Papers

This paper studies necessary optimality conditions for stochastic control problems governed by decoupled forward–backward stochastic differential equations with reflection constraints on the backward component. The cost functional is defined through the initial value of a reflected BSDE whose lower obstacle depends on the forward state. The nonsmooth nature of the reflection term prevents the direct application of classical variational methods. Using a penalization approach, spike variations, and duality arguments, we derive a penalized variational relation. A key feature of our method is to isolate the reflection contribution into a residual term, avoiding the differentiation of the nonsmooth penalization operator while preserving a Hamiltonian adjoint structure analogous to the classical stochastic maximum principle. Passing to the limit yields a reflected variational inequality of Pontryagin type, where the obstacle effect is explicitly represented through a residual contribution associated with the reflection term.

Let $k\ge2$ be a fixed integer. Bollobás and Scott proved that every $3$-uniform hypergraph with $m$ edges admits a partition of its vertex set into $k$ parts such that each part spans at most $m/k^3+O_k(m^{6/7})$ edges. Scott later suggested that the error term should be $O_k(m^{2/3})$. In this paper, we show that every $3$-uniform hypergraph with $m$ edges admits a partition into $k$ parts such that each part spans at most $m/k^3+O_k(m^{2/3})$ edges, thereby establishing the proposed error term.

Part I: Semiclassical Gravity Efficiently Solves $\mathsf{NP}$-Complete Problems Assuming the gravitational field is classical and that it couples to quantum fields via the semiclassical Einstein field equations (EFEs), we show that the weak-field dynamics of a massive and non-relativistic qubit can in principle be used to solve an $\mathsf{NP}$-complete problem in polynomial time. We attribute this vast computational power to the non-linear qubit dynamics afforded by the semiclassical EFEs. Consequently, the above two assumptions entail a violation of the Physical Extended Church–Turing Thesis, which we regard as evidence for the quantization of gravity. Part II: Spacetime Quasicrystals We generalize self-similar quasicrystals, such as the Penrose tilings, from Euclidean space to Minkowski spacetime. We construct the first examples of Lorentzian quasicrystals and identify their key novel features. We then explore possible connections to fundamental physics. First, we propose a speculative scenario in which our $(3+1)$D universe is embedded in a particularly symmetric $(9+1)$D torus $\mathbb{T}^{9,1}$, previously identified as the most symmetric toroidal compactification of the superstring. We suggest that this construction may shed light on the seesaw relation $M_{\rm Pl}M_{\rm vac} \approx M_{\rm EW}^{2}$ between the Planck, vacuum-energy, and electroweak scales. Second, we replace the random causal sets of causal set theory with ordered, highly symmetric quasi-crystalline causal sets. We find suggestive evidence that our quasicrystals outperform Poisson random sets in the number-volume correspondence in both $(1+1)$D and $(3+1)$D, where such behaviour has been conjectured to be impossible. Finally, extending hyperuniformity from space to spacetime, we show that our $(1+1)$D quasicrystal is the first demonstrated example of a spacetime-hyperuniform structure.

Following Macintyre's quantifier elimination result for semi-algebraic $p$-adic sets from 1976, Denef's cell decomposition from 1984 has played a catalyzing role for both motivic integration and Hensel minimality. Motivic integration has in turn been applied in the Langlands program via transfer principles to change the characteristic of the local field, e.g., for the fundamental lemma. Hensel minimality has been used to study rational points on definable sets, providing non-archimedean analogues of results by Pila and Wilkie in o-minimal structures. I will review some related results and open questions.

In this paper, we study observable sets for the one-dimensional Schrödinger equation associated with the operator $H=-d^2/dx^2+|x|$ on $L^2(\mathbb R)$. We prove that a measurable set is observable at every positive time if and only if it is thick. For each fixed observation time, the observability constant is uniform over all measurable sets with prescribed thickness parameters. The theorem answers the question concerning the relationship between thick and observable set raised by Gengsheng Wang [12] for this anharmonic oscillator.

We study $(-1)$-homogeneous solutions of the steady Navier-Stokes equations in $\mathbb R^n\setminus\{0\}$ with $(-3)$-homogeneous, locally Lipschitz forces. Previous work [2] established the existence of $(-1)$-homogeneous solutions without any smallness assumption in dimensions $4\leq n\leq16$. In this paper, we remove the dimension restriction and obtain the existence of self-similar solutions in all higher dimensions. The new ingredient is a dimension-independent a priori estimate for the total head pressure on the sphere, obtained by adapting the source-flux method and the weighted identity of the pressure of Wang and Yang [28] to the homogeneous setting. This combines with the structural relation between the radial velocity and the total head pressure used in [2] to close the a priori estimates for the solutions in all dimensions.

