From curves to currents
Many natural real-valued functions of closed curves are known to extend continuously to the larger space of geodesic currents. For instance, the extension of length with respect to a fixed hyperbolic metric was a motivating example for the development of geodesic currents. We give a simple criterion...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Cambridge University Press
2021-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509421000682/type/journal_article |
| _version_ | 1811156206529544192 |
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| author | Dídac Martínez-Granado Dylan P. Thurston |
| author_facet | Dídac Martínez-Granado Dylan P. Thurston |
| author_sort | Dídac Martínez-Granado |
| collection | DOAJ |
| description | Many natural real-valued functions of closed curves are known to extend continuously to the larger space of geodesic currents. For instance, the extension of length with respect to a fixed hyperbolic metric was a motivating example for the development of geodesic currents. We give a simple criterion on a curve function that guarantees a continuous extension to geodesic currents. The main condition of our criterion is the smoothing property, which has played a role in the study of systoles of translation lengths for Anosov representations. It is easy to see that our criterion is satisfied for almost all known examples of continuous functions on geodesic currents, such as nonpositively curved lengths or stable lengths for surface groups, while also applying to new examples like extremal length. We use this extension to obtain a new curve counting result for extremal length. |
| first_indexed | 2024-04-10T04:47:38Z |
| format | Article |
| id | doaj.art-db814d14d3694cc98a5a6142342d2192 |
| institution | Directory Open Access Journal |
| issn | 2050-5094 |
| language | English |
| last_indexed | 2024-04-10T04:47:38Z |
| publishDate | 2021-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj.art-db814d14d3694cc98a5a6142342d21922023-03-09T12:34:52ZengCambridge University PressForum of Mathematics, Sigma2050-50942021-01-01910.1017/fms.2021.68From curves to currentsDídac Martínez-Granado0https://orcid.org/0000-0002-1692-2144Dylan P. Thurston1https://orcid.org/0000-0003-4610-9792Department of Mathematics, University of California, Davis, One Shields Ave, Davis, 95616, USA; E-mail:Department of Mathematics, Indiana University, 831 East 3rd St., Bloomington, 47405, USA; E-mail:Many natural real-valued functions of closed curves are known to extend continuously to the larger space of geodesic currents. For instance, the extension of length with respect to a fixed hyperbolic metric was a motivating example for the development of geodesic currents. We give a simple criterion on a curve function that guarantees a continuous extension to geodesic currents. The main condition of our criterion is the smoothing property, which has played a role in the study of systoles of translation lengths for Anosov representations. It is easy to see that our criterion is satisfied for almost all known examples of continuous functions on geodesic currents, such as nonpositively curved lengths or stable lengths for surface groups, while also applying to new examples like extremal length. We use this extension to obtain a new curve counting result for extremal length.https://www.cambridge.org/core/product/identifier/S2050509421000682/type/journal_articlegeodesic currentsextremal lengthcurve counting57M5037E30 |
| spellingShingle | Dídac Martínez-Granado Dylan P. Thurston From curves to currents Forum of Mathematics, Sigma geodesic currents extremal length curve counting 57M50 37E30 |
| title | From curves to currents |
| title_full | From curves to currents |
| title_fullStr | From curves to currents |
| title_full_unstemmed | From curves to currents |
| title_short | From curves to currents |
| title_sort | from curves to currents |
| topic | geodesic currents extremal length curve counting 57M50 37E30 |
| url | https://www.cambridge.org/core/product/identifier/S2050509421000682/type/journal_article |
| work_keys_str_mv | AT didacmartinezgranado fromcurvestocurrents AT dylanpthurston fromcurvestocurrents |