arXiv subjects

Mohamed El Morsalani

Publications and source records attributed to Mohamed El Morsalani.

6 recordsLinked to original sources

arXiv2026

Inverse Problems for the Return Map in the Class ( $\mathcal{O}_C$ ): Reconstruction and Identifiability

We study the inverse problem associated with the return map $F$ on the boundary of a convex core $C$. By decomposing the return displacement into tangential and normal components, we derive an exact nonlinear reconstruction system for the unknown thickness $d$, yielding one-point uniqueness. Analyzing the linearization at fixed points introduces an operator $Q_*$: when $Q_* \neq 0$, the thickness and outer shape are uniquely recoverable via global propagation. In the degenerate regime $Q_* = 0$, local ambiguity is resolved using nonlinear Ehresmann-type transport and holonomy maps along closed paths. Finally, we establish conditional realizability and identify key topological and dynamical quantities from the return data, characterizing the commuting regime as a singular limit.

math.DS↗

arXiv2026

Geometry, Dynamics, and Topology of the Thickness Landscape: A Morse-Theoretic Analysis of the Return Map in the Class $\mathcal{O}_C$

We study the geometry, dynamics, and topology of the thickness landscape associated with the geometric return map in the class $\mathcal O_C$. For a convex core $C$ and an admissible outer domain $Ω$, the fixed points of the return map $F:\partial C\to\partial C$ coincide with the critical points of the thickness function $d$. At any such critical point $c_*$, the exact linearization reads \[ DF(c_*)=I+B_*H_*, \] where $H_*=\operatorname{Hess}_{\partial C}d(c_*)$ and $B_*=d_*(I-d_*S_*)^{-1}$ is positive definite. We show that the self-adjoint preconditioned Hessian $K_*=B_*^{1/2}H_*B_*^{1/2}$ admits an exact representation via the shape operators of the core and outer boundaries, implying that curvature modulates the return spectrum while preserving the inertia of $H_*$. In the strict thickness-ascent regime, every negative eigenvalue $μ$ of $K_*$ satisfies $μ\in (-2,0)$, yielding the stable/unstable dimension identities for any Morse equilibrium $c_Config$: \[ \dim E^s(c_*)=\operatorname{ind}_M(c_*), \qquad \dim E^u(c_*)=m-\operatorname{ind}_M(c_*). \] Since $\partial C\cong\mathbb S^m$, Morse inequalities act as equilibrium-counting laws for the return dynamics, implying an even number of equilibria. We compute the local fixed-point index $\operatorname{ind}_{\mathrm{fix}}(F,c_*)=(-1)^{m+\operatorname{ind}_M(c_*)}$, establish the global Euler balance with $°(F)=1$, and identify this index with the Euler-characteristic jump in the forward-invariant superlevel filtration. Finally, we discuss invariant manifolds, a conditional Morse--Smale structure, geometric examples, and the spectral extension to Morse--Bott manifolds, unifying boundary curvature, thickness geometry, discrete dynamics, and Morse topology.

math.DS↗

arXiv2026

Global Convergence Regimes of the Return Dynamics in the Class $\mathcal{O}_C$

This study investigates the global dynamics of discrete return maps $F: \partial C \to \partial C$ generated on the boundary of a convex core $C \subset \mathbb{R}^N$ via an outward normal displacement by a thickness function $d(c)$ followed by an inward normal return.By deriving a uniform global one-step asymptotic expansion preconditioned by the curvature operator $B_c = d(c)(I - d(c)S_c)^{-1}$, the authors establish a second-order expansion of the thickness increment governed by the preconditioned Hessian $K_c = B_c^{1/2} H_c B_c^{1/2}$. Under the spectral bound $λ_{\min}(K_c) \ge -2 + δ$, the map exhibits strict global thickness ascent, rendering $\mathcal{V}(c) = d_{\max} - d(c)$ a strict Lyapunov function. This guarantees finite dissipation, vanishing step sizes, and global convergence of orbits to unique critical points, with exponential rates at nondegenerate maxima and no nontrivial periodic behavior.Finally, numerical computations on the unit circle confirm that $λ_{\min} = -2$ marks a sharp stability threshold: crossing it leads to a period-doubling transition from curvature-preconditioned global convergence to stable period-two dynamics.