We prove invariance of the geometric Arveson-Douglas conjecture under Veronese embeddings of projective varieties. This provides new examples of singular varieties satisfying the conjecture. Furthermore, it reduces the conjecture to projective varieties whose defining ideals admit quadratic Gröbner bases, by keeping the same Schatten regularity thresholds. Our results lead to canonical examples of subproduct-systems whose Toeplitz algebras are not $KK$-equivalent to the complex numbers. Finally, we show that Douglas' fundamental class conjecture, asserting that the associated Toeplitz extension implements $\mathbb{T}$-equivariant duality in $KK$-theory, is invariant under Veronese embeddings as well. While the latter result bootstraps the fundamental class conjecture, it also leads to infinitely many counterexamples.

We prove the Hassanzadeh-Nasrollah Nejad-Simis conjecture: if $f\in (x_1,\ldots,x_n)^2\subsetneq k[[x_1,\ldots,x_n]]$ has an isolated critical point and $\operatorname{char}k=0$, then its Euler conductor $J_f:f$ is not contained in the Tjurina ideal $(J_f,f)$. More generally, if $A=R/I$ is a nonzero Noetherian $k$-algebra and $a=[f]\in A$ is nilpotent with $d_{A/k}a=0$, then $I:f\nsubseteq(I,f)$. For a local ring $(R,\mathfrak n)$ we obtain the stronger noncontainment $I:f\nsubseteq(I,f)+\mathfrak n(I:f)$. The proof reduces a hypothetical containment to a self-exact square-zero element and detects its differential by the trace of a regular representation over the dual numbers. Finally, an explicit five-variable isolated singularity satisfies $J_f:f\subseteq\overline{J_f}$, disproving the integral-closure strengthening proposed by Ma and Zuo.

A collection of $m \geq 2$ lines through the origin in $\mathbb{R}^2$ is balanced if the sum of the orthogonal projections onto these lines equals $m/2$ times the identity. A line arrangement in $\mathbb{RP}^2$, endowed with the round metric of curvature $1$, is point-balanced if the tangent lines at every vertex form a balanced collection. A line arrangement $\mathcal{A}$ in $\mathbb{RP}^2$ has the Hirzebruch property if it consists of $3k$ lines and every line contains exactly $k+1$ vertices. Using a Kempf–Ness convexity argument, we show that an irreducible arrangement $\mathcal{A}$ has the Hirzebruch property if and only if its projective equivalence class contains a point-balanced representative, unique up to orthogonal transformations. We then combine elementary properties of balanced collections of lines in $\mathbb{R}^2$ with spherical geometry to prove that every irreducible point-balanced arrangement is a reflection arrangement. This gives a new proof of Panov's classification of real Hirzebruch arrangements.

We develop the method of central extensions (the Skjelbred–Sund method) for nilpotent right alternative algebras over the field of complex numbers and apply it to obtain the algebraic classification of five-dimensional nilpotent right alternative algebras. More precisely, we classify, up to isomorphism, all complex five-dimensional nilpotent right alternative algebras that have no annihilator component (that is, which are not a direct sum of a smaller algebra and a one-dimensional algebra with zero product) and which are not $2$-step nilpotent. Every such algebra is a non-split central extension of a nontrivial nilpotent right alternative algebra of dimension three (by a two-dimensional space) or of dimension four (by a one-dimensional space). For each of the relevant three- and four-dimensional algebras we compute the second cohomology space, the automorphism group and its action on the second cohomology, and we determine all orbits that give non-split extensions. The resulting list consists of $124$ algebras and families of algebras; it contains $29$ algebras with two-dimensional annihilator and $95$ algebras with one-dimensional annihilator.

