Weil–Petersson metrics, Manhattan curves and Hausdorff dimension

Pollicott, Mark; Sharp, Richard

doi:10.1007/s00209-015-1575-8

Cited by 5 publications

(8 citation statements)

References 18 publications

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“…We prove that when we look at a path in T (S), the variation of corresponding Manhattan curves contains information on the pressure metric. As similar result has been proved by Pollicott and Sharp [PS16] when S is a closed surface. We generalize it to surfaces with punctures.…”

Section: Remark 13supporting

confidence: 86%

“…When S has no punctures, results in this work are not new. Manhattan curves and rigidity results are, for instance, discussed in [Bur93,Sha98], and the pressure metric on T (S) is discovered in [McM08] and further investigated in [PS16,BCS18]. Nevertheless, when S has punctures, especially when Fuchsian representations are not convex co-compact, far fewer results along this line are proved.…”

Section: Introductionmentioning

confidence: 99%

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Pressure metrics and Manhattan curves for Teichmüller spaces of punctured surfaces

Kao

2020

Isr. J. Math.

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In this paper, we extend the construction of pressure metrics to Teichmüller spaces of surfaces with punctures. This construction recovers Thurston's Riemannian metric on Teichmüller spaces. Moreover, we prove the real analyticity and convexity of Manhattan curves of finite area typepreserving Fuchsian representations, and thus we obtain several related entropy rigidity results. Lastly, relating the two topics mentioned above, we show that one can derive the pressure metric by varying Manhattan curves.

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Section: Remark 13supporting

confidence: 86%

Section: Introductionmentioning

confidence: 99%

Pressure metrics and Manhattan curves for Teichmüller spaces of punctured surfaces

Kao

2020

Isr. J. Math.

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show abstract

“…We prove that when we look at a path in T (S), the variation of corresponding Manhattan curves contains information of the pressure metric. Similar result has been proved by Pollicott and Sharp [PS16] when S is a closed surface. We generalize it to surfaces with punctures.…”

Section: Introductionsupporting

confidence: 86%

“…When S has no puncture, results in this work are not new. Manhattan curves and rigidity results are, for instance, discussed in [Bur93,Sha98], and the pressure metric on T (S) is discovered in [McM08] and further investigated in [PS16,BCS18]. Nevertheless, when S has punctures, especially when Fuchsian representations are not convex co-compact, much less results along this line are proved.…”

Section: Introductionmentioning

confidence: 99%

Pressure metrics and Manhattan curves for Teichmüller spaces of punctured surfaces

Kao¹

2019

Preprint

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In this paper, we extend the construction of pressure metrics to Teichmüller spaces of surfaces with punctures. This construction recovers Thurston's Riemannian metric on Teichmüller spaces. Moreover, we prove the real analyticity and convexity of Manhattan curves of finite area type-preserving Fuchsian representations, and thus we obtain several related entropy rigidity results. Lastly, relating the two topics mentioned above, we show that one can derive the pressure metric by varying Manhattan curves.

show abstract

“…One must compare this with Giulietti, Kloeckner, Lopes, and Marcon [23], a study of thermodynamical formalism in a geometric framework. See also Pollicott and Sharp [46] and Bridgeman, Canary and Sambarino [9] and the Weil-Petersson metric in the infinite-dimensional Teichmüler space in Takhtajan and Teo [55].…”

Section: Pressure Pseudo-metric On the Topological Classmentioning

confidence: 99%

Birkhoff sums as distributions II: Applications to deformations of dynamical systems

Grotta-Ragazzo,

Smania

2021

Preprint

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Often topological classes of one-dimensional dynamical systems are finite codimension smooth manifolds. We describe a method to prove this sort of statement that we believe can be applied in many settings. In this work we will implement it for piecewise expanding maps. The most important step will be the identification of infinitesimal deformations with primitives of Birkhoff sums (up to addition of a Lipschitz function), that allows us to use the ergodic properties of piecewise expanding maps to study the regularity of infinitesimal deformations.Contents 42 References 43

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Weil–Petersson metrics, Manhattan curves and Hausdorff dimension

Cited by 5 publications

References 18 publications

Pressure metrics and Manhattan curves for Teichmüller spaces of punctured surfaces

Pressure metrics and Manhattan curves for Teichmüller spaces of punctured surfaces

Pressure metrics and Manhattan curves for Teichmüller spaces of punctured surfaces

Birkhoff sums as distributions II: Applications to deformations of dynamical systems

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