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¨obiustransformations acting on the unit disk.

In all this talk the Fuchsian groups will be assumed to contain no elliptic elements.

By discreteness,Γ( z0), the orbit of z0 ∈ D under the Fuchsian group Γ, can only accumulate on the unit circle and the set of accumulation points does not depend on z0 ∈ 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¨obiustransformations acting on the unit disk.

In all this talk the Fuchsian groups will be assumed to contain no elliptic elements.

By discreteness,Γ( z0), the orbit of z0 ∈ D under the Fuchsian group Γ, can only accumulate on the unit circle and the set of accumulation points does not depend on z0 ∈ 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¨obiustransformations acting on the unit disk.

In all this talk the Fuchsian groups will be assumed to contain no elliptic elements.

By discreteness,Γ( z0), the orbit of z0 ∈ D under the Fuchsian group Γ, can only accumulate on the unit circle and the set of accumulation points does not depend on z0 ∈ 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 X (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 X (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 X (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 X (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 X (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

n Let S be an abelian covering by Z 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 n for the usual random walk on Z to be transient.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 5 / 29 The Teichm¨ullerSpace

Let S, S0 be two quasiconformally equivalent Riemann surfaces and f : S → S0 a quasiconformal homeomorphism: if Γ is the Fuchsian group uniformizing S then f has a quasiconformal lift to D whose ∂f ∂f complex dilatation µ = ∂z¯/ ∂z belongs to

0 ∞ γ M(Γ) = {µ ∈ L (D), kµk∞ < 1, ; ∀γ ∈ Γ, µ = µ ◦ γ }. γ0

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 groupΓ µ and descends to a µ quasiconformal homeomorphism between S and D/Γ . If now f , g are two quasiconformal homeomorphisms from S onto S0, S00 lifting to f µ, f ν respectively then g −1 ◦ f is homotopic to a 0 00 µ ν conformal isomorphism between S and S iff f = f on ∂D.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 6 / 29 The Teichm¨ullerSpace

Let S, S0 be two quasiconformally equivalent Riemann surfaces and f : S → S0 a quasiconformal homeomorphism: if Γ is the Fuchsian group uniformizing S then f has a quasiconformal lift to D whose ∂f ∂f complex dilatation µ = ∂z¯/ ∂z belongs to

0 ∞ γ M(Γ) = {µ ∈ L (D), kµk∞ < 1, ; ∀γ ∈ Γ, µ = µ ◦ γ }. γ0

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 groupΓ µ and descends to a µ quasiconformal homeomorphism between S and D/Γ . If now f , g are two quasiconformal homeomorphisms from S onto S0, S00 lifting to f µ, f ν respectively then g −1 ◦ f is homotopic to a 0 00 µ ν conformal isomorphism between S and S iff f = f on ∂D.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 6 / 29 The Teichm¨ullerSpace

Let S, S0 be two quasiconformally equivalent Riemann surfaces and f : S → S0 a quasiconformal homeomorphism: if Γ is the Fuchsian group uniformizing S then f has a quasiconformal lift to D whose ∂f ∂f complex dilatation µ = ∂z¯/ ∂z belongs to

0 ∞ γ M(Γ) = {µ ∈ L (D), kµk∞ < 1, ; ∀γ ∈ Γ, µ = µ ◦ γ }. γ0

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 groupΓ µ and descends to a µ quasiconformal homeomorphism between S and D/Γ . If now f , g are two quasiconformal homeomorphisms from S onto S0, S00 lifting to f µ, f ν respectively then g −1 ◦ f is homotopic to a 0 00 µ ν conformal isomorphism between S and S iff f = f on ∂D.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 6 / 29 The Teichm¨ullerSpace

Let S, S0 be two quasiconformally equivalent Riemann surfaces and f : S → S0 a quasiconformal homeomorphism: if Γ is the Fuchsian group uniformizing S then f has a quasiconformal lift to D whose ∂f ∂f complex dilatation µ = ∂z¯/ ∂z belongs to

0 ∞ γ M(Γ) = {µ ∈ L (D), kµk∞ < 1, ; ∀γ ∈ Γ, µ = µ ◦ γ }. γ0

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 groupΓ µ and descends to a µ quasiconformal homeomorphism between S and D/Γ . If now f , g are two quasiconformal homeomorphisms from S onto S0, S00 lifting to f µ, f ν respectively then g −1 ◦ f is homotopic to a 0 00 µ ν conformal isomorphism between S and S iff f = f on ∂D.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 6 / 29 The Teichm¨ullerSpace

We can then define the Teichm¨ullerspace T (Γ) as either the set of homeomorphisms of the circle {f µ, µ ∈ M(Γ)} or the set of equivalence classes {[µ]} where µ ∼ ν in M(Γ) iff f µ and f ν agree on the unit circle.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 7 / 29 Mostow property

