Papers

The modular Mandelbrot set $M_\Gamma$, the connectedness locus of the modular mating family of 2 : 2 holomorphic correspondences $\mathcal{F}_a$ on the Riemann sphere, is homeomorphic to the classical Mandelbrot set $M$. The Klein combination locus $\mathcal{K}$ (the "discreteness locus" of the family $\mathcal{F}_a$) is a pinched neighborhood of $M_\Gamma$ in the $a$-plane, pinched at the root point. We construct a canonical map $\Psi$ from $\mathcal{K}\setminus M_\Gamma$ into the hyperbolic plane $\mathbb{H}$, inspired by the construction of Douady and Hubbard for their celebrated conformal bijection $\Phi: \mathbb{C}\setminus M \to \mathbb{C}\setminus \overline{\mathbb{D}}$, and we prove that $\Psi$ is analytic. This map $\Psi$ induces a tessellation of $\mathcal{K}\setminus M_\Gamma$ by pulling back a tessellation of $\mathbb{H}$ invariant under the modular group. We develop a series of conjectures concerning the structure of $\mathcal{K}$, its boundary, and $\Psi(\mathcal{K}) \subset \mathbb{H}$.

We explore the problem of asymptotically stabilizing a class of interacting particle systems to a prescribed particle configuration with zero velocity. The class of interacting particle systems is motivated by the Cucker-Smale model on the mean-field level. Rather than designing a feedback by manually solving an IDA-PBC matching equation, we derive a feedback using an instantaneous rolling horizon control algorithm. We show that this feedback preserves the port-Hamiltonian structure of the interacting particle system and therefore allows for an interpretation in the IDA-PBC framework. Additionally, the energy balance of the closed-loop Hamiltonian provides an important tool to prove convergence of the particles' velocities and interaction forces in discrete and continuous time. If the control operator is surjective, we additionally prove convergence of the particle positions to the prescribed configuration.

We resolve two questions posed by Huang and Ye concerning full-scattering systems. First, we construct a four-symbol subshift which is $\Delta^*$-mixing but not full-scattering, thereby showing that $\Delta^*$-mixing does not imply full-scattering. Second, we prove that every finite product of full-scattering systems is full-scattering.

Contact Hamiltonian dynamics gives dissipative mechanics an intrinsic action variable, but explicit contact splittings reach only kinetic energies whose terms are exactly integrable: frozen-coordinate diagonal metrics (the spherical pendulum, a torus particle) are included, while dense metrics with momentum cross terms, with the double pendulum as flagship, are not. We introduce a projected Pihajoki-contact integrator for this non-separable setting, combining phase-space duplication, symmetric projection onto the physical diagonal, and constant-friction damping half-steps, with the action factor carried by an exact Herglotz update. As in the projected extended-phase-space framework it builds on, the construction needs no binding parameter, returns the copies to the diagonal at every step, and confines the nonlinear solve to the $2n$ projection variables. For constant friction the step rescales $\omega=d\eta$ by the exact factor $e^{-\gamma\tau}$ when the projection is solved exactly (a classical conformally symplectic identity, realized here for this class), while time-symmetry, consistency, and smoothness yield an $O(\tau^3)$ one-step contact-form residual, a bound not specific to the contact form. On the damped double pendulum, spherical pendulum, and torus particle the method is second-order accurate, reproduces the contact decay law, and controls long-time energy and contact drift in coarse or stiff regimes where the Tao baseline and the unprojected average lose the solution. A head-to-head with exact-contactomorphism splittings delimits the niche: where a frozen-coordinate splitting exists it preserves the contact form exactly and wins at matched cost; for the dense double-pendulum metric the realizable alternative is first-order with a prohibitive constant and the projected method prevails. The contact-form estimate is local, one-step, and constant-friction.

