Pub Date : 2021-09-28DOI: 10.2422/2036-2145.202110_010
Przemysław Berk
We study the combinatorial and dynamical properties of translations surfaces with horizontal saddle connections from the point of view of backward Rauzy-Veech induction. Namely, we prove that although the horizontal saddle connections do not rule out existence of the infinite orbit under backward Rauzy-Veech algorithm, they disallow the ∞completeness of such orbit. Furthermore, we prove that if an orbit under backward Rauzy-Veech algorithm is infinite, then the minimality of the horizontal translation flow is equivalent to the eventual appearance of all horizontal saddle connections as sides of the polygonal represenation of a surface. The main goal of this note is to study the relations between horizontal saddle connections and the combinatorics of the inverse RauzyVeech algorithm for translation surfaces as well as dynamics of the horizontal translation flows. In [2] (Proposition 4.3) Marmi, Ulcigrai and Yoccoz prove that if a translation surface does not have horizontal saddle connections, then its backward Rauzy-Veech induction orbit is indefinitely well-defined and ∞-complete, that is every symbol is a backward winner infinitely many times. In the same article the authors pose a question, whether these two conditions are equivalent. We answer affirmatively to this question in Theorem 11. The proof utilizes only combinatorics and geometry of translation surfaces. However, before proving Theorem 11, we prove Proposition 7 which states that, typically, possessing horizontal saddle connections does not rule out that the backward orbit with respect to the inverse RauzyVeech algorithm is well defined. Moreover, in Theorem 12 we prove that appearance of horizontal connections as sides of polygonal representations of translations surfaces is closely tied to the minimality of the horizontal translation flow. More precisely, we show that the horizontal translation flow is minimal if and only if all (if any) horizontal saddle connections appear as sides of a polygonal representation of a surface after applying a finite number of backward Rauzy-Veech induction steps. MSC CLASSIFICATION: 37E05, 37E35 1 ar X iv :2 10 9. 13 69 1v 2 [ m at h. D S] 2 3 Fe b 20 22 P. BERK HORIZONTAL SADDLE CONNECTIONS Acknowledgments: The author would like to thank Corinna Ulcigrai for pointing out the problem and her continuous support and Frank Trujillo for many useful remarks. The research that lead to this result was supported by Swiss National Science Foundation Grant 200021 188617/1 and Narodowe Centrum Nauki Grant OPUS 142017/27/B/ST1/00078. 1. Interval exchange transformations and translations surfaces We recall first basic notions and properties related to IETs and translation surfaces. Let A be an alphabet of #A ≥ 2 elements. For more information and basic properties, including the ergodic properties of interval exchange transformations, translation surfaces and Rauzy-Veech algorithm we refer the reader e.g. to [4] and [5]. Let SA 0 :={π = (π0, π1) : A → {1, . . . ,
{"title":"Backward Rauzy-Veech algorithm and horizontal saddle connections","authors":"Przemysław Berk","doi":"10.2422/2036-2145.202110_010","DOIUrl":"https://doi.org/10.2422/2036-2145.202110_010","url":null,"abstract":"We study the combinatorial and dynamical properties of translations surfaces with horizontal saddle connections from the point of view of backward Rauzy-Veech induction. Namely, we prove that although the horizontal saddle connections do not rule out existence of the infinite orbit under backward Rauzy-Veech algorithm, they disallow the ∞completeness of such orbit. Furthermore, we prove that if an orbit under backward Rauzy-Veech algorithm is infinite, then the minimality of the horizontal translation flow is equivalent to the eventual appearance of all horizontal saddle connections as sides of the polygonal represenation of a surface. The main goal of this note is to study the relations between horizontal saddle connections and the combinatorics of the inverse RauzyVeech algorithm for translation surfaces as well as dynamics of the horizontal translation flows. In [2] (Proposition 4.3) Marmi, Ulcigrai and Yoccoz prove that if a translation surface does not have horizontal saddle connections, then its backward Rauzy-Veech induction orbit is indefinitely well-defined and ∞-complete, that is every symbol is a backward winner infinitely many times. In the same article the authors pose a question, whether these two conditions are equivalent. We answer affirmatively to this question in Theorem 11. The proof utilizes only combinatorics and geometry of translation surfaces. However, before proving Theorem 11, we prove Proposition 7 which states that, typically, possessing horizontal saddle connections does not rule out that the backward orbit with respect to the inverse RauzyVeech algorithm is well defined. Moreover, in Theorem 12 we prove that appearance of horizontal connections as sides of polygonal representations of translations surfaces is closely tied to the minimality of the horizontal translation flow. More precisely, we show that the horizontal translation flow is minimal if and only if all (if any) horizontal saddle connections appear as sides of a polygonal representation of a surface after applying a finite number of backward Rauzy-Veech induction steps. MSC CLASSIFICATION: 37E05, 37E35 1 ar X iv :2 10 9. 