Papers

In this paper we study the Kirchhoff-Pohozaev equation introduced in \cite{P2} and its constants of motion (see \cite{BoitiManfrin2025}, \cite{BoitiManfrin2026}) on $n$ dimensional tori. We show that these Hamiltonians are all in involution and we prove that they are generated by an infinite list of constants of motion which are all defined and in involution on a fixed phase space. Then we study the Kirchhoff-Pohozaev equation restricted to a finite Fourier support. In dimension $n=1$ we show that such finite dimensional reduction is always completely integrable and provide an analytic Brikhoff normal form in a neighborhood of the origin. We also give sufficient conditions for integrability for $n>1$. We finally show that the formal Birkhoff Normal form of the Kirchhoff-Pohozaev equation is integrable for $n=1$.

We give a variational derivation of the central configuration equations in pair space, where the relative position vectors between pairs of bodies serve as the primary variables. Vector Lagrange multipliers enforce the linear triangle relations required for these pair vectors to be realizable in the plane. For arbitrary $N$, we show that the constrained Hessian decomposes as $H_{\mathcal C}=L^\Delta+\widetilde L$, where the gap Laplacian $L^\Delta$ is positive semidefinite and the transverse Laplacian $\widetilde L$ is signed and contains all possible negative directions. For non-collinear planar four-body central configurations, the signed part has rank two. This yields a new proof of the known bound that the Morse index is at most two. We then characterize positive definiteness and degeneracy of the Hessian by the eigenvalues of an explicit $2\times2$ matrix or, equivalently, by two generalized eigenvalues of a pencil of symmetric matrices. We illustrate the general criterion using the equal-mass square central configuration. For reflection-symmetric configurations, the generalized eigenvalue problem decomposes into two independent smaller problems. This reduction applies to kite and isosceles trapezoidal configurations. For the trapezoid, however, the reflection acts non-orthogonally on the pair coordinates, giving a different, non-orthogonal reduction. Applying the reflection-symmetry reduction to the rhombus family, we prove that the Hessian is positive definite modulo the rotational zero mode; equivalently, every rhombus central configuration is nondegenerate modulo rotations and has Morse index zero.

In this paper, we prove that for every integer $k\geq2$ and every $c>1$ sufficiently close to $1$, there is $\kappa>0$ such that every sufficiently large subset of $\{1,\ldots,N\}$ of density at least $(\log\log N)^{-\kappa}$ contains \[ x,\quad x+\lfloor n^c\rfloor,\quad x+\lfloor n^c\rfloor^2, \quad\ldots,\quad x+\lfloor n^c\rfloor^k. \] We also prove pointwise almost-everywhere convergence of the associated multiple ergodic averages.

We survey biLipschitz (BL) and bounded displacement (BD) equivalence of Delone sets, with an emphasis on examples arising in aperiodic order. After recalling the classical results of Burago-Kleiner, McMullen, and Laczkovich, we discuss criteria for rectifiability, uniform spreadness, and BD equivalence, and how these relate to discrepancy and point-counting estimates. We then focus on Delone sets associated with substitution tilings, where these questions can often be studied through the combinatorial and spectral properties of the underlying substitution rules. We also consider selected classes of cut-and-project sets and briefly discuss extensions beyond BL and BD equivalence.