If Gamma is co-finite then the one-dimensional version of Mostow rigidity theorem asserts that if [µ] is in T (Γ) then either f µ is singular wrt Lebesgue measure or it is a M¨obiustransformation. This theorem has been generalized to divergent type groups by Agard and Kuusalo. Theorem (Astala-Z.) Mostow property does not hold when Γ is of convergent type.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 8 / 29 Mostow property

If Gamma is co-finite then the one-dimensional version of Mostow rigidity theorem asserts that if [µ] is in T (Γ) then either f µ is singular wrt Lebesgue measure or it is a M¨obiustransformation. This theorem has been generalized to divergent type groups by Agard and Kuusalo. Theorem (Astala-Z.) Mostow property does not hold when Γ is of convergent type.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 8 / 29 Mostow property

If Gamma is co-finite then the one-dimensional version of Mostow rigidity theorem asserts that if [µ] is in T (Γ) then either f µ is singular wrt Lebesgue measure or it is a M¨obiustransformation. This theorem has been generalized to divergent type groups by Agard and Kuusalo. Theorem (Astala-Z.) Mostow property does not hold when Γ is of convergent type.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 8 / 29 Mostow property

If Gamma is co-finite then the one-dimensional version of Mostow rigidity theorem asserts that if [µ] is in T (Γ) then either f µ is singular wrt Lebesgue measure or it is a M¨obiustransformation. This theorem has been generalized to divergent type groups by Agard and Kuusalo. Theorem (Astala-Z.) Mostow property does not hold when Γ is of convergent type.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 8 / 29 Mostow property Idea of proof: In the convergent case there exist µ’s in M(Γ) such that |µ(z)|2 dzdz¯ 1 − |z| is a Carleson measure, and in this case it is known that f µ is absolutely continuous (even equivalent) to Lebesgue measure.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 9 / 29 Mostow property Idea of proof: In the convergent case there exist µ’s in M(Γ) such that |µ(z)|2 dzdz¯ 1 − |z| is a Carleson measure, and in this case it is known that f µ is absolutely continuous (even equivalent) to Lebesgue measure.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 9 / 29 Bers embedding

Let Γ be a Fuchsian group and µ ∈ M(Γ) we extend µ to the whole plane by setting µ = 0 outside the unit disk and call fµ the plane quasiconformal homeomorphism with dilatation this extended µ and normalized by fixing 0, 1, ∞.

fµ conjugates Γ to a Kleinian groupΓ µ whose limit set is included in the quasicircle image of the unit circle by fµ: we say that Γµ is a quasi-Fuchsian group.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 10 / 29 Bers embedding

Let Γ be a Fuchsian group and µ ∈ M(Γ) we extend µ to the whole plane by setting µ = 0 outside the unit disk and call fµ the plane quasiconformal homeomorphism with dilatation this extended µ and normalized by fixing 0, 1, ∞.

fµ conjugates Γ to a Kleinian groupΓ µ whose limit set is included in the quasicircle image of the unit circle by fµ: we say that Γµ is a quasi-Fuchsian group.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 10 / 29 Bers embedding

Let Γ be a Fuchsian group and µ ∈ M(Γ) we extend µ to the whole plane by setting µ = 0 outside the unit disk and call fµ the plane quasiconformal homeomorphism with dilatation this extended µ and normalized by fixing 0, 1, ∞.

fµ conjugates Γ to a Kleinian groupΓ µ whose limit set is included in the quasicircle image of the unit circle by fµ: we say that Γµ is a quasi-Fuchsian group.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 10 / 29 Bers embedding

µ fµ depends only on [µ] and f is the conformal welding of the above defined quasicircle. If f µ descends to a quasiconformal homeomorphism between two Riemann surfaces then the above quasifuchsian group simultaneously uniformizes S and S0. Bers embedding allows to equip T (Γ) with a complex structure.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 11 / 29 Bers embedding

µ fµ depends only on [µ] and f is the conformal welding of the above defined quasicircle. If f µ descends to a quasiconformal homeomorphism between two Riemann surfaces then the above quasifuchsian group simultaneously uniformizes S and S0. Bers embedding allows to equip T (Γ) with a complex structure.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 11 / 29 Bers embedding

µ fµ depends only on [µ] and f is the conformal welding of the above defined quasicircle. If f µ descends to a quasiconformal homeomorphism between two Riemann surfaces then the above quasifuchsian group simultaneously uniformizes S and S0. Bers embedding allows to equip T (Γ) with a complex structure.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 11 / 29 The Bowen property

Bowen has proven that if Γ is co-finite then, if µ ∈ M(Γ), either Λ(Γµ) is a circle or else its Hausdorff dimension is > 1. We say that co-finite groups have Bowen property. What are the first kind groups having Bowen’s property?

Theorem (Astala-Z.) No first kind convergence type group has Bowen property.

