Papers

We prove that completely parabolic translation surfaces are Veech, answering a question of Smillie-Weiss.

For a fixed matrix-polynomial trajectory in $\operatorname{SL}_3(\mathbb R)/\operatorname{SL}_3(\mathbb Z)$, we give a necessary and sufficient intrinsic Diophantine condition for its time averages to equidistribute at a power rate on its homogeneous orbit closure. The condition uses a lattice in the space of polynomial functions on the normalized orbit-closure group, each admitting a representative of total degree at most two in the matrix entries. These polynomial tests have a common degree bound of two, independent of the curve degree and optimal in general. The proof begins with a classification of the possible orbit-closure groups and derives lower bounds for polynomial coefficient vectors from the intrinsic condition. It combines quantitative nilmanifold equidistribution, unipotent tangent approximation, and effective flow estimates, with arithmetic separation and quantitative nondivergence controlling cusp excursions and periodic-orbit obstructions.

For fixed positive degrees $n,m$, alternating monic reciprocal transformations define a rational discrete dynamical system on the finite-dimensional coefficient space of polynomial pairs. This system arises from congruence jumping in polynomial divisibility problems, where a quotient move changes the polynomial state, but its coefficient dynamics can be studied independently. We prove that every state with nonzero constant terms returns after four alternating updates to its rational gauge classes and classify when this return orbit is realized over $\mathbb Z$. The local obstruction is encoded by endpoint $p$-adic valuation defects. After the first integral block, constant-term valuations evolve by an explicit matrix $M_{n,m}\in\operatorname{SL}_2(\mathbb Z)$ whose trace depends only on $s=nm$. This yields a finite-order, elliptic-type regime for $s<4$, a parabolic dilation regime for $s=4$, and a hyperbolic regime for $s>4$. In the hyperbolic case, invariant thresholds, irrational eigencones, and discrete valley formulas give finite criteria for forward and bi-infinite integral orbits; nevertheless, failure can be delayed arbitrarily long. At the pointwise level, separated auxiliary-relation transport factors through the jumped-variable projection. One jump preserves the visible support rank, while two alternating jumps collapse it to dimension at most one; saturation cannot restore rank. In the quadratic specialization, a rank-two $p$-adic relation lattice gives an exact local cancellation-depth formula and a finite criterion on periodic Vieta orbits. We classify the fixed auxiliary relations that saturate indefinitely on an irreducible six-cycle and construct arbitrarily long persistence on parabolic and fixed hyperbolic orbits.

The goal of this work is to calculate the mapping class groups of solenoidal manifolds. We introduce the dynamical commensurator group associated to an odometer action, and develop methods to calculate the mapping class group using the dynamical commensurator. Our results generalize the previous results of Odden for the mapping class group of the universal solenoid over a Riemann surface, and of Bering and Studenmund for the pointed mapping class group of the universal solenoid over a compact aspherical manifold. We include a selection of examples.

We study maximal subgroups of topological full groups of irreducible one-sided shifts of finite type whose associated groupoids have Cantor unit space. For each wide open subgroupoid of such a groupoid, we construct a subgroupoid graph and identify the subgroupoid with a full compact open reduction of the corresponding countable graph groupoid. Every minimal wide open subgroupoid is either AF or purely infinite. A proper wide open subgroupoid is prime if it admits no proper intermediate wide open subgroupoid. We prove a trichotomy for prime subgroupoids: they are minimal, arise from a clopen bipartition, or have a unique nonempty proper closed invariant set, which is nowhere dense. For a prime subgroupoid, the normalizer of its topological full group is maximal in the ambient topological full group precisely when it maps onto the ambient abelianization. An analogous criterion holds in every subgroup containing the ambient commutator subgroup. For a minimal prime subgroupoid, its topological full group is maximal precisely when it equals its normalizer and the inclusion induces a surjection on first groupoid homology. We also describe all fully amplified maximal subgroups, namely those closed under local agreement. For Thompson's group $V$, the stabilizer of every simple type system is maximal, extending results of Belk, Bleak, Quick and Skipper for finite type systems and certain classes of infinite ones. Finally, every subgroup containing the ambient commutator subgroup has continuum many pairwise nonisomorphic maximal subgroups acting minimally on the unit space.