The hypothalamic pituitary adrenal (HPA) axis is a key regulatory system coordinating endocrine responses to physiological and psychological stress. While the ACTH stimulation test remains a cornerstone of adrenal function assessment, its interpretation is complicated by the dynamic and adaptive nature of the HPA axis under chronic stress exposure. In particular, prolonged stress induces glandular remodeling, glucocorticoid receptor (GR) resistance and delayed feedback recovery, all of which may alter test outcomes without indicating primary adrenal failure. In this study, we present a mechanistic modeling framework that integrates hormonal kinetics, feedback inhibition and functional mass adaptation of the corticotroph and adrenal compartments. We simulate the HPA axis over $180$ days encompassing three phases - baseline, chronic stress and recovery, while introducing a time varying GR resistance function to mimic feedback desensitization and its resolution. Using this framework, we evaluate both low dose ($1 \mu g)$ and high dose ($250 \mu g)$ ACTH stimulation tests across physiological phases. Our simulations show that cortisol responses are highly sensitive to both the magnitude and timing of stress exposure and that ACTH responsiveness is phase dependent and often blunted during recovery due to persistent feedback resistance. Low dose ACTH testing more reliably reflects partial adrenal adaptation, while high dose tests risks masking dysfunction due to supraphysiological drive. These results highlight the limitations of static testing paradigms and suggest that accounting for glandular plasticity and GR feedback dynamics is essential for effective endocrine diagnosis particularly in stress related or treatment induced adrenal disorders.

Interval translation maps are piecewise translations for which the images of distinct continuity intervals may overlap. We study when their measured dynamics can be represented by finite or countable interval exchange transformations. First, we give a direct entropy-based proof of the known fact that an interval translation map is invertible almost everywhere with respect to every nonatomic invariant probability measure. The proof uses a polynomial upper bound for the complexity of the natural branch coding. We then use a consequence of a theorem of Arnoux, Ornstein, and Weiss: every nonatomic measure-preserving automorphism of a standard probability space admits a countable interval exchange transformation model. Applied to the almost-everywhere invertible core, this gives an abstract countable interval exchange transformation model for every measured interval translation map. We next study the canonical order-preserving coordinate defined by the distribution function of the invariant measure. For each branch, we introduce a positive defect measure recording the excess measure carried by its direct image. Its push-forward to the distribution coordinate gives a canonical cut set such that, away from this set, the induced map is locally a translation. Finite defect support yields a finite interval exchange transformation model, while a Lebesgue-null defect cut set yields a countable model. Under branchwise nonsingularity, the defect support is the closure of the cuts arising from active gaps in the support, giving an equivalent geometric characterization of the finite case. Finally, for a self-similar infinite-type Bruin-Troubetzkoy map, we compute the canonical defect cut set explicitly. It is countably infinite and Lebesgue-null, yielding a genuinely countable, non-finite interval exchange transformation model.

Relative periodic orbits (RPOs) are solutions of the three-body problem that are periodic in a uniformly rotating reference frame and, in general, quasi-periodic in inertial coordinates. We present a numerical procedure for computing and continuing one-parameter families of RPOs of the planar Newtonian three-body problem. The method exploits consecutive syzygies, understood here as configurations in which the three bodies are aligned and their velocities satisfy the corresponding symmetry conditions. Matching the positions and momenta at two consecutive alignments reduces the computation of RPOs to a low-dimensional nonlinear problem. Its solutions are then numerically continued, and linear stability is determined from the nontrivial eigenvalues of the rotated monodromy matrix after removing the neutral directions associated with conserved quantities and continuous symmetries. The procedure is applied to several mass distributions and initial configurations, producing families of Poincaré, Hill, and binary-type solutions. These families exhibit transitions from nearly circular to highly eccentric motion, changes of stability near resonances and turning points, and absolute periodic solutions when the rotation angle is a rational multiple of 2{\pi}. In the Hill families, the continuation connects satellite configurations with circumstellar motion as the smallest body loses its gravitational binding to the intermediate body. Circumbinary and circumstellar configurations are also obtained in the binary regime. The results illustrate the dynamical diversity of RPOs and provide coherent three-body motions that can be used as prescribed trajectories in restricted four-body models.

