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In this work, we study rates of mixing for small independent and identically distributed random perturbations of contracting Lorenz maps sufficiently close to a Rovella parameter. By using a random Young tower construction, we prove that this random system has exponential decay of correlations.
We establish the pointwise equidistribution of self-similar measures in the complex plane. Let $\beta \in \mathbb Z[\mathrm{i}]$, whose complex conjugate $\overline{\beta}$ is not a divisor of β, and $T \subset \mathbb Z[\mathrm{i}]$ a finite subset. Let µ be a non-atomic self-similar measure with respect to the IFS $\big\{f_{t}(z)=\frac{z+t}{\beta}\colon t\in T\big\}$. For $\alpha \in \mathbb Z[\mathrm{i}]$, if α and β are relatively prime, then we show that the sequence $(\alpha^n z)_{n\ge 1}$ is equidistributed modulo one for µ-almost everywhere $z \in \mathbb{C}$. We also discuss normality of radix expansions in Gaussian integer base, and obtain pointwise normality. Our results generalize partially the classical results in the real line to the complex plane.
This is a corrigendum to ‘An uncountable Furstenberg–Zimmer structure theory’ [Ergod. Th. & Dynam. Sys.43(7) (2023), 2404–2436]. We report two issues in that paper. First, Lemma A.5 and Proposition A.6 in the Appendix, which supported a spectral analysis of conditional Hilbert–Schmidt operators, are incorrect. These results were used in the proof of Lemma 4.4, which establishes part of the equivalences in Theorem 4.1. We provide a correction for this issue here. While the proof strategy of Lemma 4.4 remains valid, the details have been revised using known auxiliary results in the non-commutative setting of tracial von Neumann algebras, replacing the faulty arguments from the Appendix. Second, the proof of the implication $\mathrm{(iii)} \Rightarrow \mathrm{(iii)}'$ in Lemma 4.10 is incorrect. We supply a new argument to address this. We also take this opportunity to correct several minor issues that have come to our attention since the paper’s publication. A fully revised version, including these corrections, as well as updated references and some fixed typos, is now available on arXiv.
which is a one-dimensional Kirchhoff-like equation with a nonlocal convolution coefficient. The novelty of our work involves allowing a variable growth term in the nonlocal coefficient. By relating the variable growth problem to a constant growth problem, we are able to deduce the existence of at least one positive solution to the differential equation when equipped with boundary data. Our methodology relies on topological fixed point theory. Because our results treat both the convex and concave regimes, together with both the variable growth and constant growth regimes, our results provide a unified framework for one-dimensional Kirchhoff-type problems.
The sharpness of various Hardy-type inequalities is well-understood in the reversible Finsler setting; while infinite reversibility implies the failure of these functional inequalities, cf. Kristály et al. [Trans. Am. Math. Soc., 2020]. However, in the remaining case of irreversible manifolds with finite reversibility, there is no evidence on the sharpness of Hardy-type inequalities. In fact, we are not aware of any particular examples where the sharpness persists. In this paper, we present two such examples involving two celebrated inequalities: the classical/weighted Hardy inequality (assuming non-positive flag curvature) and the McKean-type spectral gap estimate (assuming strong negative flag curvature). In both cases, we provide a family of Finsler metric measure manifolds on which these inequalities are sharp. We also establish some sufficient conditions, which guarantee the sharpness of more involved Hardy-type inequalities on these spaces. Our relevant technical tool is a Finslerian extension of the method of Riccati pairs (for proving Hardy inequalities), which also inspires the main ideas of our constructions.
In this paper, we prove the existence of minimizers for the sharp stability constant of Caffarelli–Kohn–Nirenberg inequality near the new curve $b^*_{\mathrm{FS}}(a)$ (which lies above the well-known Felli–Schneider curve $b_{\mathrm{FS}}(a)$), extending the work of Wei and Wu [Math. Z., 2024] to a slightly larger region. Moreover, we provide an upper bound for the Caffarelli–Kohn–Nirenberg inequality with an explicit sharp constant, which may have its own interest.
The partial transposition from quantum information theory provides a new source to distill the so-called asymptotic freeness without the assumption of classical independence between random matrices. Indeed, a recent paper [10] established asymptotic freeness between partial transposes in the bipartite situation. In this paper, we prove almost sure asymptotic freeness in the general multipartite situation and establish a central limit theorem for the partial transposes.
Let $ K $ be a compact subset of the d-torus invariant under an expanding diagonal endomorphism with s distinct eigenvalues. Suppose the symbolic coding of K satisfies weak specification. When $ s \leq 2 $, we prove that the following three statements are equivalent: (A) the Hausdorff and box dimensions of $ K $ coincide; (B) with respect to some gauge function, the Hausdorff measure of $ K $ is positive and finite; (C) the Hausdorff dimension of the measure of maximal entropy on $ K $ attains the Hausdorff dimension of $ K $. When $ s \geq 3 $, we find some examples in which statement (A) does not hold but statement (C) holds, which is a new phenomenon not appearing in the planar cases. Through a different probabilistic approach, we establish the equivalence of statements (A) and (B) for Bedford–McMullen sponges.