We give an elementary proof that if $f$ is a locally constant function on a topologically mixing subshift of finite type, then the equilibrium measures for $tf$ converge as $t\to\infty$, a result due to Brémont. The proof shows that the limit measure assigns an algebraic number to every cylinder.

We investigate the existence of (metrically) expansive operators on Fréchet spaces without continuous norm. We first solve a problem posed by Bernardes, Caraballo, Darji, Fávaro, and Peris in 2025, by showing that there is no expansive operator on the Fréchet space $\omega$ of all sequences. We prove this fact by investigating the existence of orbits remaining cofinitely inside some fixed closed subspace of finite codimension. We then give examples of expansive operators on some other Fréchet spaces without continuous norm.

We study the discrete Huygens phase map on the ring $C_n$, the simplest oscillator network in which the coupling can be frustrated. Reciprocal coupling favours phase opposition between neighbours, while the edge phase differences must sum to zero modulo $2\pi$ around the cycle; for odd $n$ these two requirements are incompatible. The parity of $n$ determines the absolute minima of the potential. When $4\mid n$, the critical set also contains circles of critical points, all of which meet at two completely degenerate configurations. After quotienting by the common phase, the map is a gradient step for an intrinsic potential on an $(n-1)$-torus, and the cycle spectrum gives an explicit interval of interaction strengths on which every orbit converges to a critical point. Our main result is that, on this whole interval and for every $n\ge3$, almost every orbit converges to a uniform twisted minimum, including at interaction strengths for which the map is not invertible. When $4\mid n$, the non-minimal points of the critical circles are treated by a centre-stable manifold argument. At the two degenerate intersections, a local blow-up reduces the problem to the cubic leading term of the potential and its restriction to the unit sphere. Orbits approaching along directions where this restriction is positive form a null set, because the cubic is harmonic in the intrinsic metric. No orbit approaches along its zero level, where the critical points are non-degenerate saddles on invariant rays. Hence the degenerate equilibria also have basins of zero measure.

Let $f(z,w)=(p(z),q(z,w))$ be a polynomial skew product of $\mathbb{C}^2$ of degree $d\geqslant 2$ and let $\mu$ and $\mu_p$ be the Green measures associated to $f$ and $p$ respectively. Let $\lambda_1 > \lambda_2$ and $\lambda_p$ be the Lyapunov exponents of $(f,\mu)$ and $(p,\mu_p)$ respectively. We show that if $\lambda_2 = \lambda_p$, then $\mu$ is exact dimensional and: \begin{equation*}\dim_H\mu =\frac{\log d}{\lambda_1}+\frac{\log d}{\lambda_2}.\end{equation*} The proof relies on the continuity of the conditional measures obtained by Jonsson, on the fact that $\mu$ has constant Jacobian $d^2$, and on a fibered version of a normal form theorem for the inverse branches of the iterates of $f$.

We obtain a uniform lower bound for the drift of random walks on hyperbolic spaces in terms of joint minimal displacement. Our main theorem applies to weakly acylindrical actions on Gromov-hyperbolic spaces and is quantitative in all relevant constants. For discrete isometry groups of Hadamard manifolds with pinched negative curvature, the required assumptions are automatically satisfied by the Margulis lemma. This yields uniform gaps at zero in drift spectra for pinched negatively curved Hadamard manifolds, hyperbolic groups, and mapping class groups acting on curve graphs.

