Papers

In this paper, we introduce and study a family of diffusion processes associated with the Pitman-Yor process and the two-parameter Poisson-Dirichlet distribution. The diffusion coefficients indexed by a non-negative parameter $\gamma$ are smaller than the corresponding one-parameter models in terms of quadratic forms or bilinear forms when $\gamma$ is positive. The well known Petrov's diffusion corresponds to $\gamma=0$ among the unlabelled diffusions. If $\gamma$ is the same as the stable parameter $\alpha$ in the Pitman-Yor process, we obtain both labelled and unlabelled reversible diffusion processes with the Pitman-Yor process and the two-parameter Poisson-Dirichlet distribution as the corresponding reversible measures. We construct these processes analytically through Dirichlet forms. In comparison with existing models in the literature, our models possess two fundamental new features. Firstly, our labelled model is characterized by an explicit generator, which is the first among all models studied so far. This makes it possible to establish the crucial integration-by-parts formula. Secondly, a novel foundational structure of the two-parameter distributions is the existence of a diversity index (a multiple of local time) for the positive stable parameter. By slowing down the diffusion in our model, the essential role of the diversity index is revealed in the evolution of the population. Additionally, we also obtain properties including ergodicity, path behaviour, and finite dimensional approximations.

We prove strong ill-posedness of the 2d incompressible Euler Equation for velocity field in the critical Besov Spaces $B^{1}_{\infty, q}$ for $1<q<\infty$.

We show that generic Maxwell fields can be represented within the matter-space framework by introducing two independent matter-space flows. In the one-flow formulation the electromagnetic field strength is the pull-back of a two-form on a three-dimensional matter space and therefore satisfies $F\wedge F=0$, so that a single flow captures only a degenerate, helicity-carrying sector. The minimal completion is obtained by writing $$F=G^{(1)}+G^{(2)},$$ where each sector field $G^{(I)}$ is the pull-back of a matter-space two-form from an independent flow. Each sector is individually degenerate, $G^{(I)}\wedge G^{(I)}=0$, while the full field satisfies $F\wedge F=2G^{(1)}\wedge G^{(2)}$; the invariant excluded by the one-flow theory is thus recovered as an inter-flow quantity. We interpret this structure in terms of mutual helicity: the total helicity decomposes into two self-helicity contributions and a mutual term whose exterior derivative is $F\wedge F$. Hence a configuration with vanishing mutual helicity lies in the degenerate sector, whereas a nondegenerate Maxwell field necessarily carries a nontrivial mutual helicity. A variational principle for the total field recovers the sourced Maxwell equation, the two matter-space variations combining into the full equation whenever the sector kernels intersect trivially. Generic electromagnetism is thereby reconstructed as the minimal coupled completion of two individually degenerate matter-space sectors.

Guiduli asked in 1996 the following problem concerning the maximum spectral radius of a graph under hereditary density constraints. If an $n$-vertex graph $G$ satisfies $e(H)\le c|V(H)|^2$ for every subgraph $H$ of $G$, must one have $\lambda(G)\le 2cn$? More generally, what remains true when the exponent $2$ is replaced by a constant less than $2$? We study the natural power-law version of this question for all $1<p\le2$. For $1<p\le 2$, define \[ d_p(G)=\max_{\varnothing\ne S\subseteq V(G)}\frac{e(G[S])}{|S|^p}. \] We determine the sharp asymptotic upper bound for $\lambda(G)$ in terms of $d_p(G)$ and $n$. More precisely, every $n$-vertex graph $G$ with at least one edge satisfies \[ \lambda(G)\le \begin{cases} \left(\left(\max_{t\in\mathbb N_{\ge1}}\dfrac{t}{(t+1)^p}\right)^{-1}+o(1)\right)d_p(G)\sqrt n,&1<p<3/2,\\[0.4em] \left(\dfrac{3\sqrt3}{4}+o(1)\right)d_p(G)\sqrt{n\log n},&p=3/2,\\[0.4em] (\mathfrak C_p+o(1))d_p(G)n^{p-1},&3/2<p<2, \end{cases} \] and each constant here is best possible. Here $\mathfrak C_p$ is characterized by an exact variational problem over finite kernels. We apply a sparse graphon operator estimate to convert hereditary $p$-density bounds into sharp spectral bounds, and this estimate also explains the transition at the critical exponent $p=3/2$. For the endpoint $p=2$, Wilf's theorem gives the exact finite-$n$ bound $\lambda(G)\le 2d_2(G)n$, with equality for $K_n$. Thus Guiduli's power-law problem is resolved in its sharp asymptotic form for every $1<p\leq2$, including exact leading constants.

