Papers

Recurrent fractal interpolation extends the usual fractal interpolation framework by introducing a recurrence structure. Fractal histopolation, on the other hand, is concerned with the area-matching properties of fractal functions. In this paper, we combine these two approaches and introduce a recurrent fractal histopolants.

In this note, we consider the advection-diffusion equation on the $d$-dimensional torus, where $d$ is arbitrary. First, for the associated transport equation, we provide an explicit example of a space-time smooth, divergence-free, random velocity field that is a universal exponential mixer. Second, we set the molecular diffusivity to a (small) positive value, and we show that the same velocity field is dissipation enhancing with optimal rate. Our construction is based on the Random Dynamical Systems viewpoint, and relies on the uniform ergodicity of the two-point Markov process associated with the two-dimensional Pierrehumbert model.

When studying the dynamical Mordell-Lang conjecture by using the $p$-adic method, experts have found that return sets of the $p$-adic version of the DML problem can be very sparse. In this article, we investigate the similar phenomenon for a complex analytic version of the DML problem and the DML problem for backward orbits.

Every solution for the classical four body problem in space which ends in total collision has a limiting shape. We prove that when that shape is planar the entire solution lies in a fixed plane and when that shape is collinear the entire solution lies in a fixed line.

Physical systems evolve continuously in time, yet their states are typically observed only at discrete times. Generating trajectories consistent with their probability densities from such observations therefore requires capturing the continuous-time evolution rather than only learning transition mappings between consecutive observations. We propose PhiBE-Flow, a framework that directly estimates the probability velocity field induced by the stochastic differential equation (SDE) which governs this continuous-time distributional evolution. PhiBE-Flow learns from discrete observations using a model-free approach requiring neither known SDE coefficients nor score estimation. We establish convergence guarantees for the method, accounting for both time-discretization and finite-sample errors. We evaluate PhiBE-Flow on systems of increasing complexity, from controlled stochastic numerical systems to Navier–Stokes dynamics and real-world videos. Our results show that PhiBE-Flow accurately recovers probability flows of stochastic dynamics, preserves multiscale physical statistics, and improves video generation performance over representative baselines. The code is available at https://github.com/R1fe/PhiBE-Flow.

We generalize the concept of orbit-breaking from the case of a homeomorphism to the case of a surjective local homeomorphism. This method creates open subgroupoids of the Deaconu–Renualt groupoid associated to the surjective local homeomorphism. When the map satisfies natural assumptions (including being locally expanding) the resulting groupoid $\mathrm{C}^*$-algebras are unital UCT Kirchberg algebras. A specific class of Deaconu–Renault systems are constructed, each modelling the Cuntz algebra $\mathcal{O}_2$. The $\mathrm{C}^*$-algebras obtained from orbit-breaking for such systems come with explicit embeddings into $\mathcal{O}_2$. We also introduce a method for orbit-breaking after removing a fixed point. Using this method, we can realize all stable UCT Kirchberg algebras with each coming with an embeddings into $\mathcal O_2\otimes\mathcal K$. Moreover, we show that, given a homomorphism between countable graded abelian groups, there is a $^*$-homomorphism between orbit-breaking algebras obtained from a groupoid inclusion. In particular, we obtain explicit $^*$-homomorphisms between stable UCT Kirchberg algebras from any group homomorphism between their $K$-groups.

The Koopman operator is a well-established framework for lifting nonlinear dynamical systems to an infinite-dimensional space where dynamics are linear. Extended Dynamic Mode Decomposition (EDMD) is a widely used method in data-driven dynamics as it provides a Galerkin approximation of the Koopman operator on the finite-dimensional span of a prescribed dictionary of observable functions. In particular, EDMD only requires samples from the dynamical system, describing the Koopman operator without knowledge of the underlying dynamics. While many studies analyze asymptotic convergence results for EDMD in terms of data, the question of more practical finite-data convergence results remains incomplete in general settings. In this work, we prove bounds for the finite-data EDMD Galerkin approximation of the Koopman operator by a rate proportional to the inverse of the root of the number of samples (the Monte Carlo rate). These results apply to multiple classes of problems, i.e., for discrete or continuous dynamics, including stochastic systems. In certain more structured setting, we demonstrate super-polynomial rates are achievable both theoretically and computationally. Finally, we derive recovery guarantees for an EDMD variant in the undersampled regime inspired by compressed sensing techniques.

