Pub Date : 2023-10-25DOI: 10.2422/2036-2145.202301_018
Karma Dajani, Slade Sanderson
We study a one-parameter family of interval maps ${T_alpha}_{alphain[1,beta]}$, with $beta$ the golden mean, defined on $[-1,1]$ by $T_alpha(x)=beta^{1+|t|}x-tbetaalpha$ where $tin{-1,0,1}$. For each $T_alpha, alpha>1$, we construct its unique, absolutely continuous invariant measure and show that on an open, dense subset of parameters $alpha$, the corresponding density is a step function with finitely many jumps. We give an explicit description of the maximal intervals of parameters on which the density has at most the same number of jumps. A main tool in our analysis is the phenomenon of matching, where the orbits of the left and right limits of discontinuity points meet after a finite number of steps. Each $T_alpha$ generates signed expansions of numbers in base $1/beta$; via Birkhoff's ergodic theorem, the invariant measures are used to determine the asymptotic relative frequencies of digits in generic $T_alpha$-expansions. In particular, the frequency of $0$ is shown to vary continuously as a function of $alpha$ and to attain its maximum $3/4$ on the maximal interval $[1/2+1/beta,1+1/beta^2]$.
{"title":"Ergodic properties of a parameterised family of symmetric golden maps: the matching phenomenon revisited","authors":"Karma Dajani, Slade Sanderson","doi":"10.2422/2036-2145.202301_018","DOIUrl":"https://doi.org/10.2422/2036-2145.202301_018","url":null,"abstract":"We study a one-parameter family of interval maps ${T_alpha}_{alphain[1,beta]}$, with $beta$ the golden mean, defined on $[-1,1]$ by $T_alpha(x)=beta^{1+|t|}x-tbetaalpha$ where $tin{-1,0,1}$. For each $T_alpha, alpha>1$, we construct its unique, absolutely continuous invariant measure and show that on an open, dense subset of parameters $alpha$, the corresponding density is a step function with finitely many jumps. We give an explicit description of the maximal intervals of parameters on which the density has at most the same number of jumps. A main tool in our analysis is the phenomenon of matching, where the orbits of the left and right limits of discontinuity points meet after a finite number of steps. Each $T_alpha$ generates signed expansions of numbers in base $1/beta$; via Birkhoff's ergodic theorem, the invariant measures are used to determine the asymptotic relative frequencies of digits in generic $T_alpha$-expansions. In particular, the frequency of $0$ is shown to vary continuously as a function of $alpha$ and to attain its maximum $3/4$ on the maximal interval $[1/2+1/beta,1+1/beta^2]$.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"110 3-4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135168690","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-25DOI: 10.2422/2036-2145.202206_016
Nuno Arala
{"title":"Lines of polynomials with alternating Galois group","authors":"Nuno Arala","doi":"10.2422/2036-2145.202206_016","DOIUrl":"https://doi.org/10.2422/2036-2145.202206_016","url":null,"abstract":"","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"5 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135169061","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-25DOI: 10.2422/2036-2145.202304_008
Keng Hao Ooi
{"title":"Fefferman-Stein type inequalities for maximal function of Choquet integrals associated with weighted capacities","authors":"Keng Hao Ooi","doi":"10.2422/2036-2145.202304_008","DOIUrl":"https://doi.org/10.2422/2036-2145.202304_008","url":null,"abstract":"","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135169062","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-25DOI: 10.2422/2036-2145.202211_001
Jingze Zhu
In this paper, we generalize a previous result to higher dimension. We prove that uniformly 3-convex translating solitons of mean curvature flow in $mathbb{R}^{n+1}$ which arise as blow up limit of embedded, mean convex mean curvature flow must have $SO(n-1)$ symmetry.
{"title":"Rotational symmetry of uniformly 3-convex translating solitons of mean curvature flow in higher dimensions","authors":"Jingze Zhu","doi":"10.2422/2036-2145.202211_001","DOIUrl":"https://doi.org/10.2422/2036-2145.202211_001","url":null,"abstract":"In this paper, we generalize a previous result to higher dimension. We prove that uniformly 3-convex translating solitons of mean curvature flow in $mathbb{R}^{n+1}$ which arise as blow up limit of embedded, mean convex mean curvature flow must have $SO(n-1)$ symmetry.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135168513","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-25DOI: 10.2422/2036-2145.202301_015
Katrin Fässler, Andrea Pinamonti, Pietro Wald
We define the Heisenberg Kakeya maximal functions $M_{delta}f$, $00$. The proof is based on a recent variant, due to Pramanik, Yang, and Zahl, of Wolff's circular maximal function theorem for a class of planar curves related to Sogge's cinematic curvature condition. As an application of our Kakeya maximal inequality, we recover the sharp lower bound for the Hausdorff dimension of Heisenberg Kakeya sets of horizontal unit line segments in $(mathbb{H}^1,d_{mathbb{H}})$, first proven by Liu.