We develop a non-asymptotic theory for nonparametric drift estimation in discretely observed multi-dimensional Lévy-driven stochastic differential equations using sparse ReLU neural networks. Under exponential \(\beta\)-mixing and an exponential-tail condition on the Lévy measure, we establish an oracle inequality for the least-squares estimator. The jump component changes both the martingale structure and the concentration regime relative to diffusion models: continuous martingales are replaced by discontinuous martingales, while the relevant empirical-process fluctuations are sub-exponential rather than sub-Gaussian. We handle these difficulties without truncating the observed increments, using an exponential-supermartingale argument for compensated Poisson integrals and \(\psi_1\)-chaining. Despite the weaker concentration, the resulting statistical bound is, up to constants, of the same order as the corresponding diffusion bound. For drift functions with hierarchical compositional structure, this yields intrinsic-dimensional convergence rates. The framework allows infinite jump activity and, in some cases, infinite variation. Finally, we establish a minimax lower bound on a fixed non-trivial compound-Poisson submodel. Together with the upper bound, this shows that the estimator is minimax optimal up to logarithmic factors.

We introduce a secondary index map from relative K-homology to the K-theory of a relative structure algebra. We prove an algebraic Mayer-Vietoris sequence for the relative structure algebra, and prove compatibility of the secondary index map with the Mayer-Vietoris boundary map. As an application, we compute the secondary index of the Dirac operator on an even-dimensional spin manifold with collared metric having uniformly positive scalar curvature on the boundary. This computation leads to a refinement of the delocalized APS-index theorem.

We consider the interaction of a viscous incompressible fluid with a flexible elastic shell occupying the boundary of the fluid's domain in two space dimensions. The shell is modelled by a perfectly elastic linearised beam equation. It is an open problem in fluid-structure interaction, even for the idealised case of a perfectly elastic plate, if a self-intersection of the structure can be excluded in the long run. We show that by adding a physically well-motived and sufficiently strong transport noise in the structure equation, the self-intersection does not occur before any given end-time with any prescribed probability smaller than 1.

In this paper we are going to discuss compactness in combinatorial Banach spaces. After discussing the definition and structure of these spaces, we proceed to propose compactness criteria for them and for their extensions. Finally, we demonstrate their connection with theory of hypergraphs.

In this paper, we prove that every separable unital AT algebra of real rank zero has complexity rank at most one. In particular, every irrational rotation algebra has complexity rank one. This answers a question of Jaime and Willett.

In the framework of multiple orthogonal polynomials (MOPs) and their extension to the multivariate setting, a unified mechanism for generating structured families of multiple orthogonal polynomials across diverse multidimensional domains is still lacking. In this work, we bridge this gap by developing an extended Koornwinder-type methodology for constructing multivariate multiple orthogonal polynomials. We introduce two distinct coupling schemes: combining a univariate MOPs family with a standard orthogonal family, which enables the definition of both Type I and Type II bivariate systems along with their dual biorthogonality relations, and coupling two univariate MOPs families to form Type II multivariate systems. Using this general framework, we provide the first explicit formulations of multiple orthogonal polynomials on a variety of bivariate regions, including bounded domains such as a parabolic domain and the square $[0,1]^2$, non-standard unbounded geometries like the positive quadrant $(\mathbb{R}_0^+)^2$ and the wedge $\mathbb{V}^2$, as well as new constructions on the triangle $T$. Furthermore, explicit multiple orthogonal systems on the $d$-dimensional simplex $T^d$ and on the $d$-dimensional cone $\mathbb{V}^d$ are given.

We present a new class of structure-preserving, arbitrarily high-order accurate space-time schemes using unstructured simplicial meshes in space for a class of nonlinear acoustic wave propagation systems, in which the nonlinearity is encoded in the constitutive relations between the state and dual variables, while the differential operators retain the standard first-order acoustic wave structure. At the fully discrete level, the schemes satisfy the following properties: I) local conservation of the state variables; II) a local energy/entropy balance on each space-time element, with a consistent, single-valued numerical energy/entropy flux and nonnegative local energy/entropy dissipation, and exact energy conservation when the stabilization terms are omitted; III) weak preservation of the curl involution of the velocity field, even in the presence of discontinuities and with the shock-capturing terms activated. The key ingredients of the new schemes are: i) a discontinuous Galerkin (DG) discretization in space, in which the dual variables are computed as the L2 projections onto the DG space of the energy derivatives with respect to the conserved variables; ii) numerical fluxes and a bubble-weighted, element-local viscous regularization expressed in terms of the dual variables; iii) a continuous-discontinuous Petrov-Galerkin method in time. Properties I)-III) are first proved mathematically and are then verified by suitable numerical tests.