math.DS↗

arXiv2026

The Return Map in the Class $\mathcal{O}_C$: Geometry, Dynamics, and Thickness Regularity

We study a discrete dynamical system $F = π\circ Φ$ on the boundary of a convex core $\partial C \subset \mathbb{R}^N$, generated by linking $C$ to an outer boundary $\partialΩ$ via normal rays and a thickness function $d$. We derive the local expansion $F(c) - c = d(c)\bigl(I - d(c)S_c\bigr)^{-1}\nabla_{\partial C}d(c) + O\bigl(\vert{}\nabla_{\partial C}d(c)\vert{}^2\bigr)$, showing how curvature precalculates thickness gradients. Fixed points correspond to critical points of $d$, with linearization $DF(c_*) = I + d_*\bigl(I - d_*S_*\bigr)^{-1}H_*$ yielding a stability classification driven by curvature and thickness variation. Results are validated on the unit circle, alongside regularity transfer and global bilipschitz bounds.

math.AP↗

arXiv2026

Geometric Gradient Flows from Elliptic Level Sets: Normal Decomposition and Reflection Dynamics

We investigate the asymptotic geometry of shifting superlevel sets $Ω_t = \{x \in Ω: u(x) > t\}$ generated by solutions to the elliptic Dirichlet problem $-Δu = f$ in $Ω$, where the non-negative source $f \not\equiv 0$ is compactly supported within a strictly convex inner core $C \subset Ω$. Under a quantitative radial monotonicity condition, each boundary $\partialΩ_t$ is a smooth normal graph over $\partial C$ characterized by a thickness function $d_t \in C^{1,α}(\partial C)$ tracking $d_0$ as $t \to 0$.A central contribution is a rigorous decomposition of the inward unit normal field along the level surfaces: $\mathbf{n}_{Ω_t} = ν- \nabla_{\partial C} d_t + \mathcal{G} + \mathcal{P}$, where $ν$ is the static radial normal, $-\nabla_{\partial C} d_t$ is the kinematic driving vector, and $\mathcal{G}, \mathcal{P}$ are curvature and PDE Hessian remainder operators.In a thin-shell configuration $(\Vert{}d_0\Vert{}_{C^1} \ll 1)$, we formalize a discrete specular point-reflection mapping $F_n$ on $\partial C$. We prove the tangential displacement satisfies $F_n(p) - p = -2d_n(p)\nabla_{\partial C}d_n(p) + R_n(p)$, with quadratic control $\Vert{}R_n\Vert{}_{L^\infty} \le C\Vert{}d_n\Vert{}_{C^1}^2$. Using the energy $\mathcal{E}(t) = \int_{\partial C} d_t^2 \, d\mathcal{H}^{N-1}$, we show these orbits approximate a continuous gradient flow driven by $+\nabla_{\partial C} d_{\tilde{t}}(p)$ to first order. Finite element computations (FEniCS) validate these convergence rates.

math.AP↗

arXiv2026

From Tangency to Fractals: Quadratic Dynamics in Nested Convex Geometry

We study the dynamics generated by return maps associated with nested convex bodies and growing domains satisfying the geometric normal property in the plane. These maps are defined by transporting boundary points along normal directions to the surrounding domain and projecting them back onto the boundary of a subsequent convex set. We introduce a tangency condition between consecutive convex sets and show that it cancels the linear term in the local expansion of the transition operators. As a result, the dynamics near tangency points is governed by a quadratic normal form with an explicit coefficient depending on curvature and second order geometric data. This quadratic tangency law constitutes the central mechanism of the system. We prove that this nonlinear contraction leads to super exponential convergence toward the tangency set. In logarithmic coordinates, the dynamics becomes approximately affine, which allows for an interpretation in terms of iterated function systems (IFS) and explains the emergence of fractal limit sets. The theory is illustrated by several geometric configurations. Ford circles reveal a connection with continued fractions, nested ellipses yield Cantor-type limit sets, and configurations such as stadia and rounded triangles demonstrate the coexistence of linear and quadratic regimes. In purely quadratic settings with m independent branches, the limit set has similarity dimension in logarithmic coordinates, and an estimation of its Hausdorff dimension. From a broader perspective, the combination of symbolic branching and nonlinear contraction suggests potential connections with geometry, in particular in hybrid classical quantum information processing frameworks.

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