Marked points on translation surfaces

Abstract We compute the gap distribution of directions of saddle connections for two classes of translation surfaces. One class will be the translation surfaces arising from gluing two identical tori along a slit. These yield the first explicit computations of gap distributions for non-lattice translation surfaces. We show that this distribution has support at zero and quadratic tail decay. We also construct examples of translation surfaces in any genus $d>1$ that have the same gap distribution as the gap distribution of two identical tori glued along a slit. The second class we consider are twice-marked tori and saddle connections between distinct marked points with a specific orientation. These results can be interpreted as the gap distribution of slopes of affine lattices. We obtain our results by translating the question of gap distributions to a dynamical question of return times to a transversal under the horocycle flow on an appropriate moduli space.

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Self-translation surfaces

Revista Matemática Contemporânea ◽

10.21711/231766361993/rmc55 ◽

1993 ◽

Vol 5 (5) ◽

Author(s):

María del Carmen Fuster ◽

Clin McGrory ◽

Juan Ballesteros

Keyword(s):

Translation Surfaces

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Inextensible Networks Deformed as Translation Surfaces

IMA Journal of Applied Mathematics ◽

10.1093/imamat/31.3.261 ◽

1983 ◽

Vol 31 (3) ◽

pp. 261-267 ◽

Cited By ~ 1

Author(s):

A. C. PIPKIN

Keyword(s):

Translation Surfaces

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Translation surfaces and their orbit closures: An introduction for a broad audience

EMS Surveys in Mathematical Sciences ◽

10.4171/emss/9 ◽

2015 ◽

Vol 2 (1) ◽

pp. 63-108 ◽

Cited By ~ 20

Author(s):

Alex Wright

Keyword(s):

Translation Surfaces ◽

Orbit Closures ◽

Broad Audience

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Weingarten Affine Translation Surfaces in Euclidean 3-Space

Results in Mathematics ◽

10.1007/s00025-017-0737-x ◽

2017 ◽

Vol 72 (4) ◽

pp. 1839-1848 ◽

Cited By ~ 1

Author(s):

Seoung Dal Jung ◽

Huili Liu ◽

Yixuan Liu

Keyword(s):

Translation Surfaces ◽

Affine Translation

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The Kontsevich–Zorich cocycle over Veech–McMullen family of symmetric translation surfaces

Journal of Modern Dynamics ◽

10.3934/jmd.2019002 ◽

2019 ◽

Vol 14 (1) ◽

pp. 21-54

Author(s):

Artur Avila ◽

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Carlos Matheus ◽

Jean-Christophe Yoccoz ◽

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...

Keyword(s):

Translation Surfaces

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Mosaics of canopy openness induced by tropical cyclones in lowland rain forests with contrasting management histories in northeastern Australia

Journal of Tropical Ecology ◽

10.1017/s0266467400001784 ◽

2000 ◽

Vol 16 (6) ◽

pp. 883-894 ◽

Cited By ~ 43

Author(s):

SIMON J. GROVE ◽

STEPHEN M. TURTON ◽

DANNY T. SIEGENTHALER

Keyword(s):

Forest Management ◽

Forest Canopy ◽

Canopy Openness ◽

Old Growth ◽

Lowland Rain Forest ◽

Forest Sites ◽

Local Topography ◽

Marked Points ◽

Impacted Site

Tropical Cyclone ‘Rona’ crossed the coast of the Daintree lowlands of northeastern Australia in 1999. This study reports on its impact on forest canopy openness at six lowland rain forest sites with contrasting management histories (old-growth, selectively logged and regrowth). Percentage canopy openness was calculated from individual hemispherical photographs taken from marked points below the forest canopy at nine plots per site 3–4 mo before the cyclone, and at the same points a month afterwards. Before the cyclone, when nine sites were visited, canopy openness in old-growth and logged sites was similar, but significantly higher in regrowth forest. After the cyclone, all six revisited sites showed an increase in canopy openness, but the increase was very patchy amongst plots and sites and varied from insignificant to severe. The most severely impacted site was an old-growth one, the least impacted a logged one. Although proneness to impact was apparently related to forest management history (old-growth being the most impacted), underlying local topography may have had an equally strong influence in this case. It was concluded that the likelihood of severe impact may be determined at the landscape-scale by the interaction of anthropogenic with meteorological, physiographic and biotic factors. In the long term, such interactions may caution against pursuing forest management in cyclone-prone areas.

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