Ruelle property: Old and New Shengjin Huo † , Michel Zinsmeister ‡ † Polytechnic University of Tianjin, P.R. of China ‡ University of Orl´ eans, France Don and John celebration, Seattle, august 20, 2019 Shengjin Huo † , Michel Zinsmeister ‡ Ruelle property: Old and New 1 / 29
Fuchsian Groups Definition A Fuchsian group is a discrete group of M¨ obius transformations acting on the unit disk. In all this talk the Fuchsian groups will be assumed to contain no elliptic elements. By discreteness, Γ( z 0 ), the orbit of z 0 ∈ D under the Fuchsian group Γ, can only accumulate on the unit circle and the set of accumulation points does not depend on z 0 ∈ D : it is called the limit set of Γ and denoted by Λ(Γ). Λ(Γ) is a closed subset of ∂ D and we say that Γ is of the first kind if Λ(Γ) = ∂ D , of the second kind otherwise. Shengjin Huo † , Michel Zinsmeister ‡ Ruelle property: Old and New 2 / 29
Fuchsian Groups Definition A Fuchsian group is a discrete group of M¨ obius transformations acting on the unit disk. In all this talk the Fuchsian groups will be assumed to contain no elliptic elements. By discreteness, Γ( z 0 ), the orbit of z 0 ∈ D under the Fuchsian group Γ, can only accumulate on the unit circle and the set of accumulation points does not depend on z 0 ∈ D : it is called the limit set of Γ and denoted by Λ(Γ). Λ(Γ) is a closed subset of ∂ D and we say that Γ is of the first kind if Λ(Γ) = ∂ D , of the second kind otherwise. Shengjin Huo † , Michel Zinsmeister ‡ Ruelle property: Old and New 2 / 29
Fuchsian Groups Definition A Fuchsian group is a discrete group of M¨ obius transformations acting on the unit disk. In all this talk the Fuchsian groups will be assumed to contain no elliptic elements. By discreteness, Γ( z 0 ), the orbit of z 0 ∈ D under the Fuchsian group Γ, can only accumulate on the unit circle and the set of accumulation points does not depend on z 0 ∈ D : it is called the limit set of Γ and denoted by Λ(Γ). Λ(Γ) is a closed subset of ∂ D and we say that Γ is of the first kind if Λ(Γ) = ∂ D , of the second kind otherwise. Shengjin Huo † , Michel Zinsmeister ‡ Ruelle property: Old and New 2 / 29
Fuchsian Groups and Riemann surfaces If Γ is a Fuchsian group then D / Γ is a Riemann surface. Conversely, if S is a Riemann surface whose universal covering is conformally equivalent to D then there exists a Fuchsian group Γ such that S is conformally equivalent to D / Γ. If S is compact, we say that Γ is co-compact and we say that Γ is co-finite if S has finite area. We say that Γ is of divergence type if � (1 − | γ (0) | ) = + ∞ , γ ∈ Γ and of convergence type otherwise. Co-finite groups are of divergence type and of the first kind. Groups of the second kind are of convergence type but the converse is not true. Shengjin Huo † , Michel Zinsmeister ‡ Ruelle property: Old and New 3 / 29
Fuchsian Groups and Riemann surfaces If Γ is a Fuchsian group then D / Γ is a Riemann surface. Conversely, if S is a Riemann surface whose universal covering is conformally equivalent to D then there exists a Fuchsian group Γ such that S is conformally equivalent to D / Γ. If S is compact, we say that Γ is co-compact and we say that Γ is co-finite if S has finite area. We say that Γ is of divergence type if � (1 − | γ (0) | ) = + ∞ , γ ∈ Γ and of convergence type otherwise. Co-finite groups are of divergence type and of the first kind. Groups of the second kind are of convergence type but the converse is not true. Shengjin Huo † , Michel Zinsmeister ‡ Ruelle property: Old and New 3 / 29
Fuchsian Groups and Riemann surfaces If Γ is a Fuchsian group then D / Γ is a Riemann surface. Conversely, if S is a Riemann surface whose universal covering is conformally equivalent to D then there exists a Fuchsian group Γ such that S is conformally equivalent to D / Γ. If S is compact, we say that Γ is co-compact and we say that Γ is co-finite if S has finite area. We say that Γ is of divergence type if � (1 − | γ (0) | ) = + ∞ , γ ∈ Γ and of convergence type otherwise. Co-finite groups are of divergence type and of the first kind. Groups of the second kind are of convergence type but the converse is not true. Shengjin Huo † , Michel Zinsmeister ‡ Ruelle property: Old and New 3 / 29
