GeCo GeDi Papershttps://gecogedi.dimai.unifi.it/papers/en-usFri, 11 Oct 2024 20:46:23 +0000Multipath matroids, digraph colourings, and the Tutte polynomialhttps://gecogedi.dimai.unifi.it/paper/596/L. Caputi, C. Collari, S. Di Trani.<p> We characterise the digraphs for which the multipaths, that is disjointunions of directed paths, yield a matroid. For such graphs, called MP-digraphs,we prove that the Tutte polynomial of the multipath matroid is related tocounting certain digraph colourings. Finally, we prove that, for MP-forests,the decategorification of the multipath cohomology yields a specialisation ofthe Tutte polynomial.</p>https://gecogedi.dimai.unifi.it/paper/596/On the support of measures of large entropy for polynomial-like mapshttps://gecogedi.dimai.unifi.it/paper/595/S. Bazarbaev, F. Bianchi, K. Rakhimov.<p>Let $f$ be a polynomial-like map with dominant topological degree $d_t\geq 2$and let $d_{k-1}<d_t$ be its dynamical degree of order $k-1$. We show that thesupport of every ergodic measure whose measure-theoretic entropy is strictlylarger than $\log \sqrt{d_{k-1} d_t}$ is supported on the Julia set, i.e., thesupport of the unique measure of maximal entropy $\mu$. The proof is based onthe exponential speed of convergence of the measures $d_t^{-n}(f^n)^*\delta_a$towards $\mu$, which is valid for a generic point $a$ and with a controllederror bound depending on $a$. Our proof also gives a new proof of the samestatement in the setting of endomorphisms of $\mathbb P^k(\mathbb C)$ - aresult due to de Th\'elin and Dinh - which does not rely on the existence of aGreen current.</p>https://gecogedi.dimai.unifi.it/paper/595/Pressure path metrics on parabolic families of polynomialshttps://gecogedi.dimai.unifi.it/paper/594/F. Bianchi, Y. M. He.<p>Let $\Lambda$ be a subfamily of the moduli space of degree $D\ge2$polynomials defined by a finite number of parabolic relations. Let $\Omega$ bea bounded stable component of $\Lambda$ with the property that all criticalpoints are attracted by either the persistent parabolic cycles or by attractingcycles in $\mathbb C$. We construct a positive semi-definite pressure form on$\Omega$ and show that it defines a path metric on $\Omega$. This provides acounterpart in complex dynamics of the pressure metric on cusped Hitchincomponents recently studied by Kao and Bray-Canary-Kao-Martone.</p>https://gecogedi.dimai.unifi.it/paper/594/On the \texorpdfstring{$ν$}{nu}-invariant of two-step nilmanifolds with closed \texorpdfstring{$\mathrm G_2$}{G2}-structurehttps://gecogedi.dimai.unifi.it/paper/593/A. Fino, G. Grantcharov, G. Russo.<p> For every non-vanishing spinor field on a Riemannian $7$-manifold, Crowley,Goette, and Nordstr\"om introduced the so-called $\nu$-invariant. This is aninteger modulo $48$, and can be defined in terms of Mathai--Quillen currents,harmonic spinors, and $\eta$-invariants of spin Dirac and odd-signatureoperator. We compute these data for the compact two-step nilmanifolds admittinginvariant closed $\mathrm G_2$-structures, in particular determining theharmonic spinors and relevant symmetries of the spectrum of the spin Diracoperator. We then deduce the vanishing of the $\nu$-invariants.</p>https://gecogedi.dimai.unifi.it/paper/593/A non-Standard Indefinite Einstein Solvmanifoldhttps://gecogedi.dimai.unifi.it/paper/592/F. A. Rossi.<p>We describe an example of an indefinite invariant Einstein metric on asolvmanifold which is not standard, and whose restriction on the nilradical isnondegenerate.</p>https://gecogedi.dimai.unifi.it/paper/592/A construction of Einstein solvmanifolds not based on nilsolitonshttps://gecogedi.dimai.unifi.it/paper/591/D. Conti, F. A. Rossi, R. Segnan Dalmasso.<p>We construct indefinite Einstein solvmanifolds that are standard, but not ofpseudo-Iwasawa type. Thus, the underlying Lie algebras take the form$\mathfrak{g}\rtimes_D\mathbb{R}$, where $\mathfrak{g}$ is a nilpotent Liealgebra and $D$ is a nonsymmetric derivation. Considering nonsymmetricderivations has the consequence that $\mathfrak{g}$ is not a nilsoliton, butsatisfies a more general condition. Our construction is based on the notion of nondiagonal triple on a nicediagram. We present an algorithm to classify nondiagonal triples and theassociated Einstein metrics. With the use of a computer, we obtain allsolutions up to dimension $5$, and all solutions in dimension $\leq9$ thatsatisfy an additional technical restriction. By comparing curvatures, we show that the Einstein solvmanifolds of dimension$\leq 5$ that we obtain by our construction are not isometric to a standardextension of a nilsoliton.