Let $\Phi$ be a holomorphic IFS on a bounded domain in $\mathbb{C}$. Suppose that the following conditions hold: (1) the maps in $\Phi$ do not have a common fixed point; (2) there does not exist a regular real-analytic curve which is invariant under all of the maps in $\Phi$; (3) $\Phi$ is not holomorphically conjugate to a homothetic IFS; (4) $\Phi$ is exponentially separated. Under these assumptions, we show that the dimensions of the self-conformal measures associated to $\Phi$, as well as the Hausdorff dimension of the associated self-conformal set, attain their natural upper bounds. The proof combines recently developed methods from the dimension theory of stationary fractal measures with complex-analytic arguments.

We characterize the simplicity of reduced crossed product C*-algebras in terms of stabilizer subgroups. Specifically, we prove that if $G$ is a countable group and $X$ is a minimal $G$-flow, then the reduced crossed product C*-algebra $\mathrm{C}(X) \times_\lambda G$ is simple if and only if there is a point in $X$ with a C*-simple stabilizer subgroup. Further, these conditions are equivalent to a generic point in $X$ having a C*-simple stabilizer subgroup. We also provide an example demonstrating that this result does not extend to uncountable groups. This completely resolves a question of Ozawa.

This work studies the relations between shadowing, topological stability, Lipschitz structural stability, and pseudo-hyperbolicity for linear dynamical systems on Banach spaces. The main result establishes that pseudo-hyperbolicity implies both topological stability and strong Lipschitz structural stability, whereas strong Lipschitz structural stability implies the shadowing property. In addition, a spectral characterization of pseudo-hyperbolicity is obtained, yielding an equivalence between pseudo-hyperbolicity, topological stability, strong Lipschitz structural stability, and the shadowing property for invertible operators whose eigenspaces associated with unimodular eigenvalues admit closed complements. In particular, these four notions are equivalent on Hilbert spaces. Combined with a recent result of Dragičević and Pituk, these results yield, for a large class of Banach spaces, an invertible operator that has the shadowing property but is not generalized hyperbolic, thereby answering an open problem in linear dynamics.

This paper provides a complete characterization of the sign of the propagation speed in the symmetric two-species Lotka-Volterra competition-diffusion model under strong competition. The system admits a unique bistable travelling front and the sign of its speed determines which of the two species invades the other. We prove that for every strong competition intensity greater than $1$ and every diffusion ratio $d\neq1$, the travelling front propagates so as to expand the territory occupied by the faster-diffusing species. The front has a zero speed exactly when $d=1$. We also establish the smooth dependence of the wave speed and the travelling front on the model parameters. The main step in the sign characterization is to prove that a monotone standing front cannot exist when the diffusion rates are different. Combined with continuity of the wave speed with respect to the parameters, the species-exchange symmetry of the system, and an explicit travelling front at a particular parameter value, we obtain the sign of the propagation speed throughout the entire parameter region. This establishes the “Unity is not strength” theorem, which was previously known only in restricted parameter regimes.

We establish the Moran–Hutchinson formula for attractors of finite systems of surjective similitudes on semimetric spaces. More precisely, for a complete, normal semimetric space satisfying strong regularity and geometric doubling, we prove that the open set condition implies that the Hausdorff measure of the attractor at the similarity dimension is positive and finite. We also prove the converse, specifically, positivity of the Hausdorff measure at the similarity dimension implies the open set condition. This provides a partial answer to a question posed by Bessenyei and Pénzes in 2022.

We study the Random Quadratic Form (RQF) on a sphere in the presence of random Brownian forcing. We show that the forcing does not effectively change the law of the process but affects the synchronization properties of the system. While the RQF without forcing exhibits partial synchronization due to the intrinsic symmetries, the introduction of an arbitrarily small forcing results in long-term symmetry breaking and leads to full synchronization. In this work we focus on the small forcing regime and recover the multiscale behavior of the two-point process. We show that in the first stage the model converges to an anti-polar configuration due to the symmetries of the RQF and in the second stage the two clusters meet due to the symmetry breaking phenomenon. The model is motivated by continuous-time machine learning models such as Neural ODEs and continuous-time formulations of transformers. In particular, the results of this work explain the role of the bias and the scale of its initialization.