Multipliers are operators on a Hilbert space that have been implicitly used in applications throughout the years. Multipliers are defined via three sequences and are mainly used in signal processing and psychoacoustics among many others. They modify a signal in three steps: analysis, term-wise scalar multiplication, and synthesis. While the study of multipliers stemmed from applications, its theory has earned an equally important place in the literature in the last few decades. One of the main questions in the theory of multipliers is the invertibility of multipliers. It is known that multipliers are not always invertible, and it has been shown that under mild conditions, the inverse of an invertible multiplier can be expressed as another multiplier of a very specific structure. In this work, we extend such result to the setting of possibly noninvertible multipliers. We solved the challenges of determining an appropriate generalized inverse to replace the inverse, and determining a proper duality relationship of the sequences of the resulting multiplier. We showed that these are nontrivial tasks that required the intricacies of the theory of frames for subspaces.

In 2003, Givental and the author constructed Hirota quadratic (or bilinear) equations for simple singularities – a result later utilized by Fan, Jarvis, and Ruan to establish the generalized Witten conjecture. Whether this construction extends to broader classes of singularities has since remained an open question. In this paper, we resolve this question affirmatively for simple elliptic singularities. More precisely, using mirror symmetry, we construct Hirota quadratic equations for simple elliptic singularities by showing that the Gromov–Witten invariants of the elliptic orbifold lines satisfy a system of bilinear equations.

In this paper, we give symbol-only, non-testing characterizations of bounded and compact composition operators on $\mathcal Q_p$, $0<p\le 1$, via a novel dyadic trace formulation over Carleson tents for generalized $p$-Nevanlinna counting functions. Our results resolve an open question raised in Xiao's 2001 book, as well as the diagonal case of Zhao's 2009 question, a longstanding problem in the theory of $\mathcal Q_p$ spaces. As an application of our main theorems, we also obtain boundedness and compactness criteria for the $\mathcal Q_p$-Carleson measure embedding $\textrm{id}:\mathcal Q_p\to L^2(WdA)$ for weights $W$ in the Littlewood–Paley class.

The two-dimensional border-collision normal form is a four-parameter family of continuous, piecewise-linear maps. When this form has a zero determinant, all of its nonlinear dynamics are captured a one-dimensional first return map. The first return map is discontinuous and piecewise-linear, where each piece of the map corresponds to a constant return time. We show that when the normal form has a repelling focus fixed point, the configuration of the first return map is dictated by a rational rotation number whereby the set of return times and ordering of the pieces of the map are given by the denominators of the left and right sequences of Farey parents of this number. The result is proved by characterising polygons formed from preimages of the switching manifold, and employing an inductive argument on the Farey web. Several surfaces in parameter space where the configuration changes are bifurcations from chaotic to quasiperiodic or mode-locked dynamics.

For the 1D generalized Coulomb problems–a family that includes the quantizations of the 1D generalized Kepler problems of Ma-Meng-Xiao \cite{MMX2025}–we construct two explicit unitary intertwiners $\hat\iota_{\pm}$, the quantum analogs of the classical symmetry regularization maps $\iota_{\pm}$ of \cite{MMX2025}, that unitarily identify each $H_{\kappa}$-energy-definite portion of the Hilbert space $L^{2}(\mathbb{R}_{>0},\mathrm{d}q)$ with a unitary lowest-weight representation of $\widetilde{\mathrm{SL}}(2,\mathbb{R})$.

We study finite-state finite-action Bayesian statistical decision problems. While exact error-exponent characterizations are known for several special cases, including hypothesis testing and hypothesis exclusion, the asymptotic behavior of the optimal Bayes regret is largely unknown for general decision problems. In this paper, we show that the optimal regret always decays exponentially fast and characterize its exact exponent for arbitrary loss functions. The exponent is given by the minimum multivariate Chernoff information over the minimal incompatible subsets of states, where an incompatible subset is a collection of states for which no single action is optimal for all states in the subset. Our result recovers the classical pairwise-minimum Chernoff exponent for symmetric multiple hypothesis testing and the multivariate Chernoff exponent for hypothesis exclusion, while also yielding, to the best of our knowledge, the first exact exponent characterization for list hypothesis testing.

