward regions with high scores for approximate multistationarity or multistability. Boundary refinement targets transitions between detected and undetected numerical classes. Buckingham–Pi groups are then used with CN2 rule induction to obtain interpretable empirical descriptions of detected regimes. Illustrative networks yield rules consistent with known multistationarity conditions, while the Schlögl model demonstrates adaptive enrichment of multistationarity detections. In the hybrid histidine kinase model and the sequential dual phosphorylation cycle, adaptive sampling substantially increased the number of parameter - total pairs with detected multistability relative to global sampling under equal sampling budgets. The framework complements analytical chemical reaction network theory by providing efficient, parameter-resolved discovery and characterization of multistationary and multistable regimes.

In this short note, we study the behavior of the algebraic entropy of birational automorphism groups in families of projective hyper-Kähler manifolds.

This manuscript develops time-varying coefficients neurodynamic models (TVCNDMs) to solve the inclusion problem (IPs) $0 \in \mathcal{P}(w) + \mathcal{Q}(w)$, where $\mathcal{P}$ is a maximal set-valued and $\mathcal{Q}$ is a single-valued operator. The underlying operators are assumed to fulfill a generalized monotonicity condition that is less restrictive than the classical monotonicity assumption. Within this framework, existence and uniqueness of the model trajectories are established, and fixed-time ($\mathrm{FxT}$) convergence of the proposed TVCNDM to the solution of the IPs, along with explicit estimates of the settling time. Furthermore, the robustness of the proposed TVCNDM is investigated in the presence of bounded external disturbances, demonstrating that the $\mathrm{FxT}$ convergence. Numerical simulations support the theoretical results, illustrating rapid error decay and improved transient behavior achieved through the time-varying design.

We define a renormalization operator for pairs of commuting orientation preserving circle diffeomorphisms of class $C^3$. The operator is driven by the accelerated Brun multidimensional continued fraction algorithm. For rotation vectors of simultaneously bounded type, under a summable strong convergence condition satisfied by all eventually periodic types with Pisot period matrix, the renormalizations converge exponentially fast in $C^2$ to pairs of rigid rotations. The pair is simultaneously $C^{1+\gamma}$ conjugate to the pair of rotations, with $\gamma>0$ depending only on the bounded type constant. A fast renormalization extends exponential convergence and $C^1$ rigidity to a class of rotation vectors of full Lebesgue measure, containing explicit vectors outside bounded type. We show that the operator is topologically hyperbolic. The partition of the renormalized pairs by rotation vector is an invariant lamination whose leaves are the topological conjugacy classes. All results hold for pairs of class $C^{2+\alpha}$.

The Hamilton-Jacobi-Bellman (HJB) equation of a nonlinear system is a nonlinear partial differential equation, and in that form it gives little away. This paper shows that it hides a Riccati equation. Carleman linearization reveals this structure. For scalar input-affine systems with analytic data, the lifted HJB equation is a single infinite-dimensional Riccati equation in a lower-triangular Toeplitz matrix. Its first entry is the Riccati equation of the standard linearization at equilibrium. Its remaining entries hold the nonlinear terms. Nothing about this is an approximation. Under controllability of the standard linearization and detectability of the cost, the equation has exactly one solution with positive diagonal entries, and it can be computed one entry at a time. This solution is the derivative of the value function. We prove that finite-sections of the solution are exact and converge exponentially to the derivative of the value function near the origin. The scalar case has a closed-form solution. This closed form is our benchmark: every result on the lifted equation is checked against it. Nothing in this structure depends on the state dimension. Our methodology points to a general theory of optimal control in which the HJB equation of analytic nonlinear systems becomes an infinite-dimensional Riccati equation.

We study a one-parameter three-dimensional quadratic flow, denoted ETM-3D, on the benchmark domain $\rho\geq0$. Its homogeneous quadratic triad preserves Euclidean quadratic energy, while the full vector field has constant divergence -8. A shifted quadratic Lyapunov function yields an explicit positively invariant absorbing set and therefore global ultimate boundedness. The equilibria reduce generically to a scalar quartic, with a separate singular case at $\rho=\sqrt{5}$. Continuation reveals two saddle-node bifurcations, a generic transcritical bifurcation, a supercritical Hopf bifurcation, and a fold of limit cycles. Stable and unstable period-one branches meet at the fold with their nontrivial Floquet multipliers approaching +1. Just beyond this point, a 25-point logarithmic parameter study gives a laminar-length scaling exponent of -0.551 (95% confidence interval [-0.621,-0.481]), consistent with type-I intermittency. At larger $\rho$, a chaotic attractor coexists with an asymptotically stable equilibrium and periodic windows occur. Structural invariants are used to distinguish the flow from several classical quadratic chaotic families.