The global C0 linearization theorem on Banach spaces was first proposed by Pugh [26], but it requires that the nonlinear term is globally bounded. In the present paper, we discuss global linearization of semilinear autonomous ordinary differential equations on Banach spaces assuming that the linear part is hyperbolic (including contraction as a particular case) and that the nonlinear term is only Lipschitz with a sufficiently small Lipschitz constant. To overcome the difficulties arising in this problem, in this paper, we rely on a splitting lemma to decouple the hyperbolic system into a contractive system along the stable manifold and an expansive system along the unstable manifold. We then construct a transformation to linearize a contractive/expansive system, which is defined by the crossing time with respect to the unit sphere. To demonstrate the strength of our result, we apply our results to a nonlinear Duffing oscillator without external excitation.
where $N \geq 2$, $1/2 \lt s \lt 1$, and $0 \lt \sigma \lt 2s/(N-2s)$. In the mass critical and supercritical cases, we establish a criterion for blow-up of solutions to the problem for cylindrically symmetric data. The results extend the known ones with respect to blow-up of solutions to the problem for radially symmetric data.
A semi-analytical study of oblique wave interaction with two $\boldsymbol {\sqcap }$-shaped breakwater designs—floating and bottom-fixed structures—incorporating two thin porous plates is presented using linearized theory. Wave potential for both configurations is developed using the eigenfunction expansion method, considering both progressive and evanescent wave modes. The problem of oblique wave scattering by $\boldsymbol {\sqcap }$-shaped breakwaters is reduced to a set of coupled integral equations of first kind, based on horizontal velocity components. These equations are solved using the multi-term Galerkin approximation with appropriate basis functions to handle the square-root singularities at sharp edges of the porous barriers. The performance of the models is evaluated by examining reflection, transmission and energy dissipation coefficients, along with free surface elevation and horizontal drift force. We observe that increasing the plate length of the breakwaters attenuates the incident waves more effectively than increasing the width. Additionally, the floating $\boldsymbol {\sqcap }$-shaped breakwater significantly reduces the free surface elevation in the transmitted region. The results from the developed model can provide valuable insights for the design of wave–structure systems in shallow waters.
Given the tropicalization of a complex subvariety of the torus, we define a morphism between the tropical cohomology and the rational cohomology of their respective tropical compactifications. We say that the subvariety of the torus is cohomologically tropical if this map is an isomorphism for all closed strata of the tropical compactification.
We prove that a schön subvariety of the torus is cohomologically tropical if and only if it is wunderschön and its tropicalization is a tropical homology manifold. The former property means that the open strata in the boundary of a tropical compactification are all connected and the mixed Hodge structures on their cohomology are pure of maximum possible weight; the latter property requires that, locally, the tropicalization verifies tropical Poincaré duality.
We study other properties of cohomologically tropical and wunderschön varieties, and show that in a semistable degeneration to an arrangement of cohomologically tropical varieties, the Hodge numbers of the smooth fibers are captured in the tropical cohomology of the tropicalization. This extends the results of Itenberg, Katzarkov, Mikhalkin and Zharkov.
We introduce a generating function approach to the affine Brauer and Kauffman categories, and show how it allows one to efficiently recover important sets of relations in these categories. We use this formalism to deduce restrictions on possible categorical actions and show how this recovers admissibility results that have appeared in the literature on cyclotomic Birman–Murakami–Wenzl (BMW) algebras and their degenerate versions, also known as cyclotomic Nazarov–Wenzl algebras or VW algebras.
We strengthen the maximal ergodic theorem for actions of groups of polynomial growth to a form involving jump quantity, which is the sharpest result among the family of variational or maximal ergodic theorems. As two applications, we first obtain the upcrossing inequalities with exponential decay of ergodic averages and then provide an explicit bound on the convergence rate such that the ergodic averages with strongly continuous regular group actions are metastable (or locally stable) on a large interval. Before exploiting the transference techniques, we actually obtain a stronger result—the jump estimates on a metric space with a measure not necessarily doubling. The ideas or techniques involve martingale theory, non-doubling Calderón–Zygmund theory, almost orthogonality argument, and some delicate geometric argument involving the balls and the cubes on a group equipped with a not necessarily doubling measure.
Based on the topological degree theory, we present some atypical bifurcation results in the sense of Prodi–Ambrosetti, i.e., bifurcation of T-periodic solutions from λ = 0. Finally, we propose some applications to Liénard-type equations.
Dedicated to Professor Maria Patrizia Pera on the occasion of her 70th birthday
Let Fn be the free group on $n \geq 2$ generators. We show that for all $1 \leq m \leq 2n-3$ (respectively, for all $1 \leq m \leq 2n-4$), there exists a subgroup of ${\operatorname{Aut}(F_n)}$ (respectively, ${\operatorname{Out}(F_n)}$), which has finiteness of type Fm but not of type $FP_{m+1}(\mathbb{Q})$; hence, it is not m-coherent. In both cases, the new result is the upper bound $m= 2n-3$ (respectively, $m = 2n-4$), as it cannot be obtained by embedding direct products of free noncyclic groups, and certifies higher incoherence up to the virtual cohomological dimension and is therefore sharp. As a tool of the proof, we discuss the existence and nature of multiple inequivalent extensions of a suitable finite-index subgroup K4 of ${\operatorname{Aut}(F_2)}$ (isomorphic to the quotient of the pure braid group on four strands by its centre): the fibre of four of these extensions arise from the strand-forgetting maps on the braid groups, while a fifth is related with the Cardano–Ferrari epimorphism.