This survey reviews results on the Hausdorff dimension of shrinking target sets and recurrence sets in dynamical systems. In their basic formulations, shrinking target sets concern orbits approaching a prescribed centre, whereas recurrence sets concern orbits returning close to their initial points. A central theme is the comparison of the Hausdorff dimensions of these two classes of limsup sets. The dynamical systems considered include expanding conformal systems, irrational rotations, affine iterated function systems, and matrix transformations. Related results on Hausdorff measure are also included when they provide natural refinements of the dimension theory. Finally, we review results on badly approximating orbits and non-recurrent sets, with particular attention to winning properties and intersections with self-conformal and self-affine sets.

Let $F: \mathbb{R}^2 \to \mathbb{R}^2$ be a polynomial map with nowhere zero Jacobian determinant. It is known that the global invertibility of $F$ is equivalent to the pullback of a parallel vector field being parallel. In this paper, we construct the phase portraits on the Poincaré disc of the pullbacks of the canonical annular, radial, and strip vector fields under non-injective maps $F$ that have recently appeared in the literature. Our arguments use local and global properties of vector fields as well as local and global properties of functions, thereby connecting techniques from dynamical systems and singularity theory.

We study positive limit cycles of the Abel equation $$\dot{x}=A(t)x^{3}+B(t)x^{2},$$ where $A$ and $B$ are $2\pi$-periodic functions. By developing a half-period convexity method, we establish a new criterion ensuring that the equation has at most one positive limit cycle. Moreover, if such a limit cycle exists, it is hyperbolic. As applications, we consider two classes of planar polynomial systems that can be reduced to Abel equations. For a subfamily of cubic rigid systems, we prove that there is at most one limit cycle, thereby giving a partial answer to the first problem in Gasull's list of 33 open problems. For quadratic systems, we derive a new uniqueness criterion for limit cycles surrounding the origin and provide explicit parameter regions under which the criterion is satisfied.

We study the structure of the centralizer of a non-elliptic holomorphic self-map of the unit disc and its relation with continuous dynamical structures. We prove that if the identity is not isolated in the centralizer of a non-elliptic self-map $\varphi \in \mathrm{Hol}(\mathbb D)$, then $\varphi$ is univalent and its centralizer contains a non-trivial continuous one-parameter semigroup of holomorphic self-maps. This result provides a general criterion for the existence of continuous dynamics arising from discrete iteration. Our approach is based on the theory of holomorphic models and on simultaneous linearization techniques for commuting self-maps. In particular, we exploit recent developments on simultaneous linearization for parabolic self-maps with positive hyperbolic step. As a consequence, we obtain new sufficient conditions for the embeddability of holomorphic self-maps into continuous semigroups.

Optimal Morse vector fields on closed surfaces, i.e.\ fields with the minimal possible number of critical points (a single source and a single sink), are classified up to topological trajectory equivalence by one-cycle signed chord diagrams in Kybalko, Prishlyak and Shchurko (2018). We recall this construction and the associated duality, which corresponds to reversing the direction of the flow, in a unified encoding in which both endpoints of a chord carry the same label, and we make it algorithmic. We give an explicit traversal procedure that detects one-cycle diagrams, a canonical form of an equivalence class under cyclic shifts, reversal and relabeling of chords, and a construction of the dual diagram. On this basis we describe an algorithm that enumerates all one-cycle signed chord diagrams with a given number of chords, both for orientable and for non-orientable surfaces, groups them into equivalence classes, pairs each class with its dual, and identifies the self-dual ones. The algorithm is implemented in Python and archived in Zenodo; its output is validated against the Harer–Zagier numbers in the orientable case and against orbit-size counts in both cases. We also note that, when a one-cycle diagram is read as a cellular embedding of a one-vertex graph, its dual diagram describes the embedding of the dual graph.

We carefully construct the profinite iterated monodromy group of a rational map as a quotient of the étale fundamental group of the punctured projective line. In doing so, we prove a number of folklore results regarding profinite IMGs; in particular, their topological generation by inertia concentrated in the post-critical orbit, and their relationship with the discrete iterated monodromy group over $\mathbb{C}$.