In a companion article \cite{RJnodyn} the present authors proved a form of the Rudolph–Johnson theorem which is effective at finite resolution, bounding the low-frequency Fourier coefficients of a $\times 2 \times 3$-invariant measure by its normalized entropy. Here we read that bound in the other direction, as a constraint on eigenfunction behavior in potential counterexamples to Furstenberg's $\times 2 \times 3$ conjecture. We show that a near return of the coordinate function $x \mapsto e^{2\pi ix}$ to itself under the semigroup generated by $2$ and $3$ forces the resolution entropy to be small, and that adherence of the coordinate function to eigenfunctions of $x \mapsto 2x$ forces it to decay. These rates are explicit in the rate of adherence and in the arithmetic of the eigenvalues.

This paper develops a one-dimensional model for a mobile viral component propagating through a spatially distributed susceptible population. Random viral motion produces diffusion, whereas a response to the relative gradient of susceptible density generates directed transport. The existence of a positive entire traveling wave is proved when directed transport dominates diffusion, and the limiting case of equal transport and diffusion coefficients is constructed separately. A dimensionless transformation then converts the remaining equation into a logistic equation and supplies closed-form profiles. These formulas identify the unique viral maximum and the far-field behavior. A conservative implicit-explicit finite-difference method provides an independent numerical check. Diffusion is evaluated at the new time level, the gradient-driven flux explicitly at cell interfaces, and susceptible depletion pointwise. Exact traveling-wave values specify the initial data and time-dependent Dirichlet conditions at both endpoints. Thus, the computation tests preservation and translation of the analytical profiles. A calculation with eight hundred spatial subintervals reproduces the localized viral band and the monotone susceptible front. The largest uniform errors are approximately $1.70\cdot10^{-2}$ for viral density and $2.22\cdot10^{-3}$ for susceptible density. The numerical results corroborate the analytical construction and show that the conservative discretization captures the proposed propagation mechanism.

Let $G$ be a Lie group equipped with commuting left and right linear representations on a vector space $V$, and let $G\Join V$ be the associated centered semidirect product. We develop a reduction scheme for covariant field theories on principal bundles with structure group $G\Join V$ from both Lagrangian and Hamiltonian perspectives, obtaining the reduced Euler-Poincaré and Lie-Poisson field equations decomposed into their $\mathfrak g$- and $V$-components, where $\mathfrak g$ is the Lie algebra of $G$. We also derive the reduced multisymplectic form formula, as well as the Noether and energy-momentum balance laws. Our reduction results are compared with those for principal bundles with standard semidirect-product structure groups, and the role of the centered variables is investigated: they retain the original two-sided geometric data and separate the left and right contributions via the heart and diamond operators. The framework is then applied to a two-sided matrix microstructural model and to the second-order jet group on $\mathbb R^r$, $G_r^2$. For the matrix model, the conjugation representation decomposes $\operatorname{Mat}(3)$ into scalar, vector, and symmetric-traceless sectors, clarifying its precise relation to director and alignment-tensor variables without identifying it with standard nematodynamics. For the one-dimensional jet group on a $(1+1)$-dimensional Minkowski base, we obtain a semilinear symmetric-hyperbolic system and an exact traveling conversion front with explicit energy-momentum exchange between the linear- and quadratic-jet sectors. Its hyperbolic-target interpretation yields a conserved positive energy for the linearized perturbations. A nonlinear numerical experiment illustrates the reduced evolution and reconstruction of the group-valued field, with consistency assessed through mesh refinement and an independent wave-map computation.

Based on our previously proposed direct approach to central configurations in the full configuration space, we study the degeneracy of planar four-body kite central configurations with arbitrary positive masses both in kite symmetry restricted subspace and in full configuration space. We derive a factorization of the determinant of the Jacobian that substantially simplifies the subsequent analysis. The methods used to study the singularity of the Jacobian include rigorous interval arithmetic, the Krawczyk operator, a quantitative implicit function theorem, and computer-assisted proofs. Both convex and concave kite central configurations are considered. In the convex case, we prove the nondegeneracy in both the restricted subspace and the full configuration space. In the concave case, there is a unique smooth degeneracy curve in the restricted subspace; an additional degeneracy curve emerges in the full space setting, which is hidden in the subspace of kite configurations and makes it unavoidable to study the degeneracy in the full configuration space. We also establish several properties of these degeneracy curves. As an illustration to this complete characterization, we exam the well-known central configuration of equilateral triangle with equal masses at the vertices and a fourth mass at the center, which reveals additional features of degeneracy captured by our full-space framework.