The main result of the paper states that if $0\neq v\in\mathbb C^m\otimes\mathbb C^n$, $d=\min\{m,n\}$, and $p=SR(v)$, then for every $1\le q\le d$, \begin{equation*} \min\left\{OSR(\mathbf U): \begin{array}{l} \mathbf U\in\mathbb M_m\otimes\mathbb M_n\text{ is unitary}, SR(\mathbf Uv)=q \end{array}\right\} = \left\lceil\max\left\{\frac pq,\frac qp\right\}\right\rceil . \end{equation*} Here $OSR(\mathbf U)$ denotes the operator Schmidt rank of $\mathbf U$. Furthermore, if $\mathbf U\in\mathbb M_m\otimes\mathbb M_n$ is unitary and $OSR(\mathbf U)\le k$, then, for every unit vector $x\in\mathbb C^m\otimes\mathbb C^n$ and every $1\le r\le d$, \begin{equation*} F_{\min\{kr,d\}}(\mathbf Ux)\ge F_r(x) \quad\text{and}\quad F_{\min\{kr,d\}}(x)\ge F_r(\mathbf Ux), \end{equation*} where $F_r(x)$ denotes the sum of the $r$ largest squared Schmidt coefficients of $x$. This result is stronger than rank reachability as it controls the entire cumulative Schmidt profile. We also establish a majorization relation and use it to obtain sharp bounds on Schatten-norm distortion and on the change of Rényi entropies under unitaries with bounded operator Schmidt rank.

For a positive real lacunary sequence and any probability measure $\mu$ with polynomial Fourier decay, we determine, for $\mu$-almost every $x$, the asymptotic maximal volume of convex bodies avoiding the first $N$ dilates of $x$ modulo $\mathbb Z^d$. Among homothets of a fixed convex body, the maximal volume is asymptotic to $(\log N)/N$. When all convex bodies in the unit cube are allowed, it is asymptotic to $d(\log N)/N$. For independent uniform points in a fixed convex body $\Omega\subset\mathbb R^d$, the maximal volume of an empty convex body is asymptotic to $d\operatorname{vol}(\Omega)(\log N)/N$ almost surely. We also show, by constructing a one-dimensional counterexample, that lacunarity cannot be replaced by sparsity on sublacunary scales.

As in all fields of engineering, control engineers constantly face a dichotomy between achieving high performance and the limited availability of resources for sensing, actuation, and control. This tension often translates into formalizing design problems with sparsity constraints; that is, the design involves decision variables whose number, spatial reach, temporal spread, or other physical expressions are much smaller than in the ideal scenario of unconstrained optimization of a task-related performance metric. Even though this fundamental limitation underlines a plethora of methodological approaches and application domains, the control literature is fragmented into several lines of work tailored to specific scenarios. This survey fills the gap by bringing together several lines of work on handling sparsity in relevant control problems such as actuator scheduling, minimal-energy controllability, maximum hands-off control, and online convex optimization, and weaving a red thread through theoretical foundations and algorithms. Beyond establishing a coherent, structured narrative of problems and solutions for handling sparsity in control scenarios, we discuss open areas and relevant directions for future research, motivated by the growing complexity of control systems and their relations with key domains such as pervasive computing, distributed learning, cyberphysical security, and ecological footprint.

Mourad E.H. Ismail and Kenneth S. Miller in 1982 conjectured that for $\nu>\mu\geq0$ and $b>a>0$ the function $$x\mapsto G(x;\mu,\nu)=\left(\frac ba\right)^{\mu-\nu}\frac{I_\mu(a\sqrt{x})I_\nu(b\sqrt{x})}{I_\mu(b\sqrt{x})I_\nu(a\sqrt{x})},$$ where $I_\nu$ stands for the modified Bessel function of the first kind, is the Laplace transform of an infinitely divisible probability distribution. Their conjecture is equivalent to the monotonicity property of some exponential sums over the zeros $j_{\nu,n}$ with respect to the order, where $j_{\nu,n}$ denotes the $n$th positive zero of the Bessel function of the first kind of order $\nu$. In this paper our aim is to reduce the problem further to the monotonicity, with respect to $\nu$, of \[Q_\nu(t)=\sum_{n\geq1}j_{\nu,n}^2e^{-t j_{\nu,n}^2},\qquad t>0,\] and to prove this monotonicity. In this way we show that indeed the above function $x\mapsto G(x;\mu,\nu)$ is the Laplace transform of an infinitely divisible probability distribution, and the above conjecture is true. The key ingredient in our proof is an integral representation for the derivative of a quotient of modified Bessel functions of the first kind with respect to the order, combined with Bernstein's theorem and a probability tail argument.