Idea of proof: it is a continuation of the idea for Mostow property. Let µ ∈ M(Γ) be such that |µ|2/(1 − |z|)dzdz¯ is a Carleson measure with small norm then Λ(Γµ) is a chord-arc curve, thus of Hausdorff dimension 1. Theorem (Bishop) All divergence-type groups have Bowen property.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 12 / 29 The Bowen property

Bowen has proven that if Γ is co-finite then, if µ ∈ M(Γ), either Λ(Γµ) is a circle or else its Hausdorff dimension is > 1. We say that co-finite groups have Bowen property. What are the first kind groups having Bowen’s property?

Theorem (Astala-Z.) No first kind convergence type group has Bowen property.

Idea of proof: it is a continuation of the idea for Mostow property. Let µ ∈ M(Γ) be such that |µ|2/(1 − |z|)dzdz¯ is a Carleson measure with small norm then Λ(Γµ) is a chord-arc curve, thus of Hausdorff dimension 1. Theorem (Bishop) All divergence-type groups have Bowen property.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 12 / 29 The Bowen property

Bowen has proven that if Γ is co-finite then, if µ ∈ M(Γ), either Λ(Γµ) is a circle or else its Hausdorff dimension is > 1. We say that co-finite groups have Bowen property. What are the first kind groups having Bowen’s property?

Theorem (Astala-Z.) No first kind convergence type group has Bowen property.

Idea of proof: it is a continuation of the idea for Mostow property. Let µ ∈ M(Γ) be such that |µ|2/(1 − |z|)dzdz¯ is a Carleson measure with small norm then Λ(Γµ) is a chord-arc curve, thus of Hausdorff dimension 1. Theorem (Bishop) All divergence-type groups have Bowen property.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 12 / 29 The Bowen property

Bowen has proven that if Γ is co-finite then, if µ ∈ M(Γ), either Λ(Γµ) is a circle or else its Hausdorff dimension is > 1. We say that co-finite groups have Bowen property. What are the first kind groups having Bowen’s property?

Theorem (Astala-Z.) No first kind convergence type group has Bowen property.

Idea of proof: it is a continuation of the idea for Mostow property. Let µ ∈ M(Γ) be such that |µ|2/(1 − |z|)dzdz¯ is a Carleson measure with small norm then Λ(Γµ) is a chord-arc curve, thus of Hausdorff dimension 1. Theorem (Bishop) All divergence-type groups have Bowen property.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 12 / 29 The Bowen property

Bowen has proven that if Γ is co-finite then, if µ ∈ M(Γ), either Λ(Γµ) is a circle or else its Hausdorff dimension is > 1. We say that co-finite groups have Bowen property. What are the first kind groups having Bowen’s property?

Theorem (Astala-Z.) No first kind convergence type group has Bowen property.

Idea of proof: it is a continuation of the idea for Mostow property. Let µ ∈ M(Γ) be such that |µ|2/(1 − |z|)dzdz¯ is a Carleson measure with small norm then Λ(Γµ) is a chord-arc curve, thus of Hausdorff dimension 1. Theorem (Bishop) All divergence-type groups have Bowen property.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 12 / 29 Ruelle property

Ruelle has proven that if Γ is a co-compact Fuchsian group then the application [µ] 7→ HD(Λ(Γµ)) is real-analytic on T (Γ). What are the Fuchsian groups which have Ruelle’s property?

Theorem (Astala,Z.) Ruelle property fails for the 3d-jungle gym and for the Riemann surface C\E where E ⊂ R has Carleson property:

∃c > 0; ∀x ∈ E, ∀r > 0, |E ∩ (x − r, x + r)| ≥ cr.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 13 / 29 Ruelle property

Ruelle has proven that if Γ is a co-compact Fuchsian group then the application [µ] 7→ HD(Λ(Γµ)) is real-analytic on T (Γ). What are the Fuchsian groups which have Ruelle’s property?

Theorem (Astala,Z.) Ruelle property fails for the 3d-jungle gym and for the Riemann surface C\E where E ⊂ R has Carleson property:

∃c > 0; ∀x ∈ E, ∀r > 0, |E ∩ (x − r, x + r)| ≥ cr.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 13 / 29 Ruelle property

Ruelle has proven that if Γ is a co-compact Fuchsian group then the application [µ] 7→ HD(Λ(Γµ)) is real-analytic on T (Γ). What are the Fuchsian groups which have Ruelle’s property?

Theorem (Astala,Z.) Ruelle property fails for the 3d-jungle gym and for the Riemann surface C\E where E ⊂ R has Carleson property:

∃c > 0; ∀x ∈ E, ∀r > 0, |E ∩ (x − r, x + r)| ≥ cr.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 13 / 29 Conical and Escaping Set

If Γ is a Fuchsian or quasi-Fuchsian group we define its conical limit setΛ c (Γ) as the set of geodesics in the limit set that return infinitely often to a fixed compact set, its complement being the escaping set Λe (Γ). −1 If Γ is a Fuchsian group, µ ∈ M(Γ), and Γµ = fµ ◦ Γ ◦ fµ then Λc (Γµ) = fµ(Λc (Γ)),Λ e (Γµ) = fµ(Λe (Γ)).