We study strictly convex central configurations of the planar Newtonian five-body problem with one dominant mass and four satellites of arbitrary positive masses. We prove that a single positive bound on the total satellite-to-primary mass ratio guarantees existence and uniqueness for every prescribed directed order of the five bodies on the convex hull, up to translations, rotations and positive scaling. The bound is independent of all satellite mass ratios, including ratios arbitrarily close to the boundary of the mass simplex. The constrained Hessian is nondegenerate on the six-dimensional shape space. We first prove uniqueness and nondegeneracy for the convex coorbital limit by an angular Hessian estimate. We then classify the possible collision limits as the satellite mass ratios vary and construct regularized local continuations for pairs, triples and pairs within triples. An exact factorization of the signed areas preserves strict convexity on independent parameter neighborhoods. These continuations connect every sufficiently small-mass configuration to the unique coorbital configuration and yield the common bound.

A weakly nonlinear stability analysis is performed to investigate the influence of thermal buoyancy on phototactic bioconvection in a suspension of swimming microorganisms confined between horizontal boundaries and subjected to bottom heating or cooling. The objective is to examine how thermal forcing modifies the post-critical evolution of bioconvective instability through its influence on bifurcation behaviour and finite-amplitude convection. Linear stability theory is first employed to determine the critical conditions for the onset of convection. The corresponding direct and adjoint eigenvalue problems, together with the second-order resolvent system, are then solved numerically to derive the Stuart-Landau amplitude equation governing the nonlinear evolution of disturbances. A systematic analysis is carried out for varying thermal Rayleigh numbers. The results show that, as the thermal Rayleigh number approaches the classical Rayleigh-Benard critical value, the critical bioconvection Rayleigh number decreases to zero, indicating a gradual transition from phototaxis-dominated to thermally driven convection. The computed positive Landau coefficient predicts supercritical pitchfork and Hopf bifurcations for stationary and oscillatory instabilities, respectively. In parameter regimes with oscillatory onset, increasing the thermal Rayleigh number drives a transition from a supercritical Hopf bifurcation to a supercritical pitchfork bifurcation. The nonlinear dynamics are further characterized through amplitude evolution, phase portraits, potential functions, and finite-amplitude flow structures.

We present a method for the data-driven inference of Hamiltonian and Lie-Poisson dynamical systems from vector-field samples, in which the Hamiltonian and the underlying Poisson tensor are estimated jointly rather than assuming the structure is known. The Hamiltonian is represented as a linear combination of random feature maps with fixed random internal weights, so its estimation is a linear regression problem for any given Poisson structure. We treat three cases of increasing generality: a known Poisson tensor, an unknown constant invertible (symplectic) structure matrix, and Lie-Poisson and affine Poisson structures, with structure matrix linear in the state. In the constant case, the Hamiltonian and structure matrix are recovered jointly from an eigenvalue problem that avoids the trivial zero solution due to the scaling symmetry of Hamiltonian systems. For Lie-Poisson and affine Poisson systems we alternate between an initial Hamiltonian estimate from a generalized eigenvalue problem for an approximate first integral, and a constrained nonlinear least-squares fit of the structure constants that enforces the Jacobi identity and, for affine structures, the cocycle condition, via continuation in a penalty parameter. Orthogonal matching pursuit selects a compact, well-conditioned subset of random features. Because the learned Hamiltonian is a sum of single-feature terms, the model admits an explicit Poisson/symplectic splitting integrator, which we use to validate the learned systems via long-time Poincare sections. We demonstrate the approach on the four- and five-dimensional Lorenz-86 model and a six-dimensional Kirchhoff rigid-body system, recovering Hamiltonians, structure constants and Casimir invariants to high accuracy, and showing the inferred Poincare sections converge to the correct invariant-tori topology as the number of random features increases.

Reservoir computing (RC) is a machine learning technique for data-driven modeling of dynamics for forecasting and control, primarily studied in computer science and physics literature with promising applications in neural network learning, physical computing, and neuroscience. Why reservoirs learn and how to choose good reservoir architectures are considered important open questions. We show that the RC problem is mathematically an extension of a classic problem in systems and control theory, the Kazantzis-Kravaris-Luenberger (KKL) observer design problem. As a consequence, many of the questions considered open for RC stand to benefit from a large body of theory in the mature KKL literature, non-exhaustively including on questions of embedding, transverse stability, local and global uniqueness guarantees, and effective data-driven solution constructions. Elaborating on this connection, we show that the surprising forecasting ability of reservoirs is in fact a direct consequence of the well-known observer internal model principle, derive an upper bound on the prediction error over a fixed forecast horizon, and provide a partial explanation for why linear readout training in RC works reasonably well. This work illustrates how classical ideas from systems and control can provide strong theoretical backing and open new questions for modern machine learning methods.

