Pub Date : 2025-01-01Epub Date: 2024-12-26DOI: 10.1007/s11784-024-01154-3
Lev Buhovsky, Shira Tanny
We study a local-to-global inequality for spectral invariants of Hamiltonians whose supports have a "large enough" disjoint tubular neighborhood on semipositive symplectic manifolds. As a corollary, we deduce this inequality for disjointly supported Hamiltonians that are -small (when fixing the supports). In particular, we present the first examples of such an inequality when the Hamiltonians are not necessarily supported in domains with contact-type boundaries, or when the ambient manifold is irrational. This extends a series of previous works studying locality phenomena of spectral invariants [9, 13, 15, 20, 25, 27]. A main new tool is a lower bound, in the spirit of Sikorav, for the energy of Floer trajectories that cross the tubular neighborhood against the direction of the negative-gradient vector field.
{"title":"A local-to-global inequality for spectral invariants and an energy dichotomy for Floer trajectories.","authors":"Lev Buhovsky, Shira Tanny","doi":"10.1007/s11784-024-01154-3","DOIUrl":"https://doi.org/10.1007/s11784-024-01154-3","url":null,"abstract":"<p><p>We study a local-to-global inequality for spectral invariants of Hamiltonians whose supports have a \"large enough\" disjoint tubular neighborhood on semipositive symplectic manifolds. As a corollary, we deduce this inequality for disjointly supported Hamiltonians that are <math><msup><mi>C</mi> <mn>0</mn></msup> </math> -small (when fixing the supports). In particular, we present the first examples of such an inequality when the Hamiltonians are not necessarily supported in domains with contact-type boundaries, or when the ambient manifold is irrational. This extends a series of previous works studying locality phenomena of spectral invariants [9, 13, 15, 20, 25, 27]. A main new tool is a lower bound, in the spirit of Sikorav, for the energy of Floer trajectories that cross the tubular neighborhood against the direction of the negative-gradient vector field.</p>","PeriodicalId":93461,"journal":{"name":"Journal of fixed point theory and Its applications","volume":"27 1","pages":"3"},"PeriodicalIF":0.0,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11671577/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142904291","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01Epub Date: 2021-09-24DOI: 10.1007/s11784-021-00899-5
Alessio Pellegrini
Let M be a closed manifold and a polytope. For each , we define a Novikov chain complex with a multiple finiteness condition encoded by the polytope . The resulting polytope Novikov homology generalizes the ordinary Novikov homology. We prove that any two cohomology classes in a prescribed polytope give rise to chain homotopy equivalent polytope Novikov complexes over a Novikov ring associated with said polytope. As applications, we present a novel approach to the (twisted) Novikov Morse Homology Theorem and prove a new polytope Novikov Principle. The latter generalizes the ordinary Novikov Principle and a recent result of Pajitnov in the abelian case.
{"title":"Polytope Novikov homology.","authors":"Alessio Pellegrini","doi":"10.1007/s11784-021-00899-5","DOIUrl":"https://doi.org/10.1007/s11784-021-00899-5","url":null,"abstract":"<p><p>Let <i>M</i> be a closed manifold and <math><mrow><mi>A</mi> <mo>⊆</mo> <msubsup><mi>H</mi> <mi>dR</mi> <mn>1</mn></msubsup> <mrow><mo>(</mo> <mi>M</mi> <mo>)</mo></mrow> </mrow> </math> a polytope. For each <math><mrow><mi>a</mi> <mo>∈</mo> <mi>A</mi></mrow> </math> , we define a Novikov chain complex with a multiple finiteness condition encoded by the polytope <math><mi>A</mi></math> . The resulting polytope Novikov homology generalizes the ordinary Novikov homology. We prove that any two cohomology classes in a prescribed polytope give rise to chain homotopy equivalent polytope Novikov complexes over a Novikov ring associated with said polytope. As applications, we present a novel approach to the (twisted) Novikov Morse Homology Theorem and prove a new polytope Novikov Principle. The latter generalizes the ordinary Novikov Principle and a recent result of Pajitnov in the abelian case.</p>","PeriodicalId":93461,"journal":{"name":"Journal of fixed point theory and Its applications","volume":"23 4","pages":"62"},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8591789/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"39642939","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01Epub Date: 2021-09-06DOI: 10.1007/s11784-021-00896-8
Armando W Gutiérrez, Anders Karlsson
This note discusses some aspects of the asymptotic behaviour of nonexpansive maps. Using metric functionals, we make a connection to the invariant subspace problem and prove a new result for nonexpansive maps of . We also point out some inaccurate assertions appearing in the literature on this topic.
{"title":"Comments on the cosmic convergence of nonexpansive maps.","authors":"Armando W Gutiérrez, Anders Karlsson","doi":"10.1007/s11784-021-00896-8","DOIUrl":"https://doi.org/10.1007/s11784-021-00896-8","url":null,"abstract":"<p><p>This note discusses some aspects of the asymptotic behaviour of nonexpansive maps. Using metric functionals, we make a connection to the invariant subspace problem and prove a new result for nonexpansive maps of <math><msup><mi>ℓ</mi> <mn>1</mn></msup> </math> . We also point out some inaccurate assertions appearing in the literature on this topic.</p>","PeriodicalId":93461,"journal":{"name":"Journal of fixed point theory and Its applications","volume":"23 4","pages":"59"},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8591709/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"39732365","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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