literatures beforehand in such a comprehensive way. The main results of this
chapter play a key role in proving Theorem 4.1 and its extension in §9. In
fact, we generalize the approach developed for simple saddles in [7] (related
to logarithmic singularity type) to multi-saddle type.

Let $M^{\prime}\subset M$ be a minimal component of a locally Hamiltonian flow
$\psi_{\mathbb{R}}$ associated with a closed 1-form $\eta$. Let $I\subset
M^{\prime}$ be its transversal curve equipped with a standard parametrization.
Recall that a parametrization of curve $\gamma:[a,b]\to M$ is _standard_ if
$\gamma:I\to M$