For any non-zero periodic function f\,: Z $\rightarrow$ C of period N ___ 1 and any non-zero algebraic number z\_0 that doesn't lie on a half-line [e^2i${\ell}$$\pi$/N , e^2i${\ell}$$\pi$/N $\infty$[, 0 ___ ${\ell}$ ___ N -1, we give a lower bound of order sqrt{s/log(s)} on the dimension of the Q(z\_0 )-vector space spanned by the numbers L(f, i, z\_0 )\,: sum\_{m=1}^$\infty$ f(m)z\_0^m/m^i. This generalizes a result Fischler proved in 2026, corresponding to f identically equal to 1 and |z\_0| ___ 1: in this case, the numbers L(f, i, z\_0 ) = Li\_i (z\_0 ) are polylogarithm values. Except for a finite number of cuts in the complex plane, our result still holds when |z\_0 | > 1, that is in a domain where the series definition above for the numbers L(f, i, z\_0 ) doesn't converge anymore, and we make sense of these numbers through analytic continuation. To obtain this result, we construct linear combinations of the numbers L(f, i, z\_0 ) using a refined version of Siegel's lemma, and we apply to them a linear independence criterion generalizing the one used by Fischler. To check the assumptions of this criterion, we rely on an integral representation of the polylogarithm functions and on a ”Shidlovskii lemma”.

We prove that every complete, connected, noncompact Riemannian three-manifold without boundary with scalar curvature at least $6$ has conjugate radius at most $2\pi/3$. This resolves Shen's conjecture on the strict conjugate-radius bound $\operatorname{conj}(M,g)<\pi$ in dimension three.

We develop a sparse domination theory that captures both local polynomial approximation and cancellation of the input below the scale of each sparse cube. Its coefficients are percentiles of localized smooth maximal functions of the input minus its local polynomial approximation, combining features of mean-oscillation sparse bounds and of the cancellative sparse bounds of Conde-Alonso, Lorist and Rey. Using wavelet representations, we characterize the Calderón-Zygmund operators whose derivatives admit these bounds by explicit polynomial testing conditions. In the homogeneous setting, the same conditions characterize weighted Hardy-Sobolev inequalities with linear dependence on the $A_\infty$ characteristic. This quantitative dependence detects the transition from BMO to $L^\infty$ testing conditions that Lerner found for pointwise bounds by Riesz potentials, a distinction that qualitative weighted estimates cannot recover. In the inhomogeneous setting, we prove corresponding weighted inequalities and show that they are strictly weaker than pointwise sparse domination from the second derivative order onward. Consequences include Hardy-Sobolev estimates below the Banach range, sharp weighted Sobolev inequalities, and pointwise bounds by Riesz potentials.

Mean field limits describe the training dynamics of wide neural networks through the evolution of the empirical distribution of their parameters. Although functional central limit theorems characterize the asymptotic fluctuations of this distribution, quantities of practical interest are typically nonlinear observables of the parameter distribution rather than the distribution itself. In this work, we show how these mean field fluctuations propagate to finite dimensional nonlinear observables for shallow neural networks trained by stochastic gradient descent. Working in the weighted Sobolev space in which the limiting fluctuation process is constructed, we apply a functional Delta method under ordinary Fréchet differentiability, without requiring Lions derivatives with respect to the measure variable. We obtain a central limit theorem for the observables and, under a suitable representation of their differentials, an explicit covariance formula inherited from the underlying mean field fluctuation theory. We also study whether prescribed quantities of interest can be recovered from the selected observations. Under a constant rank assumption, we prove that a quantity of interest factors locally through the observation functional if and only if, throughout a neighborhood, the kernel of the differential of the observation is contained in that of the quantity of interest. Thus, a differential condition expressed directly in the ambient Sobolev space yields an exact nonlinear local factorization. These results provide a framework both for quantifying finite-width uncertainty on observable, statistically or physically meaningful quantities and for assessing whether the chosen observations contain the information required to identify them.

This paper derives an equivalent formulation of the Costabel FEM-BEM coupling for the time-harmonic Maxwell transmission problem with a Lipschitz interface, and constructs a block-diagonal preconditioner for the resulting discrete system. The reformulated coupling is posed on a product of trace spaces and is obtained by introducing a trace lifting operator and the Steklov–Poincaré operator. At the algebraic level, it corresponds to eliminating the interior FEM unknowns, resulting in a discrete system that involves the Schur complement of the interior FEM block. Assuming that the frequency lies outside the resonance sets, we prove that the reformulated coupling is equivalent to the Costabel coupling at both the continuous and discrete levels. Using the operator preconditioning framework, we build a preconditioner from the Galerkin matrix induced by the Maxwell single layer boundary integral operator. The spectral condition number of the preconditioned matrix is shown to be uniformly bounded. Numerical experiments illustrate the practical effectiveness of the proposed approach.