The Hausdorff dimension of Λc (Γµ) is the Poincar´eexponent

X s δ(Γµ) = inf{s : d(γ(0), Λ(Γµ)) < +∞}. γ∈Γ

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 14 / 29 Conical and Escaping Set

If Γ is a Fuchsian or quasi-Fuchsian group we define its conical limit setΛ c (Γ) as the set of geodesics in the limit set that return infinitely often to a fixed compact set, its complement being the escaping set Λe (Γ). −1 If Γ is a Fuchsian group, µ ∈ M(Γ), and Γµ = fµ ◦ Γ ◦ fµ then Λc (Γµ) = fµ(Λc (Γ)),Λ e (Γµ) = fµ(Λe (Γ)).

The Hausdorff dimension of Λc (Γµ) is the Poincar´eexponent

X s δ(Γµ) = inf{s : d(γ(0), Λ(Γµ)) < +∞}. γ∈Γ

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 14 / 29 Conical and Escaping Set

If Γ is a Fuchsian or quasi-Fuchsian group we define its conical limit setΛ c (Γ) as the set of geodesics in the limit set that return infinitely often to a fixed compact set, its complement being the escaping set Λe (Γ). −1 If Γ is a Fuchsian group, µ ∈ M(Γ), and Γµ = fµ ◦ Γ ◦ fµ then Λc (Γµ) = fµ(Λc (Γ)),Λ e (Γµ) = fµ(Λe (Γ)).

The Hausdorff dimension of Λc (Γµ) is the Poincar´eexponent

X s δ(Γµ) = inf{s : d(γ(0), Λ(Γµ)) < +∞}. γ∈Γ

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 14 / 29 Return to Astala-Z. theorem

The Poincar´eexponent is lower semi-continuous: if a sequence of quasi-Fuchsian groups Γn converges to Γ, meanning that for every γ ∈ Γ, γ(0) is a limit of elements in ∪nΓn(0), then

lim inf δ(Γn) ≥ δ(Γ). n→∞

Theorem (Huo-Z.) Let Γ be a convergence type first kind group: then Γ fails to have Ruelle property.

To prove this theorem it suffices to produce µ ∈ M(Γ) such that 2 |µ| /(1 − |z|)dzdz¯ is a Carleson measure and δ(Γµ) > 1, since then HD(Λ(Γtµ)) > 1 for t = 1 while it is=1 on a neighbourhood of 0.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 15 / 29 Return to Astala-Z. theorem

The Poincar´eexponent is lower semi-continuous: if a sequence of quasi-Fuchsian groups Γn converges to Γ, meanning that for every γ ∈ Γ, γ(0) is a limit of elements in ∪nΓn(0), then

lim inf δ(Γn) ≥ δ(Γ). n→∞

Theorem (Huo-Z.) Let Γ be a convergence type first kind group: then Γ fails to have Ruelle property.

To prove this theorem it suffices to produce µ ∈ M(Γ) such that 2 |µ| /(1 − |z|)dzdz¯ is a Carleson measure and δ(Γµ) > 1, since then HD(Λ(Γtµ)) > 1 for t = 1 while it is=1 on a neighbourhood of 0.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 15 / 29 Return to Astala-Z. theorem

The Poincar´eexponent is lower semi-continuous: if a sequence of quasi-Fuchsian groups Γn converges to Γ, meanning that for every γ ∈ Γ, γ(0) is a limit of elements in ∪nΓn(0), then

lim inf δ(Γn) ≥ δ(Γ). n→∞

Theorem (Huo-Z.) Let Γ be a convergence type first kind group: then Γ fails to have Ruelle property.

To prove this theorem it suffices to produce µ ∈ M(Γ) such that 2 |µ| /(1 − |z|)dzdz¯ is a Carleson measure and δ(Γµ) > 1, since then HD(Λ(Γtµ)) > 1 for t = 1 while it is=1 on a neighbourhood of 0.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 15 / 29 Proof of H-Z. theorem

We first notice that S = D/Γ may be deformed to a four-punctured sphere whose Teichm¨uller space is non trivial. By lower semi-continuity, Bowen property for cofinite groups and the fact that Λ(Γ) = Λc (Γ) for such groups, one can find µ ∈ M(Γ) such that δ(Γµ) > 1.