We study diffeomorphisms of Banach spaces satisfying generalized hyperbolicity: the stable and unstable subspaces are invariant only up to inclusion, need not depend continuously on the point, and expansivity may fail. On closed manifolds, this condition is equivalent to Axiom~A with strong transversality. We introduce a continuous-selection method that converts uniform orbitwise estimates into bounded continuous solutions of cohomological equations for generalized-hyperbolic cocycles. For diffeomorphisms in the uniform global \(C^1\) category, this yields continuous semiconjugacies in both directions under sufficiently small bounded Lipschitz perturbations, and topological stability in the sense of Walters. For the infinite product of one-dimensional Morse-Smale systems on \(\ell^p(\mathbb Z)\), \(1\leq p\leq\infty\), we construct compatible forward and reverse cohomological solution operators and prove structural stability, including for perturbations that couple coordinates, although the product is not expansive. For \(p=\infty\) this gives a structurally stable diffeomorphism with uncountably many hyperbolic fixed points, in contrast with the finitely many basic sets of Axiom~A diffeomorphisms. The cocycle analysis yields Lipschitz shadowing for the induced operator on bounded continuous vector fields.

We study central configurations for $n$ bodies interacting through pair potentials \[ U_\sigma = \frac{\kappa}{\sigma} \sum_{i<j}m_i m_j r_{ij}^{\sigma}, \] with logarithmic limit at $\sigma=0$. For the Newtonian exponent $\sigma=-1$, the classical Chazy–Wintner–Smale finiteness problem asks whether, for prescribed positive masses, there are only finitely many central configurations up to similarity. Finiteness is known in several important special cases and under additional dimensional or genericity hypotheses. It is therefore natural to ask whether an analogous finiteness principle continues to hold for other homogeneous exponents. We show that for every positive even exponent \[ \sigma=2k, \qquad k\ge2, \] the answer is emphatically negative. In fact, these exponents admit an abundance of continuous families of pairwise non-similar central configurations. The mechanism is provided by spherical designs: on a common sphere the force kernel for $\sigma=2k$ restricts to a polynomial of degree at most $k$, and consequently every positive weighted spherical $k$-design is a central configuration. We compute the corresponding central multiplier explicitly.

In his highly influential paper, On the maximal ergodic theorem for certain subsets of the integers, Bourgain introduced the study of pointwise convergence of ergodic averages along randomly generated subsets of the integers, developing a robust \(L^p\)-theory for \(p>1\). More recently, this theory was partially complemented at the \(L^1\) endpoint by work of LaVictoire, who treated averages along random sequences that are “slightly denser" than the set of squares, for which universal \(L^1\) pointwise convergence is known to fail. In this work, we prove that, almost surely, for every measure-preserving system and every integrable function, the ergodic averages formed by sampling along random sequences with quadratic growth converge almost everywhere. Our proof combines harmonic-analytic methods developed by M.~Christ with a “random sampling" of quadratic forms, thereby overcoming the main uniformity obstruction.

We prove that for any set $A$ with positive multiplicative upper density, there exist $x$, $y$, $z$ and $w$ in $A$ such that $x^2 + y^2w^2 = z^2$, generalizing a result of Frantzikinakis, Klurman and Moreira. Our techniques apply more broadly to a wide class of systems of equations, allowing us to solve these equations simultaneously and furthermore to restrict the w variable in an entirely different multiplicatively dense set. To do this, we prove a novel multiple recurrence result for multiplicative dynamical systems.

The Izhikevich model is a computationally efficient neuron model that can exhibit a wide range of firing patterns observed in the brain. However, it is a discontinuous dynamical system, which means the methods for applying weakly coupled oscillator theory developed for continuous dynamical systems cannot be applied. Therefore, the collective behaviour of coupled Izhikevich model has not been fully studied. To our knowledge, we carry out the first computation of the phase model for an Izhikevich neuronal model with chemical synapses. Using the phase model we study the existence and stability of phase-locking states and how these vary with parameters. Moreover, we find an interesting connection between the stability of anti-phase states of the phase model and the bifurcation behaviour of the uncoupled Izhikevich model itself. The accuracy of the prediction of the phase model is supported by simulations of two weakly coupled Izhikevich neurons.