Let $H\in C^\infty(\mathbb R^n\times\mathbb T^n)$ be a periodic Tonelli Hamiltonian with critical value $c$. For each $k\in\mathbb N$, let $u_k$ be the normalized minimizer of the variational functional introduced by Evans[7], \[ I_k[w]=\int_{\mathbb T^n} e^{kH(Dw,x)}\,dx, \qquad \int_{\mathbb T^n}w\,dx=0. \] If $u_\infty$ is a uniform limit of a subsequence of $\{u_k\}$ and the Mather quotient $({A}_M,\delta_M)$ satisfies $H^1( A_M,\delta_M)=0$, then $u_\infty$ is a critical subsolution that is strict outside ${{A}}$ and \[ {A} = \{x\in\mathbb T^n\,:\,Du_\infty(x)\ \text{exists and }H(Du_\infty(x),x)=c\}=\{x\in\mathbb T^n\,:\,u_\infty(x)=u_{-}(x)\}, \] where ${A}$ is the projected Aubry set and $u_{-}$ is the backward weak KAM solution associated with $u_\infty$. In particular, by the theorem of Fathi–Figalli–Rifford[10], this conclusion holds for all smooth Tonelli Hamiltonians on $\mathbb T^n$ when $n\leq3$. This characterization also suggests a natural numerical localization principle for approximating the entire Aubry set through near-contact sets between $u_k$ and its large-time backward Lax–Oleinik evolution.

Combinatorial-topological methods for characterizing dynamics are rigorous, generalizable, computable, and they only require approximations, but the dimension of the phase space is a computational bottleneck to their wider application. Motivated by the growing number of machine learning techniques for obtaining lower-dimensional latent representations of dynamics, we present an initial study of combinatorial-topological techniques in the dimensionality reduction setting. We establish bounds under which an algebraic structure that organizes dynamics can be lifted from the latent space to the original system. As a corollary, one can conclude the existence of attractors within certain regions of the original phase space. The hypothesis of these results is expressed in terms of an approximate semiconjugacy between the original and latent dynamics. To demonstrate the ideas, we combine autoencoder-based models with Conley-Morse graph computations for Leslie population models, a thirteen-dimensional Mediterranean red coral population model, and the Chafee–Infante equation. While the lift of the Conley index is still an open question, the examples recover the expected algebraic topological invariants in several settings.

Let $\mathbb{Z}_p$ be the ring of $p$-adic integers with respect to a prime $p$ and let $d$ be a positive integer. For each $\mathbf{z}=(z_1, z_2,..., z_d) \in \mathbb{Z}_p^d$, let $T_{\mathbf{z}}: \mathbb{Z}^d \times \mathbb{Z}_p \to \mathbb{Z}_p$ be an adding-type $\mathbb{Z}^d$-action on $\mathbb{Z}_p$ defined by $T^{\mathbf{n}}_{\mathbf{z}}(x):=x+ \sum_{i=1}^d n_i z_i$ for $\mathbf{n}=(n_1,n_2, \cdots, n_d) \in \mathbb{Z}^d$ and $x \in \mathbb{Z}_p$. Under some mild assumptions on $\mathbf{z}$, the action $T_{\mathbf{z}}$ is a free $\mathbb{Z}^d$-odometer (by odometer, we mean a minimal and equicontinuous action on a Cantor space). In this paper, we derive a necessary condition for continuous orbit equivalence between such $\mathbb{Z}^d$-odometers by constructing algebraic models for them. We then study the free minimal constant speedups of these $\mathbb{Z}^d$-odometers. It turns out that such a speedup of $T_{\mathbf{z}}$ is again an adding-type $p$-adic $\mathbb{Z}^d$-odometer $T_{\mathbf{w}}$ for some $\mathbf{w} \in \mathbb{Z}_p^d$. However, the necessary condition above may not hold for the speedup. This provides the first known examples of free minimal bounded speedups (of free $\mathbb{Z}^d$-odometers) which are not continuously orbit equivalent to the original ones and hence disproves a conjecture by Johnson and McClendon. Our result also indicates that continuous orbit equivalence is a rare phenomenon for free minimal constant speedups of $p$-adic $\mathbb{Z}^d$-odometers of adding type when $d \geqslant 2$.