Let $\Gamma$ be a countable discrete group acting on a tracial von Neumann algebra $(Q,\tau_Q)$ in a trace-preserving way. We study invariant von Neumann subalgebras of $Q\rtimes\Gamma$ and of $L(\Gamma)\bar\otimes Q$, where the latter is equipped with the diagonal action. We show that, under the pointwise weak mixing assumption, an invariant subalgebra whose image under the canonical conditional expectation onto $Q$ is scalar is necessarily contained in $L(\Gamma)$. Further, under relative weak mixing and central freeness assumptions, we determine the structure of invariant subalgebras of the crossed product with non-trivial intersection with $Q$. For p.m.p. actions of groups with trivial amenable radical and the invariant subalgebras rigidity property, pointwise relative weak mixing gives a complete classification of invariant subalgebras in the tensor product case. We construct essentially free, prime, pointwise weakly mixing p.m.p. actions of $\mathbb{F}_n$ for every $2\leq n\leq\infty$, as well as non-prime examples with explicitly described factors.

We analyze exponential mixing for analytic observables and show that mixing at a double exponential rate occurs for genuinely nonlinear systems. Neither its presence nor its sharp exponent is stable under arbitrarily small area-preserving perturbations in the real-analytic topology. We construct two families of area-preserving Anosov Blaschke product diffeomorphisms that converge in the real-analytic topology to the same hyperbolic toral automorphism. For nonzero parameters, both deformations are topologically conjugate to the linear model but not $C^1$-conjugate. For all nonzero parameter values, the maps in the first family are mixing at a double exponential rate with the same sharp exponent, strictly smaller than that of the linear map. The sharp exponent is therefore not lower semicontinuous at the linear map. In the second family, mixing at a double exponential rate is lost altogether, although the maps remain exponentially mixing. We also establish double exponential bounds for every ergodic toral endomorphism and uniform bounds of the same type for totally ergodic higher-rank toral actions. Our method combines the Fourier escape argument with Koopman operator estimates from a Hilbert space of analytic functions to its dual space of hyperfunctions.

We introduce a spectrum of dimensions of measures, the packing intermediate dimensions $\dim_{P,\theta}\mu$, $\theta\in(0,1]$, of a Borel probability measure $\mu$ on $\mathbb{R}^n$ and the quasi-Hausdorff dimension $\dim_{qH}\mu$. We show that this spectrum encodes the information on $\mu$ that governs the packing dimension of its projections and Hölder images. First, $\dim_{qH}\mu$ is the exact threshold for the packing dimension profiles of Falconer and Howroyd: for every real $0<s\leq n$, $\dim_P^s\mu=s$ if and only if $s\leq\dim_{qH}\mu$. Second, we prove a lower bound for the profiles in terms of $\dim_H\mu$, $\dim_{P,\theta}\mu$ and the Assouad dimension of $\operatorname{supp}\mu$, which improves the bound of Falconer and Mattila, the best previously known general lower bound for the packing dimension of projected measures. The improvement is strict whenever $\dim_H\mu<\dim_P\mu$ and $\dim_A(\operatorname{supp}\mu)<n$, and when $\dim_A(\operatorname{supp}\mu)=n$ it occurs exactly when $\dim_{P,\theta}\mu$ exceeds an explicit universal lower bound for some $\theta$ in an explicit range. We present examples of measures with $\dim_H\mu<\dim_P\mu<\dim_A(\operatorname{supp}\mu)=n$ for which our bound gives the exact value $\dim_P\mu_V=\min\{\dim_P\mu,m\}$ for almost every $m$-dimensional subspace $V$, whereas the bound of Falconer and Mattila is strictly smaller. As applications, for a large family of random $\alpha$-Hölder maps $f_\omega:\mathbb{R}^n\to\mathbb{R}^m$, we prove that, for compactly supported $\mu$, $\dim_P\mu_{f_\omega}=m$ almost surely if and only if $\alpha m\leq\dim_{qH}\mu$, and that the packing analogue of Kahane's formula holds for fractional Brownian motions if and only if $\dim_{qH}\mu=\dim_P\mu$. Analogous results are obtained for analytic sets.