In this paper, we study the ferromagnetic nearest-neighbor Ising model on rooted spherically symmetric trees with one or \(q\) children per vertex, according to a prescribed sequence of branching levels. Unbounded runs of one-child levels force uniqueness at every nonzero homogeneous field and every finite inverse temperature. We quantify this effect through the locations of sufficiently long nonbranching stretches. Conversely, uniqueness need not entail any uniform rate: for each prescribed vanishing sequence, we construct a density-one tree whose finite-volume boundary influence decays more slowly along a subsequence. The tree retains branching number \(q\), although its nonzero-field coexistence region disappears. For eventually bounded gaps, we prove an optimal uniform comparison with the periodic tree having the largest allowed gap and obtain an explicit interval of coexistence fields. Thus bounded gaps are equivalent, within this family, to coexistence at some finite temperature and nonzero field. For eventually periodic gaps, Gibbs uniqueness is equivalent to fixed-point uniqueness of the full-period return map. A negative Schwarzian derivative yields a complete classification: in the supercritical regime the map has three fixed points inside a closed coexistence interval, two at its endpoints, and one outside. The nonzero endpoints are automatically nondegenerate folds, with square-root splitting of the merging branches.

We study flows on finite volume homogeneous manifolds arising as directional restrictions of time changes of higher rank abelian unipotent actions. We establish Ratner-type rigidity results in this setting. For first row actions on finite volume quotients of $\operatorname{SL}_n(\mathbb{R})$, we show that natural quantitative assumptions on the cocycle force the flow to be algebraic and its ergodic invariant measure to be homogeneous. These results extend to higher dimensional subactions and to a general abstract setting. We also show that, on compact quotients of $\operatorname{SL}_3(\mathbb{R})$, the smooth time change is smoothly conjugate to a linear algebraic action by a leaf preserving diffeomorphism if and only if the cocycle of the directional flow has bounded deviations along one direction. We also obtain precise asymptotics for compact root increments. The principal thresholds in our results are sharp.

A shift space is said to have typical periodic optimization (TPO) if the set of Lipschitz functions whose unique maximizing measure is supported on a periodic orbit contains an open dense subset of the space of Lipschitz functions. We show that beta-shifts whose lexicographically largest point is a Sturmian sequence do not have TPO: on each such beta-shift there is a non-empty open set of Lipschitz functions, all of which have the Sturmian measure as their unique maximizing measure. These are the first known examples of beta-shifts without TPO, and, since they have the specification property, the first known examples of shift spaces with specification but without TPO. The corresponding beta-transformations do not have TPO in any space of Hölder functions on the interval. The main ingredient in the proof of these results is a rigidity property of Sturmian subshifts: modulo constants and Lipschitz coboundaries, the space of Lipschitz functions on such a subshift is one-dimensional.

This is a very first draft of a manuscript about existence of solutions of differential inclusions involving chattering.

We develop a parity-sensitive admissibility theory for topologically expansive decreasing Lorenz maps. By embedding them into the standard circle map $m_{-2}$, we realize every decreasing-admissible pair. We then study the negative doubling map $T_{-2}$ with a hole $(a,b)$, where $0<a\leq1/2\leq b<1$. Using the alternating lexicographic order, we classify plateaux of the symbolic survivor spaces. Under weak admissibility and strict shifted ordering, the boundary subshifts are realized as kneading spaces of decreasing Lorenz maps. We also establish entropy correspondences across countable itinerary discrepancies in completed survivor systems. For fixed $a$, we prove that $b\mapsto\dim_{\mathrm H}S_{-2}(a,b)$ is a devil's staircase, possibly constant. Under the pullback-null condition~(P), we obtain the corresponding survivor-entropy result for expansive decreasing Lorenz maps. We verify this condition for the piecewise-linear family in Example 5.1.

We investigate the weakly nonlinear evolution of phototaxis-induced bioconvection in an absorbing and isotropically scattering algal suspension subjected to oblique and normal collimated irradiation. Two hydrodynamic configurations are considered, a rigid lower boundary with either a stress-free or a rigid upper surface. A multiple-scales expansion is carried to third order, and the Fredholm solvability condition yields a Stuart–Landau equation for the amplitude of the critical mode. Oblique irradiation affects the onset threshold in opposite ways in the two configurations. It progressively lowers the critical Rayleigh number beneath a stress-free surface and progressively raises it beneath a rigid surface. Beneath a stress-free surface the onset remains stationary, but the pitchfork bifurcation changes from supercritical to subcritical and becomes strongly subcritical at large incidence angles, where the preferred pattern wavelength also increases markedly. Beneath a rigid surface the critical mode changes from stationary to oscillatory at moderate obliquity, giving a supercritical Hopf bifurcation, and returns to stationary at large obliquity, where the pitchfork bifurcation becomes subcritical.