For birth–death processes with a strong Allee effect, long survival near a positive stable equilibrium does not by itself determine the principal spectral gap. We identify its large-population limit when the birth and death rates at population size $n$ are $n\widetilde\lambda(n/K)$ and $n\widetilde\mu(n/K)$, where $K$ is the population scale and $\widetilde\lambda,\widetilde\mu$ are the per-capita rates. Write $V(x)=x(\widetilde\lambda(x)-\widetilde\mu(x))$ for the deterministic drift, with unstable threshold $x_1$ and positive stable equilibrium $x_2$. For the two smallest eigenvalues $\rho_1(K)<\rho_2(K)$ of the negative generator killed at extinction, we prove \[ \lim_{K\to\infty}\bigl(\rho_2(K)-\rho_1(K)\bigr) =\min\{\widetilde\mu(0)-\widetilde\lambda(0),V'(x_1),-V'(x_2)\}, \] so the limit depends on the linearization rates at all three equilibria. Each term can be the unique minimum even when $x_1$ and $x_2$ are fixed. The proof combines a frozen boundary model with two local oscillator limits through discrete Ismagilov–Morgan–Simon (IMS) localization. A uniform comparison of the global and stable local ground states controls the orthogonality constraint in the variational lower bound; projected local trial vectors give the matching upper bound. The result determines the limiting $L^2$ spectral gap and optimal Poincaré constant of the $Q$-process, which describes conditioning on indefinite survival. Combined with principal-eigenvalue asymptotics from a companion work, it separates an exponentially growing quasi-stationary mean extinction time from a spectral relaxation time with a finite positive limit.

We study density-dependent birth–death processes with a strong Allee effect and absorbing extinction. The deterministic system is bistable: extinction and a positive equilibrium are locally asymptotically stable, separated by an unstable Allee threshold. Let $K$ be the population-size scaling parameter. As $K\to\infty$, we derive a sharp Eyring–Kramers-type asymptotic formula, including the leading prefactor, for the principal eigenvalue of the killed generator, and a uniform approximation of the corresponding positive eigenvector. Consequently, we obtain a sharp asymptotic formula for the mean extinction time under the quasi-stationary distribution (QSD). In the bistable regime, the initial population determines the competition between early extinction and positive metastability. After initial relaxation, the law is approximated in total variation by an initial-state-dependent convex combination of the QSD and the Dirac mass at extinction. The mixing coefficient is identified as the probability of reaching the positive stable region before absorption. We identify a probabilistic transition layer of width $O(\sqrt K)$ around the Allee threshold, where the mixing coefficient converges to an explicit Gaussian profile. Notably, a population starting at the threshold reaches the positive metastable region or becomes extinct with asymptotically equal probabilities. We establish existence and uniqueness of the QSD, quantitative convergence to the quasi-stationary regime, and a discrete Gaussian approximation of the QSD centered at the positive stable level. Our analysis combines a self-adjoint spectral realization of the killed generator, matched asymptotic analysis of the associated second-order recurrence, and discrete Laplace estimates for reversible weights.

We generalise a mixing result due to Pommerenke for non-autonomous dynamical systems of forward compositions of the boundary values of centred inner functions; the proof depends on a precise bound on the orbits of such forward compositions in the unit disc. This enables us to give new criteria for such systems to have positive metric entropy, lower entropy and topological entropy. In the opposite direction, we construct a forward composition of centred degree two Blaschke products with topological entropy zero.

This proof-of-concept paper demonstrates how the Kalman filter, as a brain imaging method, can be enhanced by incorporating raw diffusion tensor imaging (DTI) data without tractography, in addition to EEG recordings. An efficient algorithm is designed to streamline the otherwise tedious process of inferring connectivity between brain regions from DTI data. The DTI-based evolution model is applied to the Kalman filter and the Standardized Kalman filter, and these are compared with their conventional counterparts using the random walk evolution model. The numerical experiments use synthetic somatosensory and auditory evoked potentials. The results show that the DTI-modelled Kalman filter can obtain deep activity in the brainstem and thalamus. Moreover, the connection model limits the estimated spreads to the correct or nearest anatomical brain region.