Recently there has been some progress in understanding the density of a subset of $[N]^2$ that avoids polynomial patterns. Kravitz, Kuca, and Leng showed that if $P\in\mathbb{Z}[z]$ satisfies certain conditions, then any set $A\subseteq[N]^2$ does not contain $(x,y),(x+P(z),y),(x,y+P(z))$, we must have \[ |A|\ll_P\frac{N^2}{(\log\log\log N)^c} \] for some small constant $c$. In this article, we show a similar result in $(\mathbb{F}_p)^2$ where we get a better bound on the density of a set $A\subseteq (\mathbb{F}_p)^2$ not containing $(x,y),(x+P(z),y),(x,y+P(z))$ with some conditions on $P\in \mathbb{F}_p[z]$.

In this paper, we prove the existence of an $SO(3) \times SO(8)$-invariant Einstein metric with positive scalar curvature on $S^3 \times S^7$.

Let $X$ be an $n$-dimensional Alexandrov space with curvature $\ge -1$, and let $\eta > 0$. Define $\mathcal{S}^{k}_\eta(X)$ as the set of $(k,\eta)$-singular points in $X$ whose tangent cones are $\eta$-away from splitting off $\mathbb{R}^{k+1}$ isometrically. For a point $p \in X$, assume that $M = B_2(p) \setminus (\mathcal{S}^{n-2}_\eta(X) \cup \partial X)$ is a smooth manifold equipped with the Riemannian metric induced by $X$. We prove that the integral of the scalar curvature of $M$ over $B_1(p)$ is bounded from above by a constant depending only on $n$ and $\eta$. As a special case, this extends Petrunin's bounded curvature integral result for complete manifolds with lower sectional curvature bound to the setting of open manifolds and smooth manifolds with boundary, provided that these manifolds are Alexandrov spaces.

Fixed point theory is central to nonlinear analysis, yet classical results often assume strict contractivity and guarantee uniqueness, limiting applicability to mappings with inherently non-unique fixed points. This paper addresses this gap by introducing a modified C-class function and establishing new non-unique fixed point theorems for Ciric-type mappings in complete metric spaces. We construct a generalized contractive inequality using a three-variable modifed C-class function combined with an altering distance function. Under orbital continuity and orbital completeness, we prove that Picard iteration converges to a fixed point without requiring global Lipschitz conditions or strict contractivity. Carefully designed examples illustrate the sharpness and generality of the framework, strictly extending classical Ciric and Chatterjea contractions as well as hybrid $({\Phi},{\psi})$-contractions. Our main contributions are: (i) a unified framework encompassing classical and generalized C-class contractions, (ii) existence results for non-unique fixed points, (iii) a simple criterion for uniqueness when desired, and (iv) practical validation through nontrivial examples. These results provide a robust foundation for further research in nonlinear operator theory, iterative methods, and applications in optimization and variational problems.

The analysis of non stationary signals in complex physical systems often relies on Time Frequency distributions. Among these, the Pseudo Wigner Ville Distribution (PWVD) stands out for its superior resolution but is mathematically challenging due to its inherent quadratic nonlinearity. This nonlinearity generates complex interference artifacts and cross terms in the phase space, potentially obscuring the physical features of the signal, particularly for nonlinear chirps. In this work, we establish a mathematically grounded framework for the PWVD for general windowed nonlinear chirps. By leveraging the theory of oscillatory integrals with coalescing stationary points, we derive a uniform asymptotic expansion that bridges the gap between heuristic signal processing and semiclassical geometric approaches (Berry's chord construction). The resulting closed form representation, expressed in terms of symmetric incomplete Airy functions, provides a unified description of the nonlinear transform's behavior, regularizing the transition across the instantaneous frequency caustics. While the framework is general, we show its power on two illustrative examples: the high precision nonlinear chirps of coalescing binaries in gravitational-wave astronomy and radar nonlinear chirps for pulse compression applications. The analytical results successfully predict the structure of interference patterns and quantify the systematic bias in peak based frequency estimation. Therefore, this study establishes a systematic bridge between nonlinear mathematical analysis and precision experimental physics, validating the PWVD as a robust tool for detailed source characterization in high signal to noise regimes.

