Sharpness for C¹ linearization of planar hyperbolic diffeomorphisms

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Planar hyperbolic diffeomorphisms can be referred to two cases: Poincaré domain (both eigenvalues lie inside the unit circle S¹) and Siegel domain (one eigenvalue inside S¹ but the other outside S¹). In Poincaré domain it was proved that C¹,α smoothness with α₀:=1-|λ₂|/|λ₁|<α≤ 1, where λ₁ and λ₂ are both eigenvalues such that 0<|λ₁|<|λ₂|<1, admits C¹ linearization and the linearization is actually C¹,β. While a sharp Hölder exponent β>0 is given, an interesting problem is: Is the exponent α₀ also sharp? On the other hand, in Siegel domain we only know that C¹,α smoothness with α∈ (0,1] admits C¹ linearization. In this paper we further study the sharpness for C¹ linearization in both cases.