We classify common invariant probabilities for two beta-transformations with a rational affine relation between their bases. Let $\beta>1$ be irrational and $\gamma=a\beta+c>1$, where $a,c\in\mathbb Q$ and $a+c\ne1$. Every finite nonnegative measure that is singular with respect to Lebesgue measure and invariant under both maps is a multiple of the point mass at zero. No entropy, ergodicity or multiplicative independence assumption is needed. The proof constructs a finite signed measure invariant under an irrational rotation and relates its mass to the first moment of the original measure. Combining this result with the known classification of coincident absolutely continuous invariant probabilities gives all common invariant probabilities in this family. The only additional measures are mixtures with a common absolutely continuous probability for adjacent bases in a known quadratic family. We also determine all common fixed points on the excluded line $a+c=1$.

We present a systematic approach to characterize Devaney chaos in the solution of homogeneous one dimensional constant-coefficients PDEs on Herzog spaces. We show that for equations where the highest-order time derivative lacks mixed spatial derivatives, there always exists a weighted space in which the solution is chaotic. This analysis is extended to include mixed partial derivatives in the highest-order temporal term. Moreover, we link the chaotic behavior in Herzog spaces to the semidiscretization of the equations by identifying isometric isomorphisms between the two contexts, providing also a characterization for chaos on these semidiscretizations. The work concludes with two applications that provide new insights on the dynamics of the fourth-order Moore-Gibson-Thompson equation and the viscous van Wijngaarden-Eringen equation.

This work addresses the computation of extreme values of observables along trajectories of dynamical systems over an \ks{infinite time horizon}. The setting is intrinsically global: trajectories may exhibit large transient excursions, approach their largest values only asymptotically as time tends to infinity, or escape to infinity in finite time. The proposed framework treats these \mk{ scenarios} in a unified way, so that the resulting extreme values reflect both the transient and the asymptotic behavior of the dynamics. The main difficulty is that neither the time interval nor the state space is assumed to be compact. We address this by combining a radial compactification of the state space with a compactified clock variable and discounted occupation measures. This yields a description of trajectory ensembles on compact sets via occupation measures and the associated Liouville's equation. Extreme values are then characterized through a family of reachability problems for level sets of the observable, which can be solved by a bisection procedure. On the theoretical side, we prove a superposition principle for an infinite-horizon Liouville's equation with a time-distributed terminal measure. This result shows that the measure formulation represents ensembles of stopped admissible trajectories, including trajectories \ks{that never stop}, and establishes equivalence between the trajectory-level and measure-level descriptions.

We study third-order linear differential equations through their associated Pfaffian and projective foliations. In the regular-singular setting, we introduce a Fuchs-adapted Pfaffian model in which the indicial polynomial appears directly in the special-fiber singular scheme, and Frobenius resonance acquires a projective geometric interpretation.

Extreme events in chaotic systems are difficult to learn from short trajectories because they are controlled by transient finite-time instability rather than by frequently observed bulk dynamics. We propose a mechanism-aware conditioning plug-in framework that turns a nudged coarse ensemble into a non-intrusive sensor of local instability geometry. In the small-noise regime, the ensemble covariance aggregates the same finite-time deformation kernels that govern local instability, providing a Jacobian-free proxy for the local amplification structure around a synchronized coarse trajectory. A small FiLM module injects statistics of this ensemble geometry into an otherwise unchanged backbone while leaving the coarse simulator unchanged. We demonstrate this interface in two distinct pipelines: a Transformer-style residual-attention corrector for a controlled low-dimensional chaotic system and a probabilistic recurrent STORN corrector for topographic two-layer quasi-geostrophic (QG) flow. In the low-dimensional benchmark, ensemble covariance directions co-activate with OTD modes and FiLM conditioning improves 99th-percentile exceedance-frequency errors over an identical no-context Transformer baseline. In QG, a fixed ensemble-conditioned FiLM-STORN model trained on only \(50\) time units substantially improves long-horizon rare-event statistics in the data-limited regime, including density-tail errors, exceedance frequencies, and spatial exceedance-area distributions relative to an unconditioned STORN trained on the same data; on averaged high-threshold exceedance diagnostics, it also outperforms the baseline STORN trained with $20$ times more high-resolution data. These results show that local instability geometry is not merely interpretable post hoc, but an actionable conditioning signal for data-efficient rare-event emulation.