The notion of fidelity for quantum states is a measure of how much two states overlap. In the matrix formalism of quantum mechanics, states are represented by density operators, i.e., positive semi-definite matrices with trace equal to 1 in a complex Euclidean space $M_n(\mathbb{C})$. Felipe-Sosa and Felipe (2022) introduced the notion of quantum states on certain Krein spaces with indefinite metric induced by a fundamental symmetry $J$, calling these $J$-states. We define an analogous notion of measurement for $J$-states to the regular quantum theory and use it to show that a notion of fidelity holds in the Krein setting. We also show that an analogous result to the Fuchs-Caves measurement holds in this setting. Following the developments of Bag, Rohilla, and Trivedi (2024), we then extend this definition of fidelity to $U$-quantum states on $S$-spaces. We demonstrate that the analogous geometric motivation holds in the Krein and $S$-space setting, as holds for quantum fidelity and geometric means of operators.

Bioelectrical signals are increasingly acquired at scales that challenge the bandwidth of brain-computer interfaces. However, their compression is still often framed as a problem of waveform preservation, limited by the entropy of the raw signal. Here we propose an information-theoretic framework in which the effective information of bioelectrical data is determined not only by signal fidelity, but also by physiological structure, model capacity and downstream task requirements. We formulate bioelectrical compression as a three-level hierarchy. At the signal level, noise is reduced to the information they carry about latent physiological sources. At the physiological level, parametric encoders map purified signals into compact, structured and quantized representations. At the semantic level, task-irrelevant information is discarded, while deep learning models exploit causal dependencies to replace marginal entropy with conditional entropy. This perspective reframes the compression limit of bioelectrical signals as a model- and task-conditioned quantity rather than a fixed property of the waveform. As increasingly expressive models become integrated with neural and physiological interfaces, bioelectrical compression may shift from transmitting signals to transmitting only the residual information required for task-level interpretation.

We study a rate-one Poissonization of the one-dimensional elephant random walk killed upon reaching $\pm N$. For every fixed memory parameter $0<p<1$, we give a self-contained proof of two-sided exponential survival bounds on the rate scale $N^{-2}$, uniformly for sufficiently large $N$ and times $t\ge TN^3$. Our approach combines the classical Dirichlet Poincaré inequality with an $O(N/t)$ estimate on the memory-dependent perturbation and a quantitative initialization of the principal-eigenvector projection. The lower bound covers the critical and superdiffusive regimes $3/4\le p<1$ in this late-time range. The confined-path comparison used for initialization also gives a direct proof of the lower bound. The spectral argument controls the exponentially small forcing terms and jumps introduced by Poisson truncation.

We propose a matrix-free right-preconditioning strategy for the Generalized Minimal Residual (GMRES) method based on the Multiscale Robin Coupled Method with oversampling (MRCM-OS) for the numerical solution of elliptic problems arising in subsurface flow. The resulting preconditioner is constructed through local subdomain solves with oversampling and smoothing, and can be applied without explicit assembly of the global operator. After a careful presentation of the new procedure, it is used in extensive numerical experiments. Our results demonstrate that the proposed approach substantially reduces iteration counts across a range of challenging, high-contrast subsurface flow problems. In many cases, convergence is obtained in one or two GMRES iterations when oversampling and smoothing are employed. The results indicate that combining GMRES with multiscale Robin-based operators is a promising direction for the construction of rapidly convergent preconditioning strategies.

Let $f: X \to S$ be a quasi-projective family of varieties defined over $\overline{\mathbb{Q}} \subset \mathbb{C}$. We show that the points of $S(\overline{\mathbb{Q}})$ that are Hodge generic for the variation of Hodge structures associated to $f$ are analytically dense in $S(\mathbb{C})$. In fact, in the spirit of the Grothendieck period conjecture and under a large monodromy assumption, we prove the density of the points of $S(\overline{\mathbb{Q}})$ where the periods of the fibre do not satisfy extra relations 'up to degree $\delta$'. As a by-product, we also establish new instances of the Mumford-Tate conjecture, beyond the realm of abelian motives. When the base $S$ is a curve, we provide quantitative estimates for points satisfying these properties. The main technical contribution is a new result on relations satisfied by solutions of $G$-operators, which relies on height estimates due to Bombieri and André.