{"title":"Kakeya maximal inequality in the Heisenberg group","authors":"Katrin Fässler, Andrea Pinamonti, Pietro Wald","doi":"10.2422/2036-2145.202301_015","DOIUrl":"https://doi.org/10.2422/2036-2145.202301_015","url":null,"abstract":"We define the Heisenberg Kakeya maximal functions $M_{delta}f$, $0<delta<1$, by averaging over $delta$-neighborhoods of horizontal unit line segments in the Heisenberg group $mathbb{H}^1$ equipped with the Kor'{a}nyi distance $d_{mathbb{H}}$. We show that $$ |M_{delta}f|_{L^3(S^1)}leq C(varepsilon)delta^{-1/3-varepsilon}|f|_{L^3(mathbb{H}^1)},quad fin L^3(mathbb{H}^1),$$ for all $varepsilon>0$. The proof is based on a recent variant, due to Pramanik, Yang, and Zahl, of Wolff's circular maximal function theorem for a class of planar curves related to Sogge's cinematic curvature condition. As an application of our Kakeya maximal inequality, we recover the sharp lower bound for the Hausdorff dimension of Heisenberg Kakeya sets of horizontal unit line segments in $(mathbb{H}^1,d_{mathbb{H}})$, first proven by Liu.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"141 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135113502","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-25DOI: 10.2422/2036-2145.202302_009
Jonathan Chapman, Sam Chow
We study the Ramsey properties of equations $a_1P(x_1) + cdots + a_sP(x_s) = b$, where $a_1,ldots,a_s,b$ are integers, and $P$ is an integer polynomial of degree $d$. Provided there are at least $(1+o(1))d^2$ variables, we show that Rado's criterion and an intersectivity condition completely characterise which equations of this form admit monochromatic solutions with respect to an arbitrary finite colouring of the positive integers. Furthermore, we obtain a Roth-type theorem for these equations, showing that they admit non-constant solutions over any set of integers with positive upper density if and only if $b= a_1 + cdots + a_s = 0$. In addition, we establish sharp asymptotic lower bounds for the number of monochromatic/dense solutions (supersaturation).
{"title":"Generalised Rado and Roth Criteria","authors":"Jonathan Chapman, Sam Chow","doi":"10.2422/2036-2145.202302_009","DOIUrl":"https://doi.org/10.2422/2036-2145.202302_009","url":null,"abstract":"We study the Ramsey properties of equations $a_1P(x_1) + cdots + a_sP(x_s) = b$, where $a_1,ldots,a_s,b$ are integers, and $P$ is an integer polynomial of degree $d$. Provided there are at least $(1+o(1))d^2$ variables, we show that Rado's criterion and an intersectivity condition completely characterise which equations of this form admit monochromatic solutions with respect to an arbitrary finite colouring of the positive integers. Furthermore, we obtain a Roth-type theorem for these equations, showing that they admit non-constant solutions over any set of integers with positive upper density if and only if $b= a_1 + cdots + a_s = 0$. In addition, we establish sharp asymptotic lower bounds for the number of monochromatic/dense solutions (supersaturation).","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"17 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135166689","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-25DOI: 10.2422/2036-2145.202301_011
Stefan Ivanov, Alexander Petkov
We establish a new version of the CR almost Schur Lemma which gives an estimation of the pseudohermitian scalar curvature on a compact strictly pseudoconvex pseudohermitian manifold to be a constant in terms of the norm of the traceless Webster Ricci tensor and the pseudohermitian torsion under a certain positivity condition. In the torsion-free case, i.e. for a compact Sasakian manifold, our positivity condition coincides with the known one and we obtain a better estimate
{"title":"The CR almost Schur Lemma and the positivity conditions","authors":"Stefan Ivanov, Alexander Petkov","doi":"10.2422/2036-2145.202301_011","DOIUrl":"https://doi.org/10.2422/2036-2145.202301_011","url":null,"abstract":"We establish a new version of the CR almost Schur Lemma which gives an estimation of the pseudohermitian scalar curvature on a compact strictly pseudoconvex pseudohermitian manifold to be a constant in terms of the norm of the traceless Webster Ricci tensor and the pseudohermitian torsion under a certain positivity condition. In the torsion-free case, i.e. for a compact Sasakian manifold, our positivity condition coincides with the known one and we obtain a better estimate","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135166697","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-25DOI: 10.2422/2036-2145.202301_014
Alexandru Dimca, Giovanna Ilardi
Let $V:f=0$ be a hypersurface of degree $d geq 3$ in the complex projective space $mathbb{P}^n$, $n geq 3$, having only isolated singularities. Let $M(f)$ be the associated Jacobian algebra and $H: ell=0$ be a hyperplane in $mathbb{P}^n$ avoiding the singularities of $V$, but such that $V cap H$ is singular. We related the Lefschetz type properties of the linear maps $ell: M(f)_k to M(f)_{k+1}$ induced by the multiplication by linear form $ell$ to the singularities of the hyperplane section $V cap H$. Similar results are obtained for the Jacobian module $N(f)$.