For any $t\in\mathbb{R}$, consider the Affine twist map $\rm{Aff_t}:\mathbb{T}^2\to\mathbb{T}^2$ given by $$\rm{Aff_t}(x,y)=(x+y \text{ mod 1}, y+t \text{ mod 1}).$$ The map $\rm{Aff_t}$ clearly possesses an invariant foliation by horizontal curves. If $t$ is rational, each leaf is periodic, and if $t$ is irrational, the orbit of each leaf is dense in the torus. In both cases, the vertical rotation set of the proper lift of $\rm{Aff_t}$ to the vertical cylinder is reduced to $\{t\}.$ Now, for $k\in\mathbb{R},$ define $$f_{k,t}(x,y):=(x+y+k\sin (2\pi x)\text{ mod 1},y+k\sin (2\pi x)+t\text{ mod 1}).$$ We show that: 1) when $t=p/q$ for integers $p$ and $q>0$, KAM theory implies the existence of a constant $k_{p/q}>0$ such that for $|k|<k_{p/q}$, the vertical rotation set of an adequate lift of $f_{k,p/q}$ is just $\{p/q\}.$ 2) when $t$ is irrational, for any $k\neq 0$ the vertical rotation set of the adequate lift of $f_{k,t}$ is a non-degenerate interval which contains $t$ in its interior. In other words, it is not easy to build area-preserving twist maps whose vertical rotation sets are reduced to a single irrational number. From Theorem A of \cite{eujul}, such a map needs to have an invariant foliation by Lipschitz graphs over the horizontal coordinate. In particular, all its iterates must satisfy a twist condition. This is precisely what does not hold for $f_{k,t}$, for all non-zero values of $k$.

Influenza A persists by continually changing the antigens that its host population recognizes. How fast it evolves and how many hosts it infects are not independent, because the immunity that selects new variants is produced by the infections the pathogen has already caused. We analyze a model of Andreasen et.\ al.\ that captures this eco-evolutionary feedback, and we extend it to the case where infection may kill the host. In the model infected and recovered hosts are distributions over a one-dimensional antigenic space coupled by cross-immunity. The endemic state is a traveling pulse: the infected distribution keeps a fixed shape and advances through antigenic space at constant speed. In the limit of weak antigenic mutation we construct the pulse asymptotically and find that: (i) the speed of antigenic evolution decreases as prevalence rises, (ii) less standing antigenic variation accompanies either faster or slower antigenic evolution, depending on whether pathogens differ in immune evasion or host lifespan (iii) the total number of infected hosts is largest at an intermediate virulence when the host birth rate is sufficiently low.

We study the moiré pattern formed by the intersection of a planar Sierpiński gasket and a copy rotated about its centre. At every resonant angle, we prove that the intersection is of finite type and compute its Hausdorff and Minkowski dimensions from an explicit finite matrix; for Lebesgue-almost every angle, we bound its upper Minkowski dimension by $\overline{\dim}_{\mathrm B}(\mathcal S_\theta)\le 2\dim_{\mathrm H}(S)-2$. Finally, we conjecture that both dimensions equal this bound at every non-resonant angle. Sections 2 and 3 are formalised in Lean.

The Anderson tight-binding model is a fundamental model of quantum transport and localization in disordered media. Completing a problem left open by Bourgain and Kenig, this paper proves Anderson localization near the bottom of the spectrum for the lattice Anderson model with Bernoulli potential, in any dimension $d\ge 2$. The proof uses the multiscale framework of Fröhlich-Spencer and Bourgain-Kenig, and the main new ingredient is a probabilistic discrete unique continuation principle (PDUC) for the discrete Schrödinger equation. This PDUC is established via a bootstrap argument and a key probabilistic lemma proved by adaptively revealing the random potential.

Let $f$ be a holomorphic automorphism of a compact Kähler manifold $X$ with simple cohomological action. We prove that the saddle periodic points of $f$ equidistribute with respect to the equilibrium measure. In particular, our result applies to every holomorphic automorphism of a compact Kähler surface with positive topological entropy.

This paper establishes that transcendental entire functions do not have wandering Fatou components whose forward orbit is bounded. This settles a long-standing question in transcendental dynamics. A stronger statement is proved using the same method: the orbit of every point in a wandering domain of a transcendental meromorphic function is unbounded. In particular, transcendental meromorphic functions do not admit orbitally bounded wandering domains. The proofs rely mostly on uniform estimates for hyperbolic area, following the approach of Ye [arXiv:2609.23834v1].