13 69 1v 2 [ m at h. D S] 2 3 Fe b 20 22 P. BERK HORIZONTAL SADDLE CONNECTIONS Acknowledgments: The author would like to thank Corinna Ulcigrai for pointing out the problem and her continuous support and Frank Trujillo for many useful remarks. The research that lead to this result was supported by Swiss National Science Foundation Grant 200021 188617/1 and Narodowe Centrum Nauki Grant OPUS 142017/27/B/ST1/00078. 1. Interval exchange transformations and translations surfaces We recall first basic notions and properties related to IETs and translation surfaces. Let A be an alphabet of #A ≥ 2 elements. For more information and basic properties, including the ergodic properties of interval exchange transformations, translation surfaces and Rauzy-Veech algorithm we refer the reader e.g. to [4] and [5]. Let SA 0 :={π = (π0, π1) : A → {1, . . . ,","PeriodicalId":8132,"journal":{"name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","volume":"37 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87524355","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-09-21DOI: 10.2422/2036-2145.20211_018
Rahul Gupta, A. Krishna, J. Rathore
. We prove a decomposition theorem for the cohomological Chow group of 0-cycles on the double of a quasi-projective R 1 -scheme over a field along a closed subscheme, in terms of the Chow groups, with and without modulus, of the scheme. This yields a significant generalization of the decomposition theorem of Binda-Krishna. As applications, we prove a moving lemma for Chow groups with modulus and an analogue of Bloch’s formula for 0-cycles with modulus on singular surfaces. The latter extends a previous result of Binda-Krishna-Saito.
{"title":"A decomposition theorem for 0-cycles and applications to class field theory","authors":"Rahul Gupta, A. Krishna, J. Rathore","doi":"10.2422/2036-2145.20211_018","DOIUrl":"https://doi.org/10.2422/2036-2145.20211_018","url":null,"abstract":". We prove a decomposition theorem for the cohomological Chow group of 0-cycles on the double of a quasi-projective R 1 -scheme over a field along a closed subscheme, in terms of the Chow groups, with and without modulus, of the scheme. This yields a significant generalization of the decomposition theorem of Binda-Krishna. As applications, we prove a moving lemma for Chow groups with modulus and an analogue of Bloch’s formula for 0-cycles with modulus on singular surfaces. The latter extends a previous result of Binda-Krishna-Saito.","PeriodicalId":8132,"journal":{"name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82992720","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-09-16DOI: 10.2422/2036-2145.202110_001
D. Stolyarov
We study the inequalities of the type $|int_{mathbb{R}^d} Phi(K*f)| lesssim |f|_{L_1(mathbb{R}^d)}^p$, where the kernel $K$ is homogeneous of order $alpha - d$ and possibly vector-valued, the function $Phi$ is positively $p$-homogeneous, and $p = d/(d-alpha)$. Under mild regularity assumptions on $K$ and $Phi$, we find necessary and sufficient conditions on these functions under which the inequality holds true with a uniform constant for all sufficiently regular functions $f$.
{"title":"Fractional integration of summable functions: Maz'ya's~$Phi$-inequalities","authors":"D. Stolyarov","doi":"10.2422/2036-2145.202110_001","DOIUrl":"https://doi.org/10.2422/2036-2145.202110_001","url":null,"abstract":"We study the inequalities of the type $|int_{mathbb{R}^d} Phi(K*f)| lesssim |f|_{L_1(mathbb{R}^d)}^p$, where the kernel $K$ is homogeneous of order $alpha - d$ and possibly vector-valued, the function $Phi$ is positively $p$-homogeneous, and $p = d/(d-alpha)$. Under mild regularity assumptions on $K$ and $Phi$, we find necessary and sufficient conditions on these functions under which the inequality holds true with a uniform constant for all sufficiently regular functions $f$.","PeriodicalId":8132,"journal":{"name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83137310","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-09-15DOI: 10.2422/2036-2145.202109_027
A. Tyulenev
Let $S subset mathbb{R}^{n}$ be an arbitrary nonempty compact set such that the $d$-Hausdorff content $mathcal{H}^{d}_{infty}(S)>0$ for some $d in (0,n]$. For each $p in (max{1,n-d},n]$, an almost sharp intrinsic description of the trace space $W_{p}^{1}(mathbb{R}^{n})|_{S}$ of the Sobolev space $W_{p}^{1}(mathbb{R}^{n})$ to the set $S$ is obtained. Furthermore, for each $p in (max{1,n-d},n]$ and $varepsilon in (0, min{p-(n-d),p-1})$, new bounded linear extension operators from the trace space $W_{p}^{1}(mathbb{R}^{n})|_{S}$ into the space $W_{p-varepsilon}^{1}(mathbb{R}^{n})$ are constructed.