We construct Dirac-harmonic maps from closed spin tori whose map components are not harmonic. For a closed Riemannian surface as target, we prove that such solutions exist for every source metric and every spin structure whenever the target metric is nonflat. The maps can be chosen null-homotopic, of rank one everywhere, with nowhere-vanishing tension and unbounded energy. For a strictly negatively curved closed oriented target surface, every homotopy class contains such a sequence when the source has the trivial spin structure. For the other spin structures we give a compatibility condition for the same construction. Adding a harmonic circle component yields coupled immersions into the product of any such surface with a circle. The construction combines a spinor phase that cancels the rotation of a moving target frame with a curvature balance along a closed curve. Minimizing the energy of short loops with prescribed signed curvature area produces the required curves without any symmetry assumption on the target. Their total geodesic curvature determines the spinor's periodicity. We also give a curve criterion in arbitrary target dimension, applications to totally geodesic surfaces and warped products, and a local choice of target metric that works on every closed manifold of dimension at least two.

This paper derives necessary optimality conditions for free end-time optimal control problems governed by sweeping process dynamics. In contrast to classical optimal control problems, the presence of the normal cone term in the state equation introduces discontinuities with respect to the state variable and prevents the standard Lipschitz regularity assumptions typically used in optimal control theory. As a result, both the analysis of trajectories and the formulation of optimality conditions require refined tools. We address problems with general endpoint constraints and time-dependent data, considering both the case in which the dynamics are Lipschitz continuous in time and the more challenging setting of merely measurable time dependence. Special attention is devoted to the additional transversality conditions associated with the free end-time variable. In the Lipschitz case, these conditions are expressed in terms of a function of bounded variation that coincides almost everywhere with the maximized Hamiltonian. In the measurable case, we show that a suitable interpretation can be obtained via the notions of sub- and super-essential values of the maximized Hamiltonian, recently introduced in the literature, leading to strengthened transversality conditions. These conditions also involve an additional term reflecting the interaction between the costate arc and the normal cone to the moving set at the optimal endpoints. Our approach combines suitable perturbation techniques with a penalization method tailored to sweeping processes. The moving set is modelled as a finite intersection of inequality-defined sets satisfying suitable constraint qualifications, including a positive linear independence condition and a diagonal dominance property of the associated Gramian matrix. We also introduce a local version of the constraint qualification condition. Our results apply to a broad class of problems, including those involving unbounded moving sets such as polyhedra. The resulting optimality system extends and sharpens recent results for classical free end-time problems to the non-Lipschitz, discontinuous framework of sweeping processes. The relevance of the transversality conditions obtained in this article is illustrated through two simple examples.

The necessary and sufficient conditions for the intersection of two real flag manifolds in a complex flag manifold to be discrete are stated including a previous known result. The discrete intersection of real flag manifolds is the orbit of a Weyl group, which is an antipodal set of a complex flag manifold. The triple of two real flag manifolds and a complex flag manifold as an ambient space is constructed from a compact symmetric triad. It is shown that the compact symmetric triad can be taken to be canonical.

We study iterated function systems with large overlaps via the Kajiwara-Watatani C*-algebra and its KMS states. Our main contributions are threefold. First, we introduce Markov partitions for iterated function systems and give an algorithm that constructs a finite Markov partition whenever one exists. Second, for contractive systems admitting a finite Markov partition, we construct an intrinsic analogue of the Hutchinson measure that is independent of any labelling or weighting of the maps. It is the unique Borel probability measure satisfying a self-similar integral equation dictated by the set-valued dynamics of the system, and it recovers the Hutchinson measure with uniform weights when the system satisfies the open set condition. Finally, we give a complete classification of KMS states for the gauge action on the Kajiwara-Watatani algebra of every such system. The critical inverse temperature is the exponential growth rate of the intrinsic orbits, which can be strictly smaller than the natural logarithm of the number of maps, and the unique KMS state at this temperature is given by our intrinsic Hutchinson measure. These systems include many with large overlaps not covered by the results of Izumi-Kajiwara-Watatani.