We use hyperbolic area and covering geometry to study wandering Fatou components and singular orbits. We give proofs without quasiconformal deformation of the rational no-wandering theorem and the known no-wandering results in Bergweiler's Question~9. The framework also excludes wandering for entire functions with compact singular sets whose derived sets lie in the Fatou set. For Bergweiler's Question~8, every wandering component of a transcendental meromorphic map has a subsequence of iterates converging locally uniformly to infinity. For Bergweiler's Question~4, we establish hyperbolic separation near singular-free Baker cycles after deleting any finite set of original singular values as orbit generators. For Bergweiler's Question~11, corresponding finite-deletion boundary estimates hold along wandering tails. In higher dimension, a relative-area theorem excludes bounded wandering components of polynomial skew products over bounded bases carrying an absolutely continuous invariant probability of full support. Applied to elliptic polynomial skew products over Siegel rotation domains, it yields a local no-wandering theorem without conditions on the fiber critical orbits or continuity of fiber Julia sets. A global criterion covers regular skew products on $\mathbb P^2$ whose bounded base Fatou components eventually enter periodic Siegel disks, including $(\lambda z+z^d,q(z,w))$ for Brjuno $\lambda$ and polynomial $q$ of total degree at most $d$ with a nonzero $w^d$ coefficient.

The degrees of the iterates of a projective endomorphism grow in two layers: an exponential rate measured by the dynamical degrees, and a polynomial correction carried by the peripheral Jordan blocks. Log-concavity constrains the first layer; we show that the Hodge index theorem already constrains the second, in every codimension and in arbitrary characteristic. Let $f$ be a surjective endomorphism of a normal projective $d$-fold over an algebraically closed field, with dynamical degrees $\lambda_i$, so that $\textrm{deg}_i(f^n)\asymp\lambda_i^n n^{\mu_i}$ for a unique integer $\mu_i\ge0$. We prove that these exponents are coupled across three adjacent codimensions: strict log-concavity of $(\lambda_i)_i$ at $i$ forces $\mu_i=0$, while equality gives \[ 0\ \le\ 2\mu_i-\mu_{i-1}-\mu_{i+1}\ \le\ 4 . \] Consequently, $\mu_i\le2i(d-i)$, and this is sharp for every $i$. For a zero-entropy automorphism, we prove that every mixed orbit function indexed by any weak composition $\gamma$ of $d$ is a multivariate quasipolynomial of even total degree at most $d^2-\sum_j\gamma_j^2$. For a zero-entropy holomorphic automorphism $g$ of a compact Kähler $d$-fold $Y$, the same method gives \[\left\|(g^n)^*|_{H^{p,q}(Y,\mathbb{C})}\right\| =O\bigl(n^{p(d-p)+q(d-q)}\bigr).\] Powers of elliptic curves attain all exponent bounds as well as the constant $4$.

A fixed coarse-grained description need not uniquely determine the underlying microscopic rules. Variations in these rules can alter collective behaviour while leaving the retained observables unchanged. Here we develop a method to identify and use such variations for collective-state design. We construct families of microscopic models with exactly matched coarse observables and use the response derivative within each family to identify independent first-order output changes. In a nonreciprocal active mixture, the largest first-order change in interfacial growth rate under a pointwise kernel budget also determines the leading minimax error of predictions based only on the matched data. Near a Hopf bifurcation, this response map gives an initial turning-kernel design, which is refined by jointly solving for the periodic kinetic state and the kernel. Fixed-kernel integrations realize prescribed oscillation powers and frequencies while preserving the selected angular relaxation rates. In a reaction-diffusion family, cubic amplitude equations guide the design of stripe and hexagon amplitudes while the full linear operator remains fixed. Additional constraints reduce the available response even when its rank is unchanged, whereas the matched angular moments retain their exact relaxation rates. These results provide a constructive route to collective-state design within a prescribed coarse-grained description.