For $Q_\alpha(z)=e^{2\pi i\alpha}z+z^2$ with $\alpha$ irrational, we prove that the analytic centraliser of $Q_\alpha$ is trivial whenever $Q_\alpha$ is not analytically linearizable. Together with the quadratic Brjuno–Yoccoz theorem this gives a complete dichotomy, for the centraliser is infinite cyclic at the non-Brjuno parameters and conjugate to the group of nonzero complex dilations at the Brjuno parameters. The result extends the dense-parameter conclusion of Avila–Cheraghi–Eliad to every nonlinearizable member of the quadratic family. The proof starts from Inou's graph-globalization construction, where we retain the original neutral fixed point on a Riemann surface and verify the branch topology and finite-valence estimate, including multiplicities. Nonlinearizability, through the equality case of Schwarz's lemma, forces the surface to be the plane, and a direct quadratic growth estimate then makes the two entire projections polynomial, so that the established polynomial commutant classification yields the integral-iterate conclusion. We also determine the convergence set of the formal commutants and classify the analytic iterative roots of every nonzero iterate.

We develop a new approach to the Joint Intersectivity Conjecture of Bergelson, Leibman, and Lesigne for commuting transformations. The conjecture asserts that the necessary divisibility condition is the only obstruction to polynomial multiple recurrence and, consequently, to the multidimensional polynomial Szemerédi theorem. We establish the conjecture for pairwise independent polynomials, obtaining the first general result for sequences of arbitrary length beyond the classical one-transformation setting. Our argument is inspired by the Furstenberg-Katznelson approach and its polynomial enhancement due to Bergelson-Leibman. Its main novelty, however, lies in the development of new techniques that allow us to bypass the use of the polynomial van der Waerden theorem, on which these methods traditionally rely, and for which no suitable analogue is currently available in our setting. In particular, we prove a Nil-Bohr returns theorem for compact extensions in systems with nilpotent structure, as well as a nil-plus-null decomposition for polynomial multicorrelation sequences that preserves joint intersectivity and positivity.

We prove the existence of unique solutions ${u\in H^1(0,T;L^2_\gamma(\mathbb{R}^n))\cap L^2(0,T;H^1_\gamma(\mathbb{R}^n))}$ of the transport equation $u_t=F\cdot\nabla u+f$ and the orthogonal dynamics equation ${u_t= Q_\text{Z} (F\cdot\nabla) Q_\text{Z} u+f}$, where $F$ is a time-independent vector field, $f$ is a forcing term, $P_\text{Z}=1-Q_\text{Z}$ is a Zwanzig projection (conditional expectation) on $L^2_\gamma(\mathbb{R}^n)$ defined by $P_\text{Z}h \propto \int h(\cdot ,y) \rho(\cdot,y)\rm{d}y$, and $\gamma$ is the standard Gaussian measure with density $\rho$. To this end, we develop a vanishing viscosity method based on the maximal regularity theory for the Ornstein-Uhlenbeck operator on $L^2_\gamma(\mathbb{R}^n)$. The following assumptions guarantee the existence and uniqueness of strong solutions: ${f\in H^1(0,T;L^2_\gamma(\mathbb{R}^n))\cap L^2(0,T;H^1_\gamma(\mathbb{R}^n))}$, ${g\in H^1_\gamma(\mathbb{R}^n)}$, $F\in W^{1,\infty}(\mathbb{R}^n,\mathbb{R}^n)$ and $\rm{ess\,sup}_{\mathbb{R}^n}\left(x\cdot F-\rm{div}F\right) < \infty$, where $g$ denotes the initial value function.

