The following appears in the paper "Continuity properties of entropy" by Newhouse from 1989:

Let $M$ be some smooth Riemannian compact manifold (you may assume boundary-less), and let $f\in Diff^{1+\epsilon}(M)$ for some $\epsilon>0$ (it means that $f,f^{-1}$ are differentiable, and the differential is $\epsilon$-H"older continuous).

For all $l\in\mathbb{N}$ and $\chi>0$, define

$\Lambda_{\chi,l}:=\{x\in M:$ There is a splitting $T_xM=E^s(x)\oplus E^u(x)$, and $\forall v_u\in E^u(x),v_s\in E^s(x), n\geq0$:

$|d_xf^nv_s|\leq l e^{-\chi n}|v_s|$, $|d_xf^{-n}v_s|\geq l^{-1} e^{\chi n}|v_s|$ and

$|d_xf^{-n}v_u|\leq l e^{-\chi n}|v_u|$, $|d_xf^{n}v_s|\geq l^{-1} e^{\chi n}|v_u|$ $\}$

The following line in Newhouse's paper is that it is fairly easy to see that $E^u(x)$ change continuously for $x\in \Lambda_{\chi,l}$ on each piece with $\dim E^u(x)$ constant.

I've tried showing so myself, or finding some reference online- neither yielded results. I'm not sure I understand even what continuity in this context means (I'd rather not to go into Grassmannians or tangent bundles, and keep it analytic); therefore even a reduction to the continuity of $Jac(d_xf|_{E^u(x)})$ will be greatly appreciated. It is not clear to me if the continuity is unique to the sets $\Lambda_{\chi,l}$, or is it a property for all $\chi$-hyperbolic points, or all Lyapunov regular points, etc. As well as the regularity module- could one discuss H"older continuity, uniform continuity etc. ?

Thanks ahead for anyone contributing... I'm sorry I couldn't present some more of my own progress on this question, I'm really quite lost on this one and would really like to understand his paper.

The following appears in the paper "Continuity properties of entropy" by Newhouse from 1989:

Let $M$ be some smooth Riemannian compact manifold (you may assume boundary-less), and let $f\in Diff^{1+\epsilon}(M)$ for some $\epsilon>0$ (it means that $f,f^{-1}$ are differentiable, and the differential is $\epsilon$-H"older continuous).

For all $l\in\mathbb{N}$ and $\chi>0$, define

$\Lambda_{\chi,l}:=\{x\in M:$ There is a splitting $T_xM=E^s(x)\oplus E^u(x)$, and $\forall v_u\in E^u(x),v_s\in E^s(x), n\geq0$:

$|d_xf^nv_s|\leq l e^{-\chi n}|v_s|$, $|d_xf^{-n}v_s|\geq l^{-1} e^{\chi n}|v_s|$ and

$|d_xf^{-n}v_u|\leq l e^{-\chi n}|v_u|$, $|d_xf^{n}v_u|\geq l^{-1} e^{\chi n}|v_u|$ $\}$

The following line in Newhouse's paper is that it is fairly easy to see that $E^u(x)$ change continuously for $x\in \Lambda_{\chi,l}$ on each piece with $\dim E^u(x)$ constant.

I've tried showing so myself, or finding some reference online- neither yielded results. I'm not sure I understand even what continuity in this context means (I'd rather not to go into Grassmannians or tangent bundles, and keep it analytic); therefore even a reduction to the continuity of $Jac(d_xf|_{E^u(x)})$ will be greatly appreciated. It is not clear to me if the continuity is unique to the sets $\Lambda_{\chi,l}$, or is it a property for all $\chi$-hyperbolic points, or all Lyapunov regular points, etc. As well as the regularity module- could one discuss H"older continuity, uniform continuity etc. ?

Thanks ahead for anyone contributing... I'm sorry I couldn't present some more of my own progress on this question, I'm really quite lost on this one and would really like to understand his paper.