设$S subset mathbb{R}^{n}$为任意非空紧集，使得$d$ -Hausdorff内容$mathcal{H}^{d}_{infty}(S)>0$对于某些$d in (0,n]$。对于每一个$p in (max{1,n-d},n]$，得到Sobolev空间$W_{p}^{1}(mathbb{R}^{n})$到集合$S$的迹线空间$W_{p}^{1}(mathbb{R}^{n})|_{S}$的几乎尖锐的内在描述。此外，对于每个$p in (max{1,n-d},n]$和$varepsilon in (0, min{p-(n-d),p-1})$，构造了从迹空间$W_{p}^{1}(mathbb{R}^{n})|_{S}$到迹空间$W_{p-varepsilon}^{1}(mathbb{R}^{n})$的新的有界线性扩展算子。
{"title":"Almost sharp descriptions of traces of Sobolev $W_{p}^{1}(mathbb{R}^{n})$-spaces to arbitrary compact subsets of $mathbb{R}^{n}$","authors":"A. Tyulenev","doi":"10.2422/2036-2145.202109_027","DOIUrl":"https://doi.org/10.2422/2036-2145.202109_027","url":null,"abstract":"Let $S subset mathbb{R}^{n}$ be an arbitrary nonempty compact set such that the $d$-Hausdorff content $mathcal{H}^{d}_{infty}(S)>0$ for some $d in (0,n]$. For each $p in (max{1,n-d},n]$, an almost sharp intrinsic description of the trace space $W_{p}^{1}(mathbb{R}^{n})|_{S}$ of the Sobolev space $W_{p}^{1}(mathbb{R}^{n})$ to the set $S$ is obtained. Furthermore, for each $p in (max{1,n-d},n]$ and $varepsilon in (0, min{p-(n-d),p-1})$, new bounded linear extension operators from the trace space $W_{p}^{1}(mathbb{R}^{n})|_{S}$ into the space $W_{p-varepsilon}^{1}(mathbb{R}^{n})$ are constructed.","PeriodicalId":8132,"journal":{"name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73231359","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-09-02DOI: 10.2422/2036-2145.202201_006
Ovidiu Preda, Miron Stanciu
. Vaisman’s theorem for locally conformally K¨ahler (lcK) compact manifolds states that any lcK metric on a compact complex manifold which admits a K¨ahler metric is, in fact, globally conformally K¨ahler (gcK). In this paper, we extend this theorem to compact complex spaces with singularities. PN-III-P1-1.1-TE-2019-0262, within PNCDI III. Miron Stanciu was partially supported by a grant of Ministry of Research and Inno-vation, CNCS - UEFISCDI, project no. PN-III-P4-ID-PCE-2020-0025, within PNCDI III.
{"title":"Vaisman theorem for LCK spaces","authors":"Ovidiu Preda, Miron Stanciu","doi":"10.2422/2036-2145.202201_006","DOIUrl":"https://doi.org/10.2422/2036-2145.202201_006","url":null,"abstract":". Vaisman’s theorem for locally conformally K¨ahler (lcK) compact manifolds states that any lcK metric on a compact complex manifold which admits a K¨ahler metric is, in fact, globally conformally K¨ahler (gcK). In this paper, we extend this theorem to compact complex spaces with singularities. PN-III-P1-1.1-TE-2019-0262, within PNCDI III. Miron Stanciu was partially supported by a grant of Ministry of Research and Inno-vation, CNCS - UEFISCDI, project no. PN-III-P4-ID-PCE-2020-0025, within PNCDI III.","PeriodicalId":8132,"journal":{"name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","volume":"228 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82265728","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-08DOI: 10.2422/2036-2145.202109_004
L. Hoang
This paper studies, in fine details, the long-time asymptotic behavior of decaying solutions of a general class of dissipative systems of nonlinear differential equations in complex Euclidean spaces. The forcing functions decay, as time tends to infinity, in a coherent way expressed by combinations of the exponential, power, logarithmic and iterated logarithmic functions. The decay may contain sinusoidal oscillations not only in time but also in the logarithm and iterated logarithm of time. It is proved that the decaying solutions admit corresponding asymptotic expansions, which can be constructed concretely. In the case of the real Euclidean spaces, the real-valued decaying solutions are proved to admit real-valued asymptotic expansions. Our results unite and extend the theory investigated in many previous works.