Let $G$ be a group and let $A$ be a set with at least two elements. A lazy cellular automaton over $A^G$ acts as the identity except at the cells where a fixed pattern occurs, where it replaces the central symbol by a fixed different symbol. Let $M(G;A)$ be the submonoid of $\mathrm{CA}(G;A)$ generated by the lazy and the invertible cellular automata. We study when $M(G;A) = \mathrm{CA}(G;A)$. Since no lazy cellular automaton is surjective, every surjective element of $M(G;A)$ is invertible; hence $M(G;A) \neq \mathrm{CA}(G;A)$ whenever $A$ is infinite or $G$ has an element of infinite order. Moreover, if $|A| = 2$ and $G$ has a quotient isomorphic to the Klein four-group $C_2 \times C_2$, then $M(G;A) \neq \mathrm{CA}(G;A)$. For locally finite $G$, and in particular for finite $G$, these are the only obstructions: we prove that $M(G;A) = \mathrm{CA}(G;A)$ if and only if $3 \leq |A| < \infty$, or $|A| = 2$ and $G$ has no quotient isomorphic to $C_2 \times C_2$.

The stability zone of Novikov's problem of planar sections of triply periodic surfaces are labeled by indivisible triples of integers, which we call the soul of the zone. Which triples may occur is not known. We show that, if a Morse function $f$ on $\mathbb T^3$ admits a translation $\tau(x)=x+v$ with $f\circ\tau=-f$, so that the sets $\{f<0\}$ and $\{f>0\}$ are congruent, then in the generic genus-$3$ situation every topological number $L$ satisfies $\langle L,2v\rangle\equiv1\pmod2$. The mechanism is that the electron/hole type of a null-homologous closed trajectory is invariant under the period lattice and is reversed by $\tau$. For $f=\cos2\pi x+\cos2\pi y+\cos2\pi z$ and the $\mu$-cube, this means that the sum of the components of each soul is odd.

We study coding multiplicity for self-similar sets on the real line. When the similarity dimension exceeds one, almost every point with respect to the natural self-similar measure has infinitely many codings. We focus on the case when the similarity dimension is at most one. Our analysis proceeds through the recurrence properties and the size of the critical set, defined as the union of the pairwise intersections of the first-level pieces. Our main structural result shows that when the critical set is finite, unboundedness of coding multiplicity is equivalent to periodic recurrence to the critical set. Moreover, every commensurable system with a finite critical set has uniformly bounded coding multiplicity. This conclusion is sharp in the incommensurable setting. We construct a one-parameter family of four-map systems such that for Lebesgue almost every parameter, the critical set is a singleton and the open set condition fails, while the attractor contains a point with infinitely many codings. More generally, for transversal families, the set of parameters admitting an infinite coding fibre is dense among those for which the strong separation condition fails. We also obtain a quantitative lower bound for the Hausdorff dimension of this parameter set. Such parameters can form a meagre set. We continue the study of the geometric size of branching, obtaining dimension formulae for typical coding fibres and almost-sure upper bounds on the dimension of the critical set in transversal families. In contrast, we construct an exponentially separated system with no dimension drop whose critical set has the same Hausdorff dimension as the attractor and whose coding map has a fibre of continuum cardinality. Although we work in the one-dimensional setting, several results and arguments extend to higher dimensions.

We establish characterizations of general nonuniform dichotomies for linear nonautonomous dynamical systems in terms of nonuniform exponential dichotomies of appropriately time-rescaled systems. In discrete time, for a class of slowly growing rates, such a characterization requires an additional condition involving a suitable notion of nonuniform ordinary dichotomy, which can be omitted for invertible systems. In continuous time, we obtain a more general characterization for a large class of growth rates, without imposing this additional condition. Our approach does not rely on adapted Lyapunov norms and applies to evolution families on arbitrary Banach spaces. As applications, we show how several results concerning general nonuniform dichotomies can be derived from their counterparts for exponential dichotomies.