We prove that for every ergodic transformation $T$ on an infinite standard probability space both the automorphism group (i.e. centralizer) $\mathrm{Aut}(T)$ and its reversing automorphism group are realizable as symmetry groups of graphings. This is an analogue of Sabidussi's realization of arbitrary graph-automorphism groups, and provides numerous examples of graphing automorphism groups carrying no compatible compact topology, answering a question of Lovasz'. Another consequence of discussion and ensuing constructions is the existence of large mutually locally-globally equivalent graphing families with highly variable symmetry.

We present a systematic study on the role of invariant connections in geometric mechanics. We first develop a comprehensive theory of reduction under left- and right-invariant connections on Lie groups, showing that the Euler–Poincaré and Lie–Poisson equations are independent of the connection. We then introduce a novel connection-dependent variational principle, where the Lagrangian depends on the velocity parallel-transported back to the initial point of the curve. For Cartan–Schouten connections, this leads to an integro-differential Euler–Poincaré equation that exhibits two distinct sources of nonlocality in time: a path-dependent term encoded in the parallel transport and a future-dependent term arising from a curvature integral. We reformulate this equation as a two-point boundary value problem and present a two-level numerical scheme. The general theory is illustrated on the Heisenberg group, where the equations simplify due to nilpotency, and on the rotation group, where the full integro-differential structure is retained.

We prove a measurable realization theorem for hyperuniform Delone point processes. In dimensions \(d\geq2\), for every prescribed \(q\geq1\), every essentially free ergodic p.m.p.\ action of \(\mathbb R^d\) admits, at every sufficiently large prescribed intensity, a generating Delone realization whose return-time point process \(\eta\) is measurably isomorphic to the original action and whose Bartlett spectrum \(\sigma_\eta\) satisfies \[ \sigma_\eta(B_\varepsilon)=o(\varepsilon^{2q}) \qquad(\varepsilon\downarrow0). \] Thus arbitrarily high finite-order low-frequency suppression can be imposed without changing the prescribed measurable dynamics. The same realizations can be chosen with surface-order ball variance and linear rigidity to any prescribed finite order, while also being maximally rigid and almost surely bounded-displacement equivalent to a lattice. For essentially free Euclidean-motion actions whose translation subaction is ergodic, the construction can be made isotropic and \(V\)-ergodic, and hence \(V\)-weakly mixing. In dimension one, every essentially free ergodic flow admits generating Delone realizations with logarithmic interval discrepancy, maximal rigidity, and near-quadratic decay of the Bartlett spectrum at the origin.

We introduce Cayley-tree POTP, obtained by imposing the pseudo-orbit equations of a finitely generated group action only along a spanning tree of a Cayley graph. For zero-dimensional actions, we characterize this property by equicontinuity along normalized replacement paths; for subshifts, the criterion is expressed in the right-coset space of the common left-period subgroup. These criteria characterize virtual freeness and, for commensurated subgroup pairs, identifies Cayley-tree POTP of the coset full shift with relative quasi-tree geometry and a finite Bass–Serre decomposition. For infinite-index VFP pairs, Cayley-tree POTP of the coset full shift is equivalent to virtual cohomological codimension one, although ordinary POTP holds for every such shift. Finally, we prove that the strong topological Rokhlin property passes to finite-index overgroups. Consequently every finitely generated virtually free group has this property, answering the virtually cyclic case posed by Doucha, and Cayley-tree POTP is generic for its Cantor actions.

We study the dynamics of the exponential map on the complex plane. The set $\Lambda_{\mathbf{c}}$ of all points sharing a given itinerary $\mathbf{c}$ is non-empty if and only if $\mathbf{c}$ is an exponentially bounded itinerary. For such itineraries, $\Lambda_{\mathbf{c}}$ also contains a curve of escaping points, and hence its Hausdorff dimension is at least~$1$. We prove that for every exponentially bounded itinerary this dimension is in fact equal to~$1$. In comparison, for certain itineraries, the set $\Lambda_{\mathbf{c}}$ exhibits highly complicated topological structures, such as indecomposable continua.