We next exhaust the fundamental domain E of Γ by compact sets Kn

and define µn ∈ M(Γ) such that µn|E = µ1Kn . LetΓ n = Γµn : by lower semi-continuity, δ(Γn) > 1 for large n. P If Γ is a convergence group then γ∈Γ(1 − |γ(0)|)δγ(0) is a Carleson measure and this implies that any µ ∈ M(Γ) with compact support has the Carleson measure property.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 16 / 29 Proof of H-Z. theorem

We first notice that S = D/Γ may be deformed to a four-punctured sphere whose Teichm¨uller space is non trivial. By lower semi-continuity, Bowen property for cofinite groups and the fact that Λ(Γ) = Λc (Γ) for such groups, one can find µ ∈ M(Γ) such that δ(Γµ) > 1.

We next exhaust the fundamental domain E of Γ by compact sets Kn

and define µn ∈ M(Γ) such that µn|E = µ1Kn . LetΓ n = Γµn : by lower semi-continuity, δ(Γn) > 1 for large n. P If Γ is a convergence group then γ∈Γ(1 − |γ(0)|)δγ(0) is a Carleson measure and this implies that any µ ∈ M(Γ) with compact support has the Carleson measure property.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 16 / 29 Proof of H-Z. theorem

We first notice that S = D/Γ may be deformed to a four-punctured sphere whose Teichm¨uller space is non trivial. By lower semi-continuity, Bowen property for cofinite groups and the fact that Λ(Γ) = Λc (Γ) for such groups, one can find µ ∈ M(Γ) such that δ(Γµ) > 1.

We next exhaust the fundamental domain E of Γ by compact sets Kn

and define µn ∈ M(Γ) such that µn|E = µ1Kn . LetΓ n = Γµn : by lower semi-continuity, δ(Γn) > 1 for large n. P If Γ is a convergence group then γ∈Γ(1 − |γ(0)|)δγ(0) is a Carleson measure and this implies that any µ ∈ M(Γ) with compact support has the Carleson measure property.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 16 / 29 A theorem by Bishop

Is Ruelle property true for for 1 or 2 − d (i.e. divergent) jungle gyms?

Definition A Riemann surface S = D/Γ is said to have big deformations near infinity if ∀ε, δ > 0, for any compact set K ⊂ S there exists µ ∈ M(Γ) supported in S\K such that kµk∞ ≤ 1 − ε and δ(Γµ) ≥ 1 + ε.

By the previous dicussions and the translation invariance of the gyms, it is clear that these surfaces have big deformations near ∞.

Theorem (Bishop) With very few exceptions, if S has big deformations near ∞ then Γ fails to have Ruelle property.

The 1 − d and 2 − d jungle gyms fall in the scope of this theorem.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 17 / 29 A theorem by Bishop

Is Ruelle property true for for 1 or 2 − d (i.e. divergent) jungle gyms?

Definition A Riemann surface S = D/Γ is said to have big deformations near infinity if ∀ε, δ > 0, for any compact set K ⊂ S there exists µ ∈ M(Γ) supported in S\K such that kµk∞ ≤ 1 − ε and δ(Γµ) ≥ 1 + ε.

By the previous dicussions and the translation invariance of the gyms, it is clear that these surfaces have big deformations near ∞.

Theorem (Bishop) With very few exceptions, if S has big deformations near ∞ then Γ fails to have Ruelle property.

The 1 − d and 2 − d jungle gyms fall in the scope of this theorem.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 17 / 29 A theorem by Bishop

Is Ruelle property true for for 1 or 2 − d (i.e. divergent) jungle gyms?

Definition A Riemann surface S = D/Γ is said to have big deformations near infinity if ∀ε, δ > 0, for any compact set K ⊂ S there exists µ ∈ M(Γ) supported in S\K such that kµk∞ ≤ 1 − ε and δ(Γµ) ≥ 1 + ε.

By the previous dicussions and the translation invariance of the gyms, it is clear that these surfaces have big deformations near ∞.

Theorem (Bishop) With very few exceptions, if S has big deformations near ∞ then Γ fails to have Ruelle property.

The 1 − d and 2 − d jungle gyms fall in the scope of this theorem.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 17 / 29 A theorem by Bishop

Is Ruelle property true for for 1 or 2 − d (i.e. divergent) jungle gyms?

Definition A Riemann surface S = D/Γ is said to have big deformations near infinity if ∀ε, δ > 0, for any compact set K ⊂ S there exists µ ∈ M(Γ) supported in S\K such that kµk∞ ≤ 1 − ε and δ(Γµ) ≥ 1 + ε.

By the previous dicussions and the translation invariance of the gyms, it is clear that these surfaces have big deformations near ∞.

Theorem (Bishop) With very few exceptions, if S has big deformations near ∞ then Γ fails to have Ruelle property.

The 1 − d and 2 − d jungle gyms fall in the scope of this theorem.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 17 / 29 A theorem by Bishop

Is Ruelle property true for for 1 or 2 − d (i.e. divergent) jungle gyms?