In this paper, we investigate the well-posedness of the three-dimensional stationary incompressible resistive-viscous Hall magnetohydrodynamic (Hall-MHD) system within the framework of scale-invariant Besov-Morrey spaces. A key component of our analysis is the derivation of product estimates in these spaces, which are instrumental in handling the system's nonlinear terms. As an application, we also establish a well-posedness result for the stationary Navier-Stokes equations in this setting, thereby broadening the admissible classes of external forces and solutions.

The Fourier neural operator (FNO) is a neural network architecture that learns mappings between function spaces. Its efficient implementation is based on the multi-dimensional Fourier transform. By deriving general regularity bounds for the FNO with respect to both the spatial and parametric variables, we prove that the generalization error of the FNO can be improved by replacing spatial tensor product grids with purpose-built rank-1 lattice points, and by using a second lattice carefully constructed as training points in the parametric space. We achieve more accurate and efficient approximations from fewer network parameters, fewer spatial points, and fewer training samples. In addition, the architecture is simplified, because the high-dimensional Fourier transform on rank-1 lattices requires only a one-dimensional fast Fourier transform, and we can use a hyperbolic cross frequency index set with lattice points. We demonstrate the benefits of our lattice-based hyperbolic-cross FNOs for an elliptic PDE on the torus.

We consider a wide class of $2D$ tight - binding models of solid state physics. These models are, in the most general case, non - homogeneous. The topological invariant ${\cal N}_3$ responsible for the quantization of the Hall conductivity, for the specific case of the integer quantum Hall effect in $2D$, is expressed through the Wigner transformation of the two-point electron Matsubara Green function. We express this invariant as a pairing of the element of the $K^{-1}$ group (generated by the Green function) with the specific element of the cyclic cohomology group $HC^3$. According to a set of local index theorems the values of ${\cal N}_3$ can be shown to be integer for a limited class of tight - binding models. We concentrate on systems with a spectral gap, with the chemical potential inside the gap and an invertible Dirac operator $\hat Q$ throughout the Matsubara-frequency domain. The mobility-gap case is not considered here and will be addressed in forthcoming publications.

We prove that the one-dimensional Dirichlet spectrum with respect to approximation in $L_1$ norm $\mathbb{D}^{[1]}$ satisfies $$ \mathbb{D}^{[1]}=\left[\frac12,1\right]. $$ This is equivalent to the fact that the Minkowski spectrum $\mathbb M$, associated with the Minkowski diagonal continued fraction, satisfies $$ \mathbb M=\left[\frac14,\frac12\right]. $$ Further, we show that level sets $$ \Theta_m=\{\alpha\in(0,1)\setminus\mathbb Q:\mathfrak m(\alpha)=m\}, $$ where $\mathfrak{m}(\alpha)$ is the Minkowski constant of $\alpha$, have Hausdorff dimension strictly greater than $1/2$ for any $m\in(1/4,1/2]$, while $\dim_H \Theta_{1/4}=\frac{1}{2}$.

The notions of provisional, negative, and apparent convergence to 0 are introduced. It is then shown that Markov's principle is equivalent, in Bishop-style constructive mathematics, to the statement `every decreasing sequence of real numbers provisionally convergent to 0 actually converges to 0', and that this equivalence holds with `provisionally' replaced by `negatively'. Finally, apparent convergence and convergence are related by means of the anti-Specker principle.

We consider the following quasilinear reaction-diffusion equation $$ \partial_tu=\Delta u^m+(1+|x|)^{\sigma}u^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty), $$ in dimension $N\geq3$ and in the range of exponents $1<p<m$ and $-\infty<\sigma<-2$. We prove that, for initial conditions $u_0$ satisfying $$ u_0\geq0, \quad u_0\not\equiv0, \quad \lim\limits_{|x|\to\infty}|x|^{-(\sigma+2)/(m-p)}u_0(x)=0, $$ the solution $u$ to the corresponding Cauchy problem remains uniformly bounded from above and below: $$ C_1\leq \|u(t)\|_{\infty}\leq C_2, \quad t\in(0,\infty), $$ for some positive constants $C_1$ and $C_2$. Under suitable conditions on $p$, we also establish the rate of expansion of the upper limit $R(t)$ of the positivity set for compactly supported data, that is, $$ At^{\beta}\leq R(t)\leq Bt^{\beta}, \quad \beta=-\frac{m-p}{\sigma(m-1)+2(p-1)}, $$ and a different time scale in outer sets, that is $$ D_1t^{-\alpha}\leq u(x,t)\leq D_2t^{-\alpha}, \quad \alpha=\frac{\sigma+2}{\sigma(m-1)+2(p-1)}, \quad {\rm if} \ |x|\geq Ct^{\beta}. $$ The boundedness is in striking contrast with the property of grow-up as $t\to\infty$ established in previous works by the authors for $\sigma>-2$, illustrating the character of threshold of the exponent $\sigma=-2$.