We study skew products of Hénon maps fibered over suitable metric spaces and investigate the analytic structure of their escaping sets. Our study extends the description of escaping sets for Hénon maps developed by Hubbard and Oberste-Vorth to skew products and provides a framework for studying their rigidity. For a compact base space, we construct an intermediate covering space of each fiberwise escaping set. When the base is the closed unit disk and the family depends holomorphically on the parameter, we obtain an analogous description of the global escaping set. We further use this covering space construction to study the relationship between biholomorphic equivalences of their global escaping sets and the underlying dynamics. Finally, for skew products over the non-compact base $\mathbb{C}$, we consider the corresponding escaping region and investigate its analytic structure.

Shared autonomy requires principled mechanisms for allocating and transferring control between a human and an autonomous agent. Existing approaches often rely on blending control inputs or heuristic switching rules, which lack theoretical guarantees and fail to account for the dynamics of authority transfer. This paper develops a cooperative game-theoretic framework for authority switching in shared autonomy. We formulate the control switching problem as an identical-interest dynamic game in which authority transitions are embedded into the system dynamics, yielding optimal switching policies rather than ad hoc rules. We establish the existence and characterization of team-optimal policies in pure strategies under stochastic human override, accounting for asymmetric authority where humans retain override capability. For linear-quadratic systems, we derive closed-form recursions for the optimal switching policies and value functions, enabling efficient computation independent of the continuous state. We validate the framework on scalar and multi-dimensional linear systems, demonstrating how optimal switching adapts to varying system dynamics, cost structures, and override probabilities. The results reveal fundamental trade-offs between human adaptability and autonomous efficiency, illustrating the practical benefits of grounding shared autonomy in cooperative game theory.

Largest Lyapunov exponents (LLEs) quantify exponential sensitivity, but data-driven estimates are often obtained from finite-precision trajectories. We show that increasing the length of a single reduced-precision chaotic record can eventually degrade a forecast-error LLE estimate. Using the logistic map at r=4, an ESP32 single-precision trajectory is reproduced bit-for-bit by NumPy float32. Across 10,000 random float32 initial conditions, every trajectory reaches an exact recurrence before iteration 7612. For one long float32 record, the estimated LLE changes from 0.6853 at N=15,000 to 0.1827 at N=20,000 and approximately zero at N=30,000 as exact train-test histories saturate. At N=100,000, the long float32 record gives 0.0016, whereas independently restarted length-100 trajectories give 0.6917; matched float64 controls remain near ln(2)=0.6931. The collapse is reproduced for 28 representative initial conditions, and its onset is strongly correlated with the recurrence scale set by transient length and digital period (Pearson r=0.982). Thus, finite-state recurrence can turn additional samples into duplicate futures rather than new dynamical information, while independent restarts substantially delay this saturation.

Let $k\ge1$ be an integer and let $\lambda$ be the Liouville function. In 1965, Chowla gave a conjecture that the values of $\lambda(n+h_1),\dots, \lambda(n+h_k)$ are asymptotically unrelated for any distinct natural numbers $h_1, \dots, h_k$. In this article, motivated by the recent work of Bergelson and Richter on the dynamical generalizations of the prime number theorem, we will show a dynamical generalization of Chowla's conjecture on average. In the proof, we follow an approach of Qi and Zheng who established a variant of Bergelson and Richter's theorem over irreducible binary cubic forms. Moreover, we will use this approach to show an analogue of the dynamical Chowla's conjecture along the primes on average. In 2016, Tao proved that the two-point logarithmic Chowla's conjecture holds. Recently, Charamaras and Richter generalized Tao's theorem to bounded arithmetic functions and proposed a conjecture that generalizes Chowla's conjecture to bounded multi-variable arithmetic functions. Building on their work, we prove a dynamical generalization of Tao's theorem and a variant for the composition of the sum-of-digits function with the prime Omega function.