We prove the Quadratic Easy Coefficients Conjecture stated as Conjecture 1 in T. W. Cusick, Recursions for quadratic rotation symmetric functions weights, Discrete Applied Mathematics 378 (2026), 93–101. For an arbitrary finite sum of quadratic monomial rotation symmetric Boolean functions, we identify the recurrent part of the rules matrix with a signed binary de Bruijn transfer matrix $B$. We then give a one-step presentation of the finite-type shift associated with the Boolean function in the symbolic-dynamics construction of Chirvasitu and Cusick. Fourier transformation in an auxiliary $\F_2$ coordinate decomposes the adjacency matrix of this shift into an unsigned de Bruijn block and the signed block $B$. Consequently the dynamical zeta function is \[ \zeta_{X_f}(z)=\frac{1}{\det(I-z\cR(f))}, \] where $\cR(f)$ is the rules matrix. This equality identifies, with their algebraic multiplicities, the characteristic values, supplied by symbolic dynamics, with the roots of the characteristic polynomial of the rules matrix. The desired easy coefficients formula follows from the trace of $B^n$. We also prove nonsingularity and justify the unique backward extension of the weight recurrence.

We introduce the notion of proper proximality for countable measured groupoids, extending the corresponding notion for countable groups. We prove that this groupoid property is equivalent to a strengthened form of relative proper proximality for the associated groupoid von Neumann algebra. We investigate permanence properties and show, in particular, that proper proximality is preserved under finite direct products, formation of transformation groupoids, and measure equivalence. We also identify broad classes of properly proximal groupoids, including transverse measured groupoids and free product groupoids, and prove that inner amenable groupoids are never properly proximal.

Early warning signals (EWS), such as increasing variance and autocorrelation, are widely used to anticipate critical transitions associated with saddle-node bifurcations. However, real-world systems are often high-dimensional and multiscale, potentially altering the classical behavior of EWS. Here, we investigate how dimensionality, inertia, and red noise influence the detectability of EWS prior to tipping points. We show that when observations are not aligned with the critical direction, stable dynamics in orthogonal directions can mask EWS until very near the bifurcation. We further demonstrate that second-order dynamics with damping modify the autocorrelation structure and may either enhance or reduce signatures of critical slowing down. Finally, we study how colored noise influences EWS. Our results show that the presence and strength of EWS are not universal properties of tipping systems but depend critically on system geometry and stochastic forcing, implying that the absence of detectable EWS does not necessarily rule out an approaching critical transition.

Permutative sliding block codes - in the sense of Hedlund - give rise to finite-to- one extensions of subshifts. We provide criteria under which minimality and unique ergodicity are preserved by this 'lifting procedure'. These findings are illustrated by means of some nat- ural example families, which we use to obtain strictly ergodic finite-to-one extensions of sub- stitution subshifts (including the case of Fibonacci, Tribonacci, silver mean and noble means substitutions). We further study the interplay between permutative sliding block codes and Toeplitz flows. In particular, we find examples which demonstrate that a permutative lift of a strictly ergodic subshift may be minimal, but not uniquely ergodic - a phenomenon which cannot occur for primitive substitution subshifts.