Definition A Riemann surface S = D/Γ is said to have big deformations near infinity if ∀ε, δ > 0, for any compact set K ⊂ S there exists µ ∈ M(Γ) supported in S\K such that kµk∞ ≤ 1 − ε and δ(Γµ) ≥ 1 + ε.

By the previous dicussions and the translation invariance of the gyms, it is clear that these surfaces have big deformations near ∞.

Theorem (Bishop) With very few exceptions, if S has big deformations near ∞ then Γ fails to have Ruelle property.

The 1 − d and 2 − d jungle gyms fall in the scope of this theorem.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 17 / 29 Second-kind Groups

By a theorem of Anderson and Rocha all finitely generated second-kind Fuchsian groups without parabolics have Ruelle’s property. What about infinitely generated second kind Fuchsian groups?

Theorem (Huo,Z.) Let Γ be a convergence-type first-kind group (for example a 3-d gym): let L be a closed geodesic not disconnecting S = D/Γ. We construct a new Riemann surface S0 by cutting S along L and gluing instead two funnels. Then the corresponding second-kind Fuchsian group fails to have Ruelle property.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 18 / 29 Second-kind Groups

By a theorem of Anderson and Rocha all finitely generated second-kind Fuchsian groups without parabolics have Ruelle’s property. What about infinitely generated second kind Fuchsian groups?

Theorem (Huo,Z.) Let Γ be a convergence-type first-kind group (for example a 3-d gym): let L be a closed geodesic not disconnecting S = D/Γ. We construct a new Riemann surface S0 by cutting S along L and gluing instead two funnels. Then the corresponding second-kind Fuchsian group fails to have Ruelle property.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 18 / 29 Second-kind Groups

By a theorem of Anderson and Rocha all finitely generated second-kind Fuchsian groups without parabolics have Ruelle’s property. What about infinitely generated second kind Fuchsian groups?

Theorem (Huo,Z.) Let Γ be a convergence-type first-kind group (for example a 3-d gym): let L be a closed geodesic not disconnecting S = D/Γ. We construct a new Riemann surface S0 by cutting S along L and gluing instead two funnels. Then the corresponding second-kind Fuchsian group fails to have Ruelle property.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 18 / 29 Sketch of Proof

As before we start by noticing that there exists a compactly supported µ ∈ M(Γ) such that δ(Γµ) > 1.

We then use the convergence type to prove that HD(Λe (Γµ)) = 1 by saying that there must exist one point in S such that the measure of the set of geodesics starting at z and NOT passing through L is > 0.

By a result of Bishop and Jones, for all t ∈ [0, 1], HD(Λe (Γtµ)) = 1. We then conclude by what preceeds and by the fact that µ has Carleson property that

HD(Λ(Γµ)) > 1 but HD(Λc (Γtµ) ≤ 1 and HD(Λe (Γtµ)) = 1

for small t.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 19 / 29 Sketch of Proof

As before we start by noticing that there exists a compactly supported µ ∈ M(Γ) such that δ(Γµ) > 1.

We then use the convergence type to prove that HD(Λe (Γµ)) = 1 by saying that there must exist one point in S such that the measure of the set of geodesics starting at z and NOT passing through L is > 0.

By a result of Bishop and Jones, for all t ∈ [0, 1], HD(Λe (Γtµ)) = 1. We then conclude by what preceeds and by the fact that µ has Carleson property that

HD(Λ(Γµ)) > 1 but HD(Λc (Γtµ) ≤ 1 and HD(Λe (Γtµ)) = 1

for small t.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 19 / 29 Sketch of Proof

As before we start by noticing that there exists a compactly supported µ ∈ M(Γ) such that δ(Γµ) > 1.

We then use the convergence type to prove that HD(Λe (Γµ)) = 1 by saying that there must exist one point in S such that the measure of the set of geodesics starting at z and NOT passing through L is > 0.

By a result of Bishop and Jones, for all t ∈ [0, 1], HD(Λe (Γtµ)) = 1. We then conclude by what preceeds and by the fact that µ has Carleson property that

HD(Λ(Γµ)) > 1 but HD(Λc (Γtµ) ≤ 1 and HD(Λe (Γtµ)) = 1

for small t.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 19 / 29 Sketch of Proof

As before we start by noticing that there exists a compactly supported µ ∈ M(Γ) such that δ(Γµ) > 1.

We then use the convergence type to prove that HD(Λe (Γµ)) = 1 by saying that there must exist one point in S such that the measure of the set of geodesics starting at z and NOT passing through L is > 0.

By a result of Bishop and Jones, for all t ∈ [0, 1], HD(Λe (Γtµ)) = 1. We then conclude by what preceeds and by the fact that µ has Carleson property that

HD(Λ(Γµ)) > 1 but HD(Λc (Γtµ) ≤ 1 and HD(Λe (Γtµ)) = 1

for small t.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 19 / 29 Sketch of Proof

Theorem (Huo,Z.) The last theorem is actually true for all infinite area surfaces S.