This paper investigates a time-periodic, spatiotemporally heterogeneous competition-diffusion system, where two species have different intrinsic growth rates but equal total resources per period. Using asymptotic analysis and principal eigenvalue theory, we systematically examine its dynamics. When both diffusion rates are small, spatial heterogeneity of the difference in time-averaged resources leads to uniform persistence, stable coexistence, and asymptotic spatial segregation; whereas its spatial uniformity produces multiple regimes–fast-diffuser selection, parameter-dependent coexistence, or slow-diffuser dominance–separated by smooth threshold curves under different structural conditions. In the regime where at least one diffusion rate is large, we obtain a detailed classification of the local dynamics, with sharp delineation of parameter regions for stable coexistence, competitive exclusion, and a novel bistable scenario, where both semi-trivial periodic solutions are linearly stable while an unstable coexistence state also exists. We further characterize the asymptotic profiles of positive solutions in the mixed-scale diffusion limit. Our results show that the interplay between temporal periodicity and spatial heterogeneity can overturn the classical "slower diffuser always prevails" principle, leading to a substantially richer range of ecological outcomes.

Oscillator Ising Machines (OIMs) have emerged as promising computational architectures for approximating solutions to combinatorial optimization problems. We derive and analyze the mean-field limit of an OIM model and show that it inherits the gradient flow structure of the finite-dimensional dynamics. We identify conditions under which this mean-field evolution admits an Eulerian formulation as a gradient flow on the Wasserstein space of probability measures and contrast this with a Lagrangian formulation which is always available. This gradient flow structure strongly constrains the long-time dynamics and enables a complete classification of limit solutions and their stability in the totally symmetric case. In particular, all limit solutions are fixed points whose phases cluster into at most four groups, and for almost all parameter values, only binarized fixed points – those with clusters at $0$ and/or $\pi$ – can be stable. Since binarized states are exactly those for which a feasible solution to the original problem can be read out, this shows that feasible solutions can almost always be recovered. We provide tight bounds on the parameter thresholds for which fixed points in this binarized family are stable, thereby identifying the threshold for binarization in this model, and present numerical evidence that the mean-field model correctly predicts behavioral regimes in certain large random networks, including Erdős-Rényi networks. Moreover, this problem provides a case study in the application of mean-field models, gradient flows, and optimal transport to the study of large-scale systems, and we highlight a number of features of this problem which are informative in this respect.

We study Gaussian random dynamical systems with coordinatewise $\tanh$ nonlinearity, where finite precision is modeled by nearest-grid rounding after each step. Gaussian symmetry reduces the dynamics to an exact Markov chain for the normalized squared radius. Rounding makes the origin absorbing, and the total variation distance to the absorbing equilibrium equals the survival probability of the absorption time. At fixed width, we identify the critical gain and prove an absorption cutoff with Gaussian profile as the mesh tends to zero. At fixed precision, global contraction yields a large-dimension absorption cutoff, while positive drift produces metastability. In the supercritical regime, we prove a large-dimension cutoff to a nonzero invariant law and show that, at fixed dimension, its mass near the repelling origin has a power-law asymptotic. All six main results are formalized in Lean 4 on top of Mathlib and independently checked against restatements that import only Mathlib.