Fuchsian Groups and Riemann surfaces If Γ is a Fuchsian group then D / Γ is a Riemann surface. Conversely, if S is a Riemann surface whose universal covering is conformally equivalent to D then there exists a Fuchsian group Γ such that S is conformally equivalent to D / Γ. If S is compact, we say that Γ is co-compact and we say that Γ is co-finite if S has finite area. We say that Γ is of divergence type if � (1 − | γ (0) | ) = + ∞ , γ ∈ Γ and of convergence type otherwise. Co-finite groups are of divergence type and of the first kind. Groups of the second kind are of convergence type but the converse is not true. Shengjin Huo † , Michel Zinsmeister ‡ Ruelle property: Old and New 3 / 29
Fuchsian Groups and Riemann surfaces If Γ is a Fuchsian group then D / Γ is a Riemann surface. Conversely, if S is a Riemann surface whose universal covering is conformally equivalent to D then there exists a Fuchsian group Γ such that S is conformally equivalent to D / Γ. If S is compact, we say that Γ is co-compact and we say that Γ is co-finite if S has finite area. We say that Γ is of divergence type if � (1 − | γ (0) | ) = + ∞ , γ ∈ Γ and of convergence type otherwise. Co-finite groups are of divergence type and of the first kind. Groups of the second kind are of convergence type but the converse is not true. Shengjin Huo † , Michel Zinsmeister ‡ Ruelle property: Old and New 3 / 29
Examples A Fuchsian group Γ is of convergence type iff S = D / Γ has a Green function, which is also equivalent to saying that Brownian motion on S is transient. So, if S = C \ E , where E is a closed set, the corresponding Fuchsian group is of convergence type iff E is non-polar (= with logarithmic capacity > 0). Notice also that it is of the second-kind only if E has a nontrivial connected component. Shengjin Huo † , Michel Zinsmeister ‡ Ruelle property: Old and New 4 / 29
Examples A Fuchsian group Γ is of convergence type iff S = D / Γ has a Green function, which is also equivalent to saying that Brownian motion on S is transient. So, if S = C \ E , where E is a closed set, the corresponding Fuchsian group is of convergence type iff E is non-polar (= with logarithmic capacity > 0). Notice also that it is of the second-kind only if E has a nontrivial connected component. Shengjin Huo † , Michel Zinsmeister ‡ Ruelle property: Old and New 4 / 29
Examples A Fuchsian group Γ is of convergence type iff S = D / Γ has a Green function, which is also equivalent to saying that Brownian motion on S is transient. So, if S = C \ E , where E is a closed set, the corresponding Fuchsian group is of convergence type iff E is non-polar (= with logarithmic capacity > 0). Notice also that it is of the second-kind only if E has a nontrivial connected component. Shengjin Huo † , Michel Zinsmeister ‡ Ruelle property: Old and New 4 / 29
Examples Let S be an abelian covering by Z n of a compact surface (we call such a surface a jungle gym after Sullivan): the corresponding Fuchsian group is of the first kind and of convergent type iff n ≥ 3 because this is the range for the usual random walk on Z n to be transient. Shengjin Huo † , Michel Zinsmeister ‡ Ruelle property: Old and New 5 / 29
The Teichm¨ uller Space Let S , S ′ be two quasiconformally equivalent Riemann surfaces and f : S → S ′ a quasiconformal homeomorphism: if Γ is the Fuchsian group uniformizing S then f has a quasiconformal lift to D whose complex dilatation µ = ∂ f z / ∂ f ∂ z belongs to ∂ ¯ M (Γ) = { µ ∈ L ∞ ( D ) , � µ � ∞ < 1 , ; ∀ γ ∈ Γ , µ = µ ◦ γ γ ′ γ ′ } . Conversely, if µ ∈ M (Γ), there exists a quasiconformal homeomorphism f µ of the disk onto itself with dilatation µ : f µ conjugates Γ to a Fuchsian group Γ µ and descends to a quasiconformal homeomorphism between S and D / Γ µ . If now f , g are two quasiconformal homeomorphisms from S onto S ′ , S ′′ lifting to f µ , f ν respectively then g − 1 ◦ f is homotopic to a conformal isomorphism between S ′ and S ′′ iff f µ = f ν on ∂ D . Shengjin Huo † , Michel Zinsmeister ‡ Ruelle property: Old and New 6 / 29
The Teichm¨ uller Space Let S , S ′ be two quasiconformally equivalent Riemann surfaces and f : S → S ′ a quasiconformal homeomorphism: if Γ is the Fuchsian group uniformizing S then f has a quasiconformal lift to D whose complex dilatation µ = ∂ f z / ∂ f ∂ z belongs to ∂ ¯ M (Γ) = { µ ∈ L ∞ ( D ) , � µ � ∞ < 1 , ; ∀ γ ∈ Γ , µ = µ ◦ γ γ ′ γ ′ } . Conversely, if µ ∈ M (Γ), there exists a quasiconformal homeomorphism f µ of the disk onto itself with dilatation µ : f µ conjugates Γ to a Fuchsian group Γ µ and descends to a quasiconformal homeomorphism between S and D / Γ µ . If now f , g are two quasiconformal homeomorphisms from S onto S ′ , S ′′ lifting to f µ , f ν respectively then g − 1 ◦ f is homotopic to a conformal isomorphism between S ′ and S ′′ iff f µ = f ν on ∂ D . Shengjin Huo † , Michel Zinsmeister ‡ Ruelle property: Old and New 6 / 29
Recommend
More recommend