The proof follows the same lines except that now, in the case of divergence type it is no longer true that the set Λe (Γ) has positive measure. We use instead a result of Fernandez and Melian asserting that, even if of measure 0, this set has dimension 1.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 20 / 29 Sketch of Proof

Theorem (Huo,Z.) The last theorem is actually true for all infinite area surfaces S.

The proof follows the same lines except that now, in the case of divergence type it is no longer true that the set Λe (Γ) has positive measure. We use instead a result of Fernandez and Melian asserting that, even if of measure 0, this set has dimension 1.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 20 / 29 Bowen’s formula How are Ruelle and Anderson-Rocha results proven? Identify the dynamics as a repelling IFS. Example: consider a linear IFS consisting of f1, ..fp mapping say a square Q to squares Qj ⊂ Q. The dimension of the corresponding Cantor set is the unique t such that p X 0 t |fi | = 1. i=1 if the IFS is only supposed to be conformal one refines the IFS at order n: to any word ω = j1..jn in the alphabet {f1, ..fp} we associate the map

fω = fjn ◦ fjn−1 ◦ ..f1

and define Pn(t) by: X 0 t exp(nPn(t)) = |fω(z0)| ω . Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 21 / 29 Bowen’s formula How are Ruelle and Anderson-Rocha results proven? Identify the dynamics as a repelling IFS. Example: consider a linear IFS consisting of f1, ..fp mapping say a square Q to squares Qj ⊂ Q. The dimension of the corresponding Cantor set is the unique t such that p X 0 t |fi | = 1. i=1 if the IFS is only supposed to be conformal one refines the IFS at order n: to any word ω = j1..jn in the alphabet {f1, ..fp} we associate the map

fω = fjn ◦ fjn−1 ◦ ..f1

and define Pn(t) by: X 0 t exp(nPn(t)) = |fω(z0)| ω . Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 21 / 29 Bowen’s formula How are Ruelle and Anderson-Rocha results proven? Identify the dynamics as a repelling IFS. Example: consider a linear IFS consisting of f1, ..fp mapping say a square Q to squares Qj ⊂ Q. The dimension of the corresponding Cantor set is the unique t such that p X 0 t |fi | = 1. i=1 if the IFS is only supposed to be conformal one refines the IFS at order n: to any word ω = j1..jn in the alphabet {f1, ..fp} we associate the map

fω = fjn ◦ fjn−1 ◦ ..f1

and define Pn(t) by: X 0 t exp(nPn(t)) = |fω(z0)| ω . Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 21 / 29 Bowen’s formula How are Ruelle and Anderson-Rocha results proven? Identify the dynamics as a repelling IFS. Example: consider a linear IFS consisting of f1, ..fp mapping say a square Q to squares Qj ⊂ Q. The dimension of the corresponding Cantor set is the unique t such that p X 0 t |fi | = 1. i=1 if the IFS is only supposed to be conformal one refines the IFS at order n: to any word ω = j1..jn in the alphabet {f1, ..fp} we associate the map

fω = fjn ◦ fjn−1 ◦ ..f1

and define Pn(t) by: X 0 t exp(nPn(t)) = |fω(z0)| ω . Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 21 / 29 Bowen’s formula How are Ruelle and Anderson-Rocha results proven? Identify the dynamics as a repelling IFS. Example: consider a linear IFS consisting of f1, ..fp mapping say a square Q to squares Qj ⊂ Q. The dimension of the corresponding Cantor set is the unique t such that p X 0 t |fi | = 1. i=1 if the IFS is only supposed to be conformal one refines the IFS at order n: to any word ω = j1..jn in the alphabet {f1, ..fp} we associate the map

fω = fjn ◦ fjn−1 ◦ ..f1

and define Pn(t) by: X 0 t exp(nPn(t)) = |fω(z0)| ω . Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 21 / 29 Bowen’s formula

It can be shown that Pn(t) converges as n → ∞ to the so-called pressure fonction P(t) and Bowen’s formula asserts that the dimension of the limit set of the IFS is the only zero of the function P. The proof of Bowen’s formula then follows from thermodynamical formalism.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 22 / 29 Bowen’s formula

It can be shown that Pn(t) converges as n → ∞ to the so-called pressure fonction P(t) and Bowen’s formula asserts that the dimension of the limit set of the IFS is the only zero of the function P. The proof of Bowen’s formula then follows from thermodynamical formalism.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 22 / 29 Parabolic points

In the same paper mentionned in the introduction, Ruelle showed that in any hyperbolic component the dimension of the Julia set of z2 + c varies real-analytically with c. In a paper with H.Akter, M.Urbanski proved that the same is true in some open set of λ0s for the family

z 7→ z(1 − z − λz2)

which possesses a persistent parabolic point.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 23 / 29 Parabolic points