We consider vacuum spatially homogeneous models in Hořava-Lifshitz gravity, which are expected to describe generic singularities for these modified gravity theories. These models perturb the well-known Bianchi models in general relativity (GR) by a parameter $v\in (0,1)$, in which GR occurs at $v=1/2$. We show that all periodic heteroclinic chains of Bianchi type II orbits are robust for small perturbations of GR, i.e. they exist for any $v\approx 1/2$. Moreover, we prove the existence of a stable manifold for each heteroclinic chain within a Cantor set for the supercritical case, $v\in (1/2,1)$. In particular, we discuss the heteroclinic chains of periods three and four in detail.

We investigate the influence of decay rates on the dynamics of three-dimensional gene regulatory networks modeled by nonlinear differential equations with logistic sigmoidal regulatory functions. Particular attention is paid to the existence, location, and stability of equilibrium points in systems involving numerous interacting parameters. An explicit analytical representation of a central equilibrium is obtained under suitable relations between the regulatory weights, thresholds, and decay coefficients, allowing its location to be determined without solving the corresponding nonlinear algebraic system numerically. General formulas for the characteristic polynomial of the linearized system are derived, and spectral conditions for instability are established for several classes of regulatory interactions. We demonstrate how variations in decay rates can change the number and stability of equilibria, transform oscillatory dynamics, and influence the emergence or suppression of complicated attractors. The possibility of chaotic behavior is investigated through eigenvalue configurations, Lyapunov exponents, and the Kaplan-Yorke dimension. Numerical examples illustrate the sensitivity of the dynamics to decay parameters, including the suppression of previously observed chaotic behavior. The results provide an analytical framework for investigating equilibrium configurations, stability transitions, and complex dynamics in parametrized gene regulatory networks.

We study spectral measures of one-dimensional quasiperiodic Schrödinger operators. For subcritical Schrödinger operator with Diophantine frequency, we prove sharp measurable-set bounds, answering a question of Avila and Jitomirskaya \cite{AJ}; we prove optimal Fourier decay and the sharp decay order of the time-averaged quantum return probability, confirming a conjecture of Aloisio, Carvalho, and de Oliveira \cite{ACO} and a prediction of Zhong and Mosseri \cite{ZhongMosseri}. For the subcritical almost Mathieu operator, we establish summable square-root envelopes for the spectral density, with an optimal density-one formulation of the lower bound, thereby answering a question of Avila \cite{Aqi}. These results follow from a general theorem for reflectionless measures under a summable square-root Craig-type condition. We establish an exact residue expansion of the squared Borel transform, which connects local density blow-up with long-time averaged return probabilities: the endpoint residues determine both the square-root edge profiles and the leading coefficient of the time-averaged return probability.

We study the topological entropy of saturated sets for continuous flows, with emphasis on singular hyperbolic dynamics. For a $C^r$-generic ($r \geq 2$) vector field $X$ carrying a geometric Lorenz attractor $\Lambda$, we prove that for every nonempty compact connected set $\mathcal{K} \subseteq \mathcal{M}(\Phi,\Lambda)$ and every open set $U \subseteq M$ with $U \cap \Lambda \neq \emptyset$, $h_{\mathrm{top}}(\Phi,G_{\mathcal{K}}(\Phi) \cap \operatorname{Trans}(\Phi,\Lambda) \cap U)=\inf_{\mu \in \mathcal{K}} h_\mu(\phi_1)$, where $\Phi=(\phi_t)_{t \in \mathbb{R}}$ is the flow induced by $X$ and $\operatorname{Trans}(\Phi,\Lambda)$ denotes the set of points whose positive orbits are dense in $\Lambda$. The same formula holds $C^1$-generically for singular hyperbolic attractors. In contrast with the uniformly hyperbolic case, we also prove that, for vector fields in a dense subset of $\mathcal{X}^r(M^3)$, geometric Lorenz attractors admit a positive-entropy invariant measure with no generic points. An analogous conclusion holds for vector fields in a dense subset of $\mathcal{X}^1(M)$ whose non-trivial singular hyperbolic attractors contain singularities and satisfy $\dim E^{cu}=2$. These examples show that the discrete-time saturated entropy theory does not extend directly to singular flows.