</p>https://gecogedi.dimai.unifi.it/paper/591/Approximation results for compact Vaisman manifoldshttps://gecogedi.dimai.unifi.it/paper/590/D. Angella, M. Miceli, G. Placini.<p>We extend the Tian approximation theorem for projective manifolds to a classof complex non-K\"ahler manifolds, the so-called Vaisman manifolds. Moreprecisely, we study the problem of approximating compact regular, respectivelyquasi-regular, Vaisman metrics by metrics induced by immersions, respectivelyembeddings, into Hopf manifolds.</p>https://gecogedi.dimai.unifi.it/paper/590/Einstein warped-product manifolds and the screened Poisson equationhttps://gecogedi.dimai.unifi.it/paper/589/A. DeBenedictis, L. Lussardi, A. Pigazzini, M. Toda.<p>We study a particular type of Einstein warped-product manifold where the warping function must satisfy the homogeneous version of the screened Poisson equation. Under these assumptions, we show that the dimension of the manifold, the (constant negative) Ricci curvature and the screened parameter are related through a quadratic equation.</p><p>This is the preprint of manuscript accepted for publication in the"Contemporary Mathematics series" of the American Mathematical Society (AMS)- Book entitled: "Recent Advances in Differential Geometry and Related Areas" to appear in 2025.</p>https://gecogedi.dimai.unifi.it/paper/589/Hölder continuity and laminarity of the Green currents for Hénon-like mapshttps://gecogedi.dimai.unifi.it/paper/588/F. Bianchi, T. C. Dinh, K. Rakhimov.<p> Under a natural assumption on the dynamical degrees, we prove that the Greencurrents associated to any H\'enon-like map in any dimension have H\"oldercontinuous super-potentials, i.e., give H\"older continuous linear functionalson suitable spaces of forms and currents. As a consequence, the unique measureof maximal entropy is the Monge-Amp\`ere of a H\"older continuousplurisubharmonic function and has strictly positive Hausdorff dimension. Underthe same assumptions, we also prove that the Green currents are woven. Whenthey are of bidegree $(1,1)$, they are laminar. In particular, our resultsgeneralize results known until now only in algebraic settings, or in dimension2.</p>https://gecogedi.dimai.unifi.it/paper/588/Bridging between überhomology and double homologyhttps://gecogedi.dimai.unifi.it/paper/587/L. Caputi, D. Celoria, C. Collari.<p> We establish an isomorphism between the 0-degree \"uberhomology and thedouble homology of finite simplicial complexes, using a Mayer-Vietoris spectralsequence argument. We clarify the correspondence between these theories byproviding examples and some consequences; in particular, we show that\"uberhomology groups detect the standard simplex, and that the doublehomology's diagonal is related to the connected domination polynomial.</p>https://gecogedi.dimai.unifi.it/paper/587/Verifying feasibility of degenerate semidefinite programshttps://gecogedi.dimai.unifi.it/paper/586/V. Kolmogorov, S. Naldi, J. Zapata.<p> This paper deals with the algorithmic aspects of solving feasibility problemsof semidefinite programming (SDP), aka linear matrix inequalities (LMI). Sincein some SDP instances all feasible solutions have irrational entries, numericalsolvers that work with rational numbers can only find an approximate solution.We study the following question: is it possible to certify feasibility of agiven SDP using an approximate solution that is sufficiently close to someexact solution? Existing approaches make the assumption that there existrational feasible solutions (and use techniques such as rounding and latticereduction algorithms). We propose an alternative approach that does not need this assumption. Morespecifically, we show how to construct a system of polynomial equations whoseset of real solutions is guaranteed to have an isolated correct solution(assuming that the target exact solution is maximum-rank). This allows, inparticular, to use algorithms from real algebraic geometry for solving systemsof polynomial equations, yielding a hybrid (or symbolic-numerical) method forSDPs. We experimentally compare it with a pure symbolic method in <a href='Henrion,Naldi, Safey El Din; SIAM J. Optim., 2016'>Henrion,Naldi, Safey El Din; SIAM J. Optim., 2016</a>; the hybrid method was able tocertify feasibility of many SDP instances on which <a href='Henrion, Naldi, Safey ElDin; SIAM J. Optim., 2016'>Henrion, Naldi, Safey ElDin; SIAM J. Optim., 2016</a> failed. We argue that our approach may have otheruses, such as refining an approximate solution using methods of numericalalgebraic geometry for systems of polynomial equations.