In this article, we consider a wave equation on a bounded three-dimensional domain, featuring a degenerate nonlocal damping mechanism that depends on the system's energy and a source term with subquintic growth. This class of models is motivated by applications in optical physics. We establish global existence in the Shatah–Strichartz sense by combining Galerkin approximations with space-time Strichartz estimates on bounded domains. A central difficulty in the long-time analysis is the interplay between three-dimensional wave dynamics, nonlocal damping, and forcing terms that are not locally Lipschitz continuous. We prove the existence of a compact global attractor and derive an upper bound for its Kolmogorov $\varepsilon$-entropy. In the non-degenerate damping case, assuming that the domain has a smooth, strictly convex boundary, we exploit the refined Strichartz estimates of Ivanovici et al.~\cite{Ivanovici-Lascar-Lebeau-Planchon-2023}, which yield, in particular, $u\in L^3_tL^{18}_x$. Under the further assumption $f\in C^2(\mathbb R)$, this improved integrability allows us to establish quasi-stability of the associated dynamical system. Consequently, the global attractor has finite fractal dimension and higher regularity, and a generalized exponential attractor exists.

The Poincaré recurrence theorem for a holomorphic correspondence provides a natural framework for studying the first return time and a dynamical system thus induced, in this setting. After establishing a version of the Kac's theorem on the average of the first return times for a holomorphic correspondence, we introduce the concept of an induced correspondence, akin to the case of dynamics of maps and prove that it inherits the ergodicity of the holomorphic correspondence. Additionally, we characterise the ergodic measures with respect to the considered holomorphic correspondence, establish the density of forward orbits under mild conditions and prove a Rokhlin-Kakutani type lemma.

Let $A\subset\mathbb{R}^d$, $d\geq3$, be the attractor of a self-similar IFS whose rotations generate a dense subgroup of $\mathrm{SO}(d)$. For every $1\leq k<\dim_{\mathrm{H}} A$ and $V\in\mathrm{Gr}(d,k)$, we prove that $\pi_VA$ has positive $k$-dimensional Lebesgue measure uniformly in $V$, and $\dim_{\mathrm{H}}(A\cap\pi_V^{-1}\{y\})=\dim_{\mathrm{H}} A-k$ for Lebesgue-almost every $y\in\pi_VA$. Moreover, if $\mu$ is a self-similar measure for such an IFS with $I_t(\mu)<\infty$ for some $k<t<d$, then every $\mu_V=(\pi_V)_*\mu$ is equivalent to the $k$-dimensional Lebesgue measure restricted to $\pi_VA$, with density $f_V$ satisfying $\sup_V\int_{\{f_V>M\}}f_V\,d\mathcal{L}_V^k\lesssim e^{-c(\log M)^{1/3}}$. Under strong separation, the conditional measures on the fibres are exact dimensional of dimension $\dim_{\mathrm{H}}\mu-k$. The proof uses self-similarity to iterate $L^1$ norms of differences of the Gaussian-smoothing of $\mu_V$ between scales. Unlike $L^q$, $q>1$, the $L^1$ norm does not grow under rescaling, allowing Varjú's $L^2$ weak gap theorem under dense support condition to be used, and avoiding to use the much stronger uniform spectral gap assumption.

Nonlinear contacts and friction strongly influence the vibration response of assembled structures, but their accurate numerical treatment is computationally demanding. The harmonic balance method is widely used to compute periodic steady-state responses, yet the required alternating frequency-time scheme becomes costly for nonsmooth and hysteretic nonlinearities and must be repeated throughout the nonlinear solution process. Here we show that this procedure can be replaced by neural networks that directly map displacement Fourier coefficients to nonlinear force coefficients and provide the corresponding Jacobian through automatic differentiation. The surrounding solver and continuation algorithms remain unchanged for the computation of frequency response curves. The neural networks exclusively learn individual nonlinear elements rather than complete system responses. Physics-based nondimensionalization and phase normalization facilitate the learning process and enable a single trained network to cover a wide range of parameter combinations. Building on the cubic spring, unilateral spring, and Jenkins elements considered here, the approach points toward a reusable library of nonlinear-element surrogates that can be combined in arbitrary number and location within a mechanical system. By bypassing the iterative force evaluation in time domain, the method offers favorable computational scaling for high-resolution analyses and systems with many nonlinear elements.

Let $(M,g)$ be a closed negatively curved surface. If $g$ is strictly $\tfrac 19$-pinched and has the same unmarked length spectrum as a hyperbolic metric, we show that $g$ is hyperbolic. As a consequence, we show that any hyperbolic metric on a surface admits a $C^2$-neighborhood in the space of metrics in which it is characterized by its unmarked length spectrum, up to isometry.