Preimage entropy measures the complexity generated by the inverse-image structure of a non-invertible dynamical system. For a continuous map $f:X\to X$ on a compact metric space, Hurley's pointwise topological preimage entropies $h_m(f)$ and $h_p(f)$ are natural invariants measuring the complexity of preimage sets. The question of whether they admit unconditional variational principles in terms of suitable measure-theoretic counterparts remains open. In this paper we resolve it by using empirical metric preimage entropies $h^*_{m,\mu}(f)$ and $h^*_{p,\mu}(f)$, defined by restricting preimage fibers to orbit segments whose empirical measures are close to a prescribed invariant measure $\mu$. We prove the variational principles $$ h_m(f)=\sup_{\mu\in\mathcal M_f(X)}h^*_{m,\mu}(f), \qquad h_p(f)=\sup_{\mu\in\mathcal M_f(X)}h^*_{p,\mu}(f) $$ for every continuous map on a compact metric space. We also show that, in general, the set of all invariant measures in these formulas cannot be replaced by the set of ergodic invariant measures. We then compare the empirical entropies with the pointwise metric preimage entropy $h_{m,\mu}(f)$. For every ergodic invariant measure $\mu$, we prove $h^*_{p,\mu}(f)\ge h_{m,\mu}(f)$. Moreover, if $f$ has uniform separation of preimages, then for every ergodic invariant measure $\mu$, $$ h^*_{m,\mu}(f)=h^*_{p,\mu}(f)=h_{m,\mu}(f). $$ Examples show that $h_{m,\mu}(f)$ is not comparable with the empirical quantities in general. We further introduce a resolving-partition property, weaker than uniform separation of preimages, under which the variational principle for $h_m(f)$ and $h_{m,\mu}(f)$ holds. Finally, we establish corresponding variational principles for preimage pressure and give an example showing that uniform separation of preimages does not imply forward expansiveness.

We study entropy and periodic-orbit growth for flows on metric spaces. First, we prove that the finite topological entropy of a flow on a compact metric space is completely carried by compact subsets of its regular set. Next, we establish a Bowen–Walters inequality for geometrically separating flows on possibly noncompact metric spaces, under uniform control at time zero and dynamical isolation at infinity. As applications, we obtain the Bowen–Walters inequality for singular suspension flows over expansive homeomorphisms, multisingular-hyperbolic sets—extending the upper-bound part of \cite{pyyz}—and asymptotically sectional-hyperbolic attractors such as Rovella's \cite{r}.

We review the current state of the study of the SRB entropy functional over open components of families of uniformly hyperbolic systems, including Anosov systems, hyperbolic attractors, expanding maps, and Markov transformations. We address problems concerning the functional's infimum, supremum, and local extrema; the gradient flow induced by the functional; and the functional's behavior near the boundary of open components.

Kaneko's val function is defined as the normalized cycle integral of the elliptic modular $j$-function along closed geodesics on the modular surface. We prove that, when primitive hyperbolic conjugacy classes are ordered by geodesic length, its values concentrate at the single point 720. More generally, an analogous concentration result holds for every weakly holomorphic modular function $f$ of weight 0, with the concentration point given by Atkin's inner product $(f, 1)_{\mathrm{At}}$. The proof combines an equidistribution theorem following Pollicott with the ergodicity of a continued-fraction suspension flow and uses the decomposition formula of Bengoechea-Imamoḡlu to construct a bounded continuous observable.

For all $n \ge 2$, we obtain a homeomorphism of Cantor space that has a distributionally $n$-scrambled Cantor set containing a distributionally $(n+1)$-scrambled uncountable set, but does not have an $(n+1)$-scrambled Cantor set.

In this paper we study the behavior of topological entropy for partially hyperbolic diffeomorphisms on $\mathbb{T}^d$ isotopic to a linear Anosov automorphism with indecomposable weak-stable subspace. In particular, we prove that for a class of DA maps associated with such linear systems, whose central behavior is sufficiently dominated by the expansion rate of the linear model, the topological entropy coincides with that of the linear map. Furthermore, we obtain uniqueness of equilibrium states for a class of low-oscillation potentials and, as an application, the uniqueness of the measure of maximal entropy. We also provide sufficient conditions for which ergodic measures arise as the unique equilibrium state of some continuous potential. Finally, on $\mathbb{T}^3$ we construct a dynamically coherent diffeomorphism in the same isotopy class whose topological entropy is strictly larger than that of the linear part.