The following appears in the paper "Continuity properties of entropy" by Newhouse from 1989:

Let $M$ be some smooth Riemannian compact manifold (you may assume boundary-less), and let $f\in Diff^{1+\epsilon}(M)$ for some $\epsilon>0$ (it means that $f,f^{-1}$ are differentiable, and the differential is $\epsilon$-H"older continuous).

For all $l\in\mathbb{N}$ and $\chi>0$, define

$\Lambda_{\chi,l}:=\{x\in M:$ There is a splitting $T_xM=E^s(x)\oplus E^u(x)$, and $\forall v_u\in E^u(x),v_s\in E^s(x), n\geq0$:

$|d_xf^nv_s|\leq l e^{-\chi n}|v_s|$, $|d_xf^{-n}v_s|\geq l^{-1} e^{\chi n}|v_s|$ and

$|d_xf^{-n}v_u|\leq l e^{-\chi n}|v_u|$, $|d_xf^{n}v_s|\geq l^{-1} e^{\chi n}|v_u|$ $\}$

The following line in Newhouse's paper is that it is fairly easy to see that $E^u(x)$ change continuously for $x\in \Lambda_{\chi,l}$ on each piece with $\dim E^u(x)$ constant.

I've tried showing so myself, or find some reference online- neither yielded results. I'm not sure I understand even what continuity in this context means (I'd rather not go into Grassmannians or tangent bundles, and keep it analytic); therefore even a reduction to the continuity of $Jac(d_xf|_{E^u(x)})$ will be greatly appreciated. It is not clear to me if the continuity is unique to the sets $\Lambda_{\chi,l}$, or is it a property for all $\chi$-hyperbolic points, or all Lyapunov regular points, etc. As well as the regularity module- could one discuss H"older continuity, uniform continuity etc. ?

Thanks ahead for anyone contributing... I'm sorry I couldn't present some more of my own progress on this question, I'm really quite lost on this one and would really like to understand his paper.

The following appears in the paper "Continuity properties of entropy" by Newhouse from 1989:

Let $M$ be some smooth Riemannian compact manifold (you may assume boundary-less), and let $f\in Diff^{1+\epsilon}(M)$ for some $\epsilon>0$ (it means that $f,f^{-1}$ are differentiable, and the differential is $\epsilon$-H"older continuous).

For all $l\in\mathbb{N}$ and $\chi>0$, define

$\Lambda_{\chi,l}:=\{x\in M:$ There is a splitting $T_xM=E^s(x)\oplus E^u(x)$, and $\forall v_u\in E^u(x),v_s\in E^s(x), n\geq0$:

$|d_xf^nv_s|\leq l e^{-\chi n}|v_s|$, $|d_xf^{-n}v_s|\geq l^{-1} e^{\chi n}|v_s|$ and

$|d_xf^{-n}v_u|\leq l e^{-\chi n}|v_u|$, $|d_xf^{n}v_s|\geq l^{-1} e^{\chi n}|v_u|$ $\}$

The following line in Newhouse's paper is that it is fairly easy to see that $E^u(x)$ change continuously for $x\in \Lambda_{\chi,l}$ on each piece with $\dim E^u(x)$ constant.

I've tried showing so myself, or finding some reference online- neither yielded results. I'm not sure I understand even what continuity in this context means (I'd rather not to go into Grassmannians or tangent bundles, and keep it analytic); therefore even a reduction to the continuity of $Jac(d_xf|_{E^u(x)})$ will be greatly appreciated. It is not clear to me if the continuity is unique to the sets $\Lambda_{\chi,l}$, or is it a property for all $\chi$-hyperbolic points, or all Lyapunov regular points, etc. As well as the regularity module- could one discuss H"older continuity, uniform continuity etc. ?

Thanks ahead for anyone contributing... I'm sorry I couldn't present some more of my own progress on this question, I'm really quite lost on this one and would really like to understand his paper.