{"title":"Lefschetz properties of Jacobian algebras and Jacobian modules","authors":"Alexandru Dimca, Giovanna Ilardi","doi":"10.2422/2036-2145.202301_014","DOIUrl":"https://doi.org/10.2422/2036-2145.202301_014","url":null,"abstract":"Let $V:f=0$ be a hypersurface of degree $d geq 3$ in the complex projective space $mathbb{P}^n$, $n geq 3$, having only isolated singularities. Let $M(f)$ be the associated Jacobian algebra and $H: ell=0$ be a hyperplane in $mathbb{P}^n$ avoiding the singularities of $V$, but such that $V cap H$ is singular. We related the Lefschetz type properties of the linear maps $ell: M(f)_k to M(f)_{k+1}$ induced by the multiplication by linear form $ell$ to the singularities of the hyperplane section $V cap H$. Similar results are obtained for the Jacobian module $N(f)$.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135168688","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-29DOI: 10.2422/2036-2145.202111_016
Florian Bertrand, Francine Meylan
We give an explicit construction of a key family of stationary discs attached to a nondegenerate model quadric in $mathbb{C}^N$ and derive a necessary condition for which (each lift) of those stationary discs is uniquely determined by its $1$-jet at a given point via a local diffeomorphism. This unique $1$-jet determination is a crucial step to deduce $2$-jet determination for CR automorphisms of generic real submanifolds in $mathbb{C}^N$.
{"title":"Explicit construction of stationary discs and its consequences for nondegenerate quadrics","authors":"Florian Bertrand, Francine Meylan","doi":"10.2422/2036-2145.202111_016","DOIUrl":"https://doi.org/10.2422/2036-2145.202111_016","url":null,"abstract":"We give an explicit construction of a key family of stationary discs attached to a nondegenerate model quadric in $mathbb{C}^N$ and derive a necessary condition for which (each lift) of those stationary discs is uniquely determined by its $1$-jet at a given point via a local diffeomorphism. This unique $1$-jet determination is a crucial step to deduce $2$-jet determination for CR automorphisms of generic real submanifolds in $mathbb{C}^N$.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"101 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135131566","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-13DOI: 10.2422/2036-2145.202111_006
David Marín, Jean-François Mattei, Éliane Salem
This work deals with the topological classification of singular foliation germs on $(mathbb C^{2},0)$. Working in a suitable class of foliations we fix the topological invariants given by the separatrix set, the Camacho-Sad indices and the projective holonomy representations and we prove the existence of a topological universal deformation through which every equisingular deformation uniquely factorizes up to topological conjugacy. This is done by representing the functor of topological classes of equisingular deformations of a fixed foliation. We also describe the functorial dependence of this representation with respect to the foliation.
{"title":"Topological moduli space for germs of holomorphic foliations II: universal deformations","authors":"David Marín, Jean-François Mattei, Éliane Salem","doi":"10.2422/2036-2145.202111_006","DOIUrl":"https://doi.org/10.2422/2036-2145.202111_006","url":null,"abstract":"This work deals with the topological classification of singular foliation germs on $(mathbb C^{2},0)$. Working in a suitable class of foliations we fix the topological invariants given by the separatrix set, the Camacho-Sad indices and the projective holonomy representations and we prove the existence of a topological universal deformation through which every equisingular deformation uniquely factorizes up to topological conjugacy. This is done by representing the functor of topological classes of equisingular deformations of a fixed foliation. We also describe the functorial dependence of this representation with respect to the foliation.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135126715","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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