For an endomorphism $\phi$ of a closed Riemannian manifold $M$, we study the pushforward action $\phi_\ast$ on the Wasserstein space $\mathcal{P}(M)$ at a measure $\mu_0$ preserved by $\phi$. We show that if $\phi$ is a $C^2$ covering map and $\mu_0$ has positive $C^1$ density, then $\phi_\ast$ is Gâteaux differentiable at $\mu_0$ along tangent directions, with derivative given by the transfer operator of $\phi$ acting on vector fields, followed by orthogonal projection onto the tangent space. The derivative is the adjoint of the Koopman operator restricted to the tangent space, and its fixed space consists of the directions in which $\mu_0$ can be deformed while preserving invariance to first order. For appropriate pairs of endomorphisms of $\mathbb{T}^d$, we compute the intersection of their fixed spaces, show that it contains an infinite family of linearly independent continuous vector fields, and construct, for every $n$, an embedded $n$-dimensional family of measures that are nearly invariant under both endomorphisms. First-order rigidity therefore fails in every dimension. In particular, rigidity phenomena such as higher-dimensional analogues of the Furstenberg conjecture, if true, are genuinely nonlinear.

We study the convergence of multiple ergodic averages involving several commuting multiplicative actions. We prove that for finitely generated systems, the averages $$\frac{1}{N}\sum_{n=1}^N S_{1,n}F_1\cdot\ldots\cdot S_{\ell,n}F_\ell$$ converge in norm, settling a conjecture of Frantzikinakis. Our methods rely on a novel generalization of Kátai's orthogonality criterion that allows us to obtain box seminorm control, the machinery of magic extensions originating in the work of Host, and a delicate induction on the complexity of the initial averages that ultimately reduces our problem to well-known mean convergence results.

A social network perspective on the human microbiome is crucial for understanding how the interplay between within-host microbial dynamics and between-host transmission influences microbial community stability, host metabolism, and population health and nutrition. Here, we provide a mathematical and numerical analysis of human-associated microbial communities in dyadic social ties and within a small network. By utilizing a generalized Lotka-Volterra model, we simulate within-host dynamics while incorporating inter-host microbial sharing driven by social interactions. Furthermore, the study accounts for individual variability in microbial behavior across different hosts. Our work identifies the conditions under which microbial sharing can have long-lasting effects, offering insights into how social networks shape the human microbiome.

We consider, for any irrational $\alpha$ and interval $I \subset \mathbb{T}$, the 2-interval coding subshift $X^{(I, \alpha)}$ induced by coding orbits under repeated rotation by $\alpha$ via membership in $I$ or $I^c$. Each sequence $c \in X^{(I, \alpha)}$ has an associated Schrödinger operator $H_c$, and in \cite{kaminaga} it was proved that if the continued fraction of $\alpha$ has digits with limsup at least $4$, then almost every $c \in X^{(I, \alpha)}$ has so-called $3$-block Gordon structure, which implies that the operator $H_c$ has no eigenvalues. We significantly improve this result by completely characterizing almost-sure $3$-block Gordon structure, proving that in fact it holds for all $(\alpha,I)$ except for a countable set of pairs $(\alpha, |I|)$ where $\alpha$ is Möbius equivalent to the silver mean and $|I| \in \mathbb{Z}\alpha + f(\alpha)$ where $f(\alpha)$ is a specific infinite series taking value either $\frac{1}{2}, \frac{\alpha}{2}$, or $\frac{\alpha+1}{2}$. We also show that for a set of $\alpha$ of full measure, and for every $I$, the set of points whose orbit codings do not have $3$-block Gordon structure has Hausdorff dimension bounded away from $1$.

Dual quasi-mixing and dual mixing of a measure preserving transformation are uniform individual ratio limit properties of the transfer operator which entail the quasi-mixing and mixing properties of Krickeberg (respectively). The real restrictions of normalised, parabolic, non-Möbius inner functions of the upper half plane preserve Lebesgue measure and are dual quasi mixing; such a restriction being dual mixing iff it is exact and singular dual quasi-mixing otherwise.