We develop a unified framework for linear-cost optimal control, finite-time optimal steering, dissipativity analysis, and zero-sum differential games for linear impulsive systems whose state is a Hermitian matrix evolving in $\mathbb{H}^{n+m}_{\succeq0}$, a class that encompasses continuous- and discrete-time linear systems and switched systems as degenerate cases, and includes the second-order moment dynamics of linear (stochastic) hybrid systems. The entire theory rests on three tools: a single key identity relating cost, trajectory, and a dual variable, an Extended Schur complement lemma, and a Schur inner-product decomposition, applied identically to the flow integral and to each jump. These yield structurally uniform sufficient and necessary conditions, dual linear matrix inequality (LMI) characterizations, and explicit optimal policies for every problem class, on both finite and infinite horizons under time-varying assumptions (without time invariance or periodicity), together with causal dwell-time policies for the problems that admit them.

In this paper we review the theory that David Goss developed, starting from 1979, to construct zeta functions around Carlitz zeta values and other remarkable formal series in local fields of positive characteristic. In the description of Goss' theory, we will see how it is primarily motivated by analogies with the classical theory of complex valued zeta and $L$-functions. We compare Goss' theory with another way of constructing zeta and $L$-functions that emerged in more recent times. The functions in the second type have as domains curves over finite fields, with the scalars extended to complete and algebraically closed fields of positive characteristic. The second type of functions interacts with Goss' functions but remains fundamentally different. We shall review a rationality theorem of Ferraro that allows, among others, to introduce some kind of analogue of the function $\xi$ of Riemann. In the path of describing Ferraro's proof, we present some essential tools useful to get into the theory: shtuka divisors and functions, special functions, Anderson motives, Drinfeld modules, among others. We discuss certain relative zeta functions that can be considered as counterparts of Dedekind zeta functions. In particular, we use methods introduced by Goss to prove that these functions extend to entire functions. The paper contains an appendix by Ferraro where the property of entireness of the above relative zeta functions is deduced (in a special case) from the conjunction of a formula by Anglès, Ngo Dac and Tavares Ribeiro and Ferraro's rationality formula. Ferraro also presents a conjecture on the order of vanishing of this function at the canonical point $\Xi$ and some numerical evidences.

The real projective line $\mathbb{R}\mathbf{P}^1$ is the boundary of $\mathbf{HP}=\{z\in \mathbb{C}\colon \Im(z)>0\}$, a model of the hyperbolic plane whose space of geodesics identifies with $\mathcal{G}(\mathbf{HP})=\mathbb{R}\mathbf{P}^1 \times \mathbb{R}\mathbf{P}^1 \setminus \mathrm{diagonal}$. The modular group $\Gamma=\operatorname{PSL}_2(\mathbb{Z})$ acts on $\mathbf{HP}$ with quotient the modular orbifold $\mathbf{M}=\Gamma\backslash \mathbf{HP}$. Consider a finite-index subgroup of the modular group $\Gamma^\prime \subset \Gamma = \operatorname{PSL}_2(\mathbb{Z})$ corresponding to a finite cover $\mathbf{M} \to \mathbf{M}^\prime$. A geodesic $(\xi^-,\xi^+)\in \mathcal{G}(\mathbf{HP})$ projects $\bmod{\Gamma^\prime}$ to a geodesic $\xi^\prime \subset \mathbf{M}^\prime$. We show that if $\xi^\prime$ is simple, then $\xi^+$ is either rational or quadratic or transcendental. In the transcendental case, we obtain bounds on the Mahler measures and show that those can be improved for geodesics fixed by pseud-Anosov maps. Finally, we also explain in detail why all this was known for the modular torus cover associated to the derived subgroup $\Gamma^\prime = [\Gamma, \Gamma]$.

In this paper, we investigate the relationship between the long time behaviour of solutions to the Kahler-Ricci flow on an asymptotically conical gradient Kahler-Ricci expander and the asymptotic behaviour of their initial data at spatial infinity.