Let $M$ be a closed smooth manifold of dimension $d\geq3$. Given a compact metrizable Choquet simplex $\mathscr S$ and a bounded nonnegative affine upper semicontinuous function $\mathfrak e$ on $\mathscr S$, we construct a $C^\infty$ diffeomorphism $h$ of $M$, isotopic to $\operatorname{id}_M$ and supported in an embedded $d$-dimensional solid torus $D^{d-1}\times S^1$, with an isolated minimal invariant Cantor set $K$. The invariant-measure simplex of $h|_K$ is affinely homeomorphic to $\mathscr S$ with entropy function $\mathfrak e$, whereas every ergodic $h$-invariant measure not supported on $K$ is a Dirac measure at a fixed point. Consequently, the set of measure-theoretic entropies of ergodic $h$-invariant probability measures and the topological entropy of $h$ are \[ \mathscr H_{\mathrm e}(h)=\{0\}\cup\mathfrak e(\operatorname{ex}\mathscr S), \qquad h_{\mathrm{top}}(h)=\max_{p\in\mathscr S}\mathfrak e(p). \] The map $h$ is $C^\infty$-approximable by zero-entropy diffeomorphisms isotopic to $\operatorname{id}_M$. Taking $\mathscr S$ to be a singleton yields counterexamples to Katok's intermediate-entropy conjecture on every such $M$. We also construct such counterexamples $h_j$ and numbers $c_j>0$ with $h_j\to\operatorname{id}_M$ in $C^\infty$, $c_j\to0$, and \[ \mathscr H_{\mathrm e}(h_j)=\{0,c_j\}, \qquad h_{\mathrm{top}}(h_j)=c_j. \] Hence the intermediate-entropy property is not $C^\infty$ open among diffeomorphisms isotopic to the identity.

Let $d\geq 1$. Let $K\subset\mathbb{R}^d$ be a non-singleton self-similar set generated by a finite strongly irreducible iterated function system satisfying the open set condition, and let $\delta=\dim_{\mathrm H} K$. For $\tau>1/d$, set \[ W_d(\tau) = \left\{ \mathbf{x}\in\mathbb{R}^d: |q\mathbf{x}-\mathbf{p}|<q^{-\tau} \text{ for infinitely many }(\mathbf{p},q)\in\mathbb{Z}^d\times\mathbb{N} \right\}. \] We prove that there exists $\varepsilon_K>0$ such that, for every $1/d<\tau<1/d+\varepsilon_K$, \[ \mathcal{H}^{s(\tau)}(K\cap W_d(\tau))=\infty, \qquad\text{with } s(\tau):=\delta+\frac{d+1}{1+\tau}-d, \] and consequently \[ \dim_{\mathrm H}(K\cap W_d(\tau)) = \delta+\frac{d+1}{1+\tau}-d. \] In dimension one, specializing to the middle-third Cantor set, this establishes the Bugeaud–Durand conjectural formula for $\tau>1$ sufficiently close to $1$.

This paper concerns the maximum number of limit cycles of generalized Abel differential equations $dx/dt = A(t)x^p + B(t)x^q$, where $A$ and $B$ belong to the linear span of a family of functions having the Chebyshev property. Motivated by a recent open problem posed by Huang et al. (Nonlinearity, 2026), we investigate whether this maximum number can be bounded in terms of $p$, $q$, and the structure of the family. Under some natural hypotheses and by means of first- and second-order analyses using Melnikov functions, we provide lower bounds for this maximum number. In contrast to previous work, no specific form for the coefficients is assumed. We then apply these estimates to Abel equations with trigonometric polynomial, polynomial, and hyperbolic coefficients. In the trigonometric polynomial case, we reestablish the results of Álvarez et al. (J. Math. Anal. Appl., 2008) and Huang et al. (SIAM J. Appl. Dyn. Syst., 2020), while in the polynomial case, we improve the classical lower bound given by Lins-Neto.

We prove various statistical properties (i.e., decay of correlations, central limit theorems with convergence rates, maximal large deviations and almost sure invariance principles) for non-smooth observables in the form of indicators functions in polynomially mixing hyperbolic dynamical systems. Such results were not known (or even expected to hold) for slowly mixing hyperbolic systems, although they are important for applications to physics and other sciences.