In this paper, we establish a new type of stability phenomenon for the topological entropy of Hamiltonian diffeomorphisms of closed surfaces. For a closed surface endowed with an area form $(\Sigma,\omega)$ and a Hamiltonian diffeomorphism $\phi$ of $(\Sigma,\omega)$, we show that for every $\varepsilon>0$ there exists $A=A(\phi,\varepsilon)>0$ such that \[ h_{\mathrm{top}}(\phi') > h_{\mathrm{top}}(\phi)-\varepsilon \] for every Hamiltonian diffeomorphism $\phi'$ obtained from $\phi$ by a deformation supported in a disjoint union of disks, each of area less than $A$. In particular, if $h_{\mathrm{top}}(\phi)>0$, then $\phi$ cannot be made to have zero entropy by an area-preserving deformation supported in disks of small area. This follows from the new braid stability result established in this paper with respect to the spectral distance recently introduced by Connery-Grigg.

We develop two methods for proving infinite girth from boundary dynamics. First, we use topologically free extreme boundary actions to give a boundary-dynamical proof of infinite girth for finitely generated acylindrically hyperbolic groups. The second combines Nakamura's criterion with, respectively, flag-space and Roller-boundary dynamics, yielding infinite-girth results for semisimple $S$-algebraic groups and for essential non-elementary actions on finite-dimensional, second countable, non-Euclidean CAT(0) cube complexes. We also prove that every finitely generated large group has infinite girth, thereby answering a question of Akhmedov and Mishra.

We investigate the fake stationarity properties of solutions to forward Stochastic Volterra Integral Equations (SVIEs) with affine drift and long-memory (regular) kernels, both on finite horizons and in the long-run regime. By either deriving explicit closed-form specifications for the deterministic initial condition $\phi$ and the mean-reversion function $\mu$ appearing in the drift, or by introducing a deterministic stabilizing factor $\varsigma$ in the diffusion coefficient associated with the kernel while keeping $\mu$ fully flexible, we show that it is possible to induce a fake stationary regime, in the sense that all marginal distributions share the same mean and variance. Afterwards, using a refined asymptotic analysis, we further establish that, in both frameworks, the time-shifted solutions of these long-memory SVIEs converge weakly, in the functional sense, toward a family of $L^2$-stationary processes sharing the same covariance structure, for suitable classes of diffusion coefficients. These results are applied to a class of exponential-fractional Stochastic Volterra Integral Equations driven by an $\alpha$-gamma fractional integration kernel, in the particular regime \(\alpha \geq 1\), which regularizes diffusion paths and invoke textit{ long-term memory}, persistence or long range dependence.

We relate the dynamics on a closed 3-manifold to the topological invariant, homology guts, proposed by Agol and Zhang. Given a closed manifold $M$ that admits a pseudo-Anosov flow $\varphi$ without perfect fits, we construct a canonical semiflow on the homology guts associated to homology classes carried by $\varphi$. We show that, up to orbit equivalent outside half annuli, the semiflow on the guts is invariant on each Thurston open cone. As an application, the fundamental group and the sutured structure of homology guts encode closed orbits of $\varphi$ with particular type. Besides, for a positive class in $H^{1}(M)$, the canonical semiflow on the guts has a well-defined growth rate.

Cherry flows are classical examples of $C^\infty$ flows on the two-dimensional torus exhibiting non-trivial recurrent dynamics. For every genus $g\geq1$ we construct a $C^\infty$ parameter family of flows whose first return map is a generalized interval exchange transformation (GIET) with flat pieces. In such family, for every interval exchange transformation $T$ satisfying the Keane property, there are parameters corresponding to a Cherry flow whose return map is semi-conjugate to $T$. In particular, for each $k=1,\ldots,g$, our family contains a Cherry flow whose unique quasi-minimal set supports exactly $k$ ergodic invariant measures. The construction relies on a Full Family Theorem for GIETs with flat pieces, which establishes the realization of every admissible Rauzy renormalization path within a specific finite-dimensional family with the optimal number of parameters.