In the same paper mentionned in the introduction, Ruelle showed that in any hyperbolic component the dimension of the Julia set of z2 + c varies real-analytically with c. In a paper with H.Akter, M.Urbanski proved that the same is true in some open set of λ0s for the family

z 7→ z(1 − z − λz2)

which possesses a persistent parabolic point.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 23 / 29 Parabolic points

In the same paper mentionned in the introduction, Ruelle showed that in any hyperbolic component the dimension of the Julia set of z2 + c varies real-analytically with c. In a paper with H.Akter, M.Urbanski proved that the same is true in some open set of λ0s for the family

z 7→ z(1 − z − λz2)

which possesses a persistent parabolic point.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 23 / 29 Parabolic points

A similar phenomenon occurs for the family fσ of Lavaurs maps after parabolic implosion for the family z2 + c when c → 1/4: M.Urbanski and Z. also proved in this case real-analyticity of the dimension despite the persitence of a parabolic point.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 24 / 29 Parabolic points

A similar phenomenon occurs for the family fσ of Lavaurs maps after parabolic implosion for the family z2 + c when c → 1/4: M.Urbanski and Z. also proved in this case real-analyticity of the dimension despite the persitence of a parabolic point.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 24 / 29 The co-finite case

The situation of a compact surface with finitely many punctures is very similar to these examples. Bowen and Series have proven that for all co-finite Fuchsian group there is a Markov map from the circle to itself which is peace-wise made of elements of the group and which represents the dynamics. In case of punctures they have built a subset K of the circle such that the first return to K is a repeller. problem: the alphabet becomes infinite.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 25 / 29 The co-finite case

The situation of a compact surface with finitely many punctures is very similar to these examples. Bowen and Series have proven that for all co-finite Fuchsian group there is a Markov map from the circle to itself which is peace-wise made of elements of the group and which represents the dynamics. In case of punctures they have built a subset K of the circle such that the first return to K is a repeller. problem: the alphabet becomes infinite.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 25 / 29 The co-finite case

The situation of a compact surface with finitely many punctures is very similar to these examples. Bowen and Series have proven that for all co-finite Fuchsian group there is a Markov map from the circle to itself which is peace-wise made of elements of the group and which represents the dynamics. In case of punctures they have built a subset K of the circle such that the first return to K is a repeller. problem: the alphabet becomes infinite.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 25 / 29 The co-finite case

The situation of a compact surface with finitely many punctures is very similar to these examples. Bowen and Series have proven that for all co-finite Fuchsian group there is a Markov map from the circle to itself which is peace-wise made of elements of the group and which represents the dynamics. In case of punctures they have built a subset K of the circle such that the first return to K is a repeller. problem: the alphabet becomes infinite.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 25 / 29 Sketch of Proof

Let θ be the smallest real number such that

X 0 t |fj (z0)| < ∞

for t > θ. the IFS is said to be regular if

X 0 t limt→θ |fj (z0)| = +∞.

Mauldin and Urbanski have shown that for a regular IFS the Bowen machinery holds.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 26 / 29 Sketch of Proof

Let θ be the smallest real number such that

X 0 t |fj (z0)| < ∞

for t > θ. the IFS is said to be regular if

X 0 t limt→θ |fj (z0)| = +∞.

Mauldin and Urbanski have shown that for a regular IFS the Bowen machinery holds.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 26 / 29 Sketch of Proof

Let θ be the smallest real number such that

X 0 t |fj (z0)| < ∞

for t > θ. the IFS is said to be regular if

X 0 t limt→θ |fj (z0)| = +∞.

Mauldin and Urbanski have shown that for a regular IFS the Bowen machinery holds.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 26 / 29 The co-finite case

The Bowen-Series Markov map gives rise to a regular IFS with Theta = 1/2. It follows that every finietly generated Fuchsian group has Ruelle’s property.

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 27 / 29 Generalization

Theorem

There exists a sequence (xn) of reals converging to ∞ such that the Riemann surface C\{xn} has Ruelle’s property.

Notice that at the opposite C\Z fails to have Ruelle property: this follows from Bishop’s results and the fact that C\Z is a Z−cover of the thrice-punctured sphere. This is the first example of an infinitely generated group with Ruelle property. Question: does there exist a Cantor set K ∈ C such that C\K has Ruelle property?

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 28 / 29 Generalization

Theorem

There exists a sequence (xn) of reals converging to ∞ such that the Riemann surface C\{xn} has Ruelle’s property.

Notice that at the opposite C\Z fails to have Ruelle property: this follows from Bishop’s results and the fact that C\Z is a Z−cover of the thrice-punctured sphere. This is the first example of an infinitely generated group with Ruelle property. Question: does there exist a Cantor set K ∈ C such that C\K has Ruelle property?

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 28 / 29 THANKS FOR YOUR ATTENTION

Shengjin Huo†, Michel Zinsmeister‡ Ruelle property: Old and New 29 / 29