{"title":"Asymptotic expansions about infinity for solutions of nonlinear differential equations with coherently decaying forcing functions","authors":"L. Hoang","doi":"10.2422/2036-2145.202109_004","DOIUrl":"https://doi.org/10.2422/2036-2145.202109_004","url":null,"abstract":"This paper studies, in fine details, the long-time asymptotic behavior of decaying solutions of a general class of dissipative systems of nonlinear differential equations in complex Euclidean spaces. The forcing functions decay, as time tends to infinity, in a coherent way expressed by combinations of the exponential, power, logarithmic and iterated logarithmic functions. The decay may contain sinusoidal oscillations not only in time but also in the logarithm and iterated logarithm of time. It is proved that the decaying solutions admit corresponding asymptotic expansions, which can be constructed concretely. In the case of the real Euclidean spaces, the real-valued decaying solutions are proved to admit real-valued asymptotic expansions. Our results unite and extend the theory investigated in many previous works.","PeriodicalId":8132,"journal":{"name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78119985","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-06DOI: 10.2422/2036-2145.202110_006
Prashanta Garain, J. Kinnunen
Abstract. We consider mixed local and nonlocal quasilinear parabolic equations of p-Laplace type and discuss several regularity properties of weak solutions for such equations. More precisely, we establish local boundeness of weak subsolutions, local Hölder continuity of weak solutions, lower semicontinuity of weak supersolutions as well as upper semicontinuity of weak subsolutions. We also discuss the pointwise behavior of the semicontinuous representatives. Our main results are valid for sign changing solutions. Our approach is purely analytic and is based on energy estimates and the De Giorgi theory.
{"title":"On the regularity theory for mixed local and nonlocal quasilinear parabolic equations","authors":"Prashanta Garain, J. Kinnunen","doi":"10.2422/2036-2145.202110_006","DOIUrl":"https://doi.org/10.2422/2036-2145.202110_006","url":null,"abstract":"Abstract. We consider mixed local and nonlocal quasilinear parabolic equations of p-Laplace type and discuss several regularity properties of weak solutions for such equations. More precisely, we establish local boundeness of weak subsolutions, local Hölder continuity of weak solutions, lower semicontinuity of weak supersolutions as well as upper semicontinuity of weak subsolutions. We also discuss the pointwise behavior of the semicontinuous representatives. Our main results are valid for sign changing solutions. Our approach is purely analytic and is based on energy estimates and the De Giorgi theory.","PeriodicalId":8132,"journal":{"name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86351491","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-03DOI: 10.2422/2036-2145.202011_018
Viorel Vîjîitu
{"title":"Holomorphic convexity of pseudoconvex surfaces","authors":"Viorel Vîjîitu","doi":"10.2422/2036-2145.202011_018","DOIUrl":"https://doi.org/10.2422/2036-2145.202011_018","url":null,"abstract":"","PeriodicalId":8132,"journal":{"name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","volume":"93 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76564710","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-03DOI: 10.2422/2036-2145.202001_010
M. Freitas, A. Ramos, M. Santos, Daniel Rocha
{"title":"On global attractors for a nonlinear porous elastic system with fractional power in the memory term","authors":"M. Freitas, A. Ramos, M. Santos, Daniel Rocha","doi":"10.2422/2036-2145.202001_010","DOIUrl":"https://doi.org/10.2422/2036-2145.202001_010","url":null,"abstract":"","PeriodicalId":8132,"journal":{"name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78857534","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-03DOI: 10.2422/2036-2145.202010_062
Gaël Rémond
{"title":"Propriétés de la hauteur de Faltings","authors":"Gaël Rémond","doi":"10.2422/2036-2145.202010_062","DOIUrl":"https://doi.org/10.2422/2036-2145.202010_062","url":null,"abstract":"","PeriodicalId":8132,"journal":{"name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78947802","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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