In his seminal paper, Kac famously asked whether "one can hear the shape of a drum" - that is whether the isometry class of a bounded domain in a Euclidean space is uniquely determined by the spectrum of its Laplace spectrum. The Laplace spectrum is closely related to the length spectrum of the associated billiard. For a convex bounded planar domain, to each billiard periodic orbit one can associate not only its length but also its rotation number. The set of pairs of length and rotation number of each periodic orbit is called the marked length spectrum. Using the marked length spectrum of the domain one can associate with it its minimal action function also known as Mathers $\beta$-function denoted by $\beta_\Omega$. Via unique quasianalytic continuation, we prove the following rigidity problem: knowing that Mather's $\beta$-functions of two billiards in two analytic planar domains coincide on a set of positive measure of KAM diophantine numbers implies that they coincide on a set of all KAM diophantine numbers. In particular, knowing that Mather's $\beta$-functions of two domains coincide on a positive measure set of KAM diophantine numbers implies that the Marvizi-Melrose invariants of these domains coincide.

We show that the stable norm of flat metrics on $H_{d}(T^n,\mathbb{R})$ is locally rigid if $1\leq d<n-1$ and locally rigid among metrics of the same volume if $d=n-1$. We also show that the stable norm on $H_{2}(T^3,\mathbb{R})$ is not locally rigid. As applications, we answer negatively a question raised by Bangert in his ICM address, prove local rigidity of the marked $k$-area spectrum of flat metrics for $1\leq k\leq n-2$, and prove a local rigidity result for the volume spectrum of flat metrics on $T^n$.

This paper develops Admissibility-Preserving Control (APC), a realization-centered safety-critical control framework for strict-feedback systems subject to asymmetric actuator limits, time-varying output constraints, and actuator-rate limitations. APC denotes the overall control architecture, whereas an Admissibility-Preserving Input Realization (APIR) denotes its constraint-realization module. Therein, the APIR dynamically generates the physical plant input while rendering its prescribed asymmetric actuator set forward invariant. In contrast to algebraic clipping and post-design saturation compensation, the actuator limits are embedded directly in a continuously differentiable dynamic realization with user-selectable regularity and interpretable tuning parameters. The APIR is integrated with recursive backstepping by treating the realized plant input as an additional state. The resulting design does not require an input-to-state stability assumption on the uncontrolled plant. Instead, the nonlinear drift terms are compensated recursively, subject to an explicit compatibility condition between the desired motion, the available control authority, and the APIR interior gain. The framework is further extended to time-varying output-safe tracking through a smooth asymmetric logarithmic barrier coordinate and its associated Lyapunov function and to simultaneous actuator-magnitude and rate constraints through a cascaded APIR. Rigorous Lyapunov and invariance analyses establish regional asymptotic tracking, forward invariance of the compatible admissible sets, and boundedness of all closed-loop signals. Numerical studies illustrate asymmetric actuator utilization, output-safety preservation, and magnitude-rate constraint enforcement.

Receptor occupancy models quantify inhibition of cancer signalling, often treating target occupancy as a proxy for downstream activity. Although this simplification yields low-dimensional models, it cannot represent delayed pathway shutdown, transient resistance, or non-monotone viability responses generated by intracellular networks. We formulate a fast–slow framework that distinguishes drug–target occupancy from downstream signalling activity. The fast variable $X$ represents occupancy, the activity variable $A$ represents pro-survival signalling, and the slow variable $B$ represents adaptive feedback including phosphatase induction, stress adaptation, or signalling rewiring. Rapid occupancy relaxation permits a quasi-steady reduction, while weak feedback guarantees global convergence to a unique equilibrium. Stronger feedback may bring the frozen activity subsystem near a saddle-node fold, producing a ghost regime in which trajectories linger near the remnant of a vanished high-activity equilibrium. We determine when slow adaptive passage preserves the inverse-square-root ghost law or, under transverse crossing, produces the dynamic delay scale $O(\varepsilon^{-1/3})$. Coupling activity to viability translates these delays into shoulders in dose–response curves, while adaptive lag may generate early-time overshoot without ad hoc forcing. A proof-of-concept fit to time-resolved viability data reproduces patterns across measured concentrations and observation times and reveals practical identifiability limitations. The framework also predicts exposure-time-dependent shifts in apparent potency, including $IC_{50}(t)$, because viability integrates signalling activity over time rather than receptor occupancy alone.