</p>https://gecogedi.dimai.unifi.it/paper/586/On the configurations of four spheres supporting the vertices of a tetrahedronhttps://gecogedi.dimai.unifi.it/paper/585/M. Longinetti, S. Naldi.<p> A reformulation of the three circles theorem of Johnson with distancecoordinates to the vertices of a triangle is explicitly represented in apolynomial system and solved by symbolic computation. A similar polynomialsystem in distance coordinates to the vertices of a tetrahedron $T \subset\mathbb{R}^3$ is introduced to represent the configurations of four spheres ofradius $R^*$, which intersect in one point, each sphere containing threevertices of $T$ but not the fourth one. This problem is related to that ofcomputing the largest value $r$ for which the set of vertices of $T$ is an$r$-body. For triangular pyramids we completely describe the set of geometricconfigurations with the required four balls of radius $R^*$. The solutionsobtained by symbolic computation show that triangular pyramids are splittedinto two different classes: in the first one $R^*$ is unique, in the second onethree values $R^*$ there exist. The first class can be itself subdivided intotwo subclasses, one of which is related to the family of $r$-bodies.</p>https://gecogedi.dimai.unifi.it/paper/585/R-hulloid of the vertices of a tetrahedronhttps://gecogedi.dimai.unifi.it/paper/584/M. Longinetti, S. Naldi, A. Venturi.<p> The $R$-hulloid, in the Euclidean space $\mathbb{R}^3$, of the set ofvertices $V$ of a tetrahedron $T$ is the mimimal closed set containing $V$ suchthat its complement is union of open balls of radius $R$. When $R$ is greaterthan the circumradius of $T$, the boundary of the $R$-hulloid consists of $V$and possibly of four spherical subsets of well defined spheres of radius $R$through the vertices of $T$. The existence of a value $R^*$ such that thesesubsets collapse into one point $O^*\not \in V$ is investigated; in such case$O^*$ is in the interior of $T$ and belongs to four spheres of radius $R^*$,each one through three vertices of $T$ and not containing the fourth one. As aconsequence, the range of $\rho$ such that $V$ is a $\rho$-body is describedcompletely. This work generalizes to three dimensions previous results, provedin the planar case and related to the three circles Johnson's Theorem.</p>https://gecogedi.dimai.unifi.it/paper/584/Cohomogeneity one RCD-spaceshttps://gecogedi.dimai.unifi.it/paper/583/D. Corro, J. Núñez-Zimbrón, J. Santos-Rodríguez.<p> We study $\mathsf{RCD}$-spaces $(X,d,\mathfrak{m})$ with group actions byisometries preserving the reference measure $\mathfrak{m}$ and whose orbitspace has dimension one, i.e. cohomogeneity one actions. To this end we prove aSlice Theorem asserting that each slice at a point is homeomorphic to anon-negatively curved $\mathsf{RCD}$-space. Under the assumption that $X$ isnon-collapsed we further show that the slices are homeomorphic to metric conesover homogeneous spaces with $\mathrm{Ric} \geq 0$. As a consequence we obtaincomplete topological structural results and a principal orbit representationtheorem. Conversely, we show how to construct new $\mathsf{RCD}$-spaces from acohomogeneity one group diagram, giving a complete description of$\mathsf{RCD}$-spaces of cohomogeneity one. As an application of these resultswe obtain the classification of cohomogeneity one, non-collapsed$\mathsf{RCD}$-spaces of essential dimension at most $4$.</p>https://gecogedi.dimai.unifi.it/paper/583/Horn maps of semi-parabolic Hénon mapshttps://gecogedi.dimai.unifi.it/paper/580/M. Astorg, F. Bianchi.<p>We prove that horn maps associated to quadratic semi-parabolic fixed pointsof H\'enon maps, first introduced by Bedford, Smillie, and Ueda, satisfy a weakform of the Ahlfors island property. As a consequence, two natural definitionsof their Julia set (the non-normality locus of the family of iterates and theclosure of the set of the repelling periodic points) coincide. As anotherconsequence, we also prove that there exist small perturbations ofsemi-parabolic H\'enon maps for which the Hausdorff dimension of the forwardJulia set $J^+$ is arbitrarily close to 4.</p>https://gecogedi.dimai.unifi.it/paper/580/Monotonicity of dynamical degrees for Hénon-like and polynomial-like mapshttps://gecogedi.dimai.unifi.it/paper/579/F. Bianchi, T. C. Dinh, K. Rakhimov.<p>We prove that, for every invertible horizontal-like map (i.e., H{\'e}non-likemap) in any dimension, the sequence of the dynamical degrees is increasinguntil that of maximal value, which is the main dynamical degree, and decreasingafter that. Similarly, for polynomial-like maps in any dimension, the sequenceof dynamical degrees is increasing until the last one, which is the topologicaldegree. This is the first time that such a property is proved outside of thealgebraic setting. Our proof is based on the construction of a suitabledeformation for positive closed currents, which relies on tools frompluripotential theory and the solution of the $d$, $\bar \partial$, and $dd^c$equations on convex domains.</p>https://gecogedi.dimai.unifi.it/paper/579/Holomorphic motions of weighted periodic pointshttps://gecogedi.dimai.unifi.it/paper/578/F. Bianchi, M. Brévard.<p> We study the holomorphic motions of repelling periodic points in stablefamilies of endomorphisms of $\mathbb P^k (\mathbb C)$. In particular, weestablish an asymptotic equidistribution of the graphs associated to suchperiodic points with respect to natural measures in the space of allholomorphic motions of points in the Julia sets.</p>https://gecogedi.dimai.unifi.it/paper/578/A Mañé-Manning formula for expanding measures for endomorphisms of $\mathbb P^k$https://gecogedi.dimai.unifi.it/paper/577/F. Bianchi, Y. M. He.<p>Let $k \ge 1$ be an integer and $f$ a holomorphic endomorphism of $\mathbbP^k (\mathbb C)$ of algebraic degree $d\geq 2$. We introduce a volume dimensionfor ergodic $f$-invariant probability measures with strictly positive Lyapunovexponents. In particular, this class of measures includes all ergodic measureswhose measure-theoretic entropy is strictly larger than $(k-1)\log d$, anatural generalization of the class of measures of positive measure-theoreticentropy in dimension 1. The volume dimension is equivalent to the Hausdorffdimension when $k=1$, but depends on the dynamics of $f$ to incorporate thepossible failure of Koebe's theorem and the non-conformality of holomorphicendomorphisms for $k\geq 2$. If $\nu$ is an ergodic $f$-invariant probability measure with strictlypositive Lyapunov exponents, we prove a generalization of the Ma\~n\'e-Manningformula relating the volume dimension, the measure-theoretic entropy, and thesum of the Lyapunov exponents of $\nu$. As a consequence, we give acharacterization of the first zero of a natural pressure function for suchexpanding measures in terms of their volume dimensions. For hyperbolic maps,such zero also coincides with the volume dimension of the Julia set, and withthe exponent of a natural (volume-)conformal measure. This generalizes resultsby Denker-Urba\'nski and McMullen in dimension 1 to any dimension $k\geq 1$. Our methods mainly rely on a theorem by Berteloot-Dupont-Molino, which givesa precise control on the distortion of inverse branches of endomorphisms alonggeneric inverse orbits with respect to measures with strictly positive Lyapunovexponents.</p>https://gecogedi.dimai.unifi.it/paper/577/Integral foliated simplicial volume and circle foliationshttps://gecogedi.dimai.unifi.it/paper/576/C. Campagnolo, D. Corro.<p> We show that the integral foliated simplicial volume of a connected compactoriented smooth manifold with a regular foliation by circles vanishes.</p>https://gecogedi.dimai.unifi.it/paper/576/Singular Riemannian Foliations, variational problems and Principles of Symmetric Criticalitieshttps://gecogedi.dimai.unifi.it/paper/575/D. Corro, Leonardo F. Cavenaghi, Marcelo K. Inagaki, Marcos M. Alexandrino.<p> A singular foliation $\mathcal{F}$ on a complete Riemannian manifold $M$ iscalled Singular Riemannian foliation (SRF for short) if its leaves are locallyequidistant, e.g., the partition of $M$ into orbits of an isometric action. Inthis paper, we investigate variational problems in compact Riemannian manifoldsequipped with SRF with special properties, e.g. isoparametric foliations, SRFon fibers bundles with Sasaki metric, and orbit-like foliations. Moreprecisely, we prove two results analogous to Palais' Principle of SymmetricCriticality, one is a general principle for $\mathcal{F}$ symmetric operatorson the Hilbert space $W^{1,2}(M)$, the other one is for $\mathcal{F}$ symmetricintegral operators on the Banach spaces $W^{1,p}(M)$. These results togetherwith a $\mathcal{F}$ version of Rellich Kondrachov Hebey Vaugon EmbeddingTheorem allow us to circumvent difficulties with Sobolev's critical exponentswhen considering applications of Calculus of Variations to find solutions toPDEs. To exemplify this we prove the existence of weak solutions to a class ofvariational problems which includes $p$-Kirschoff problems.</p>https://gecogedi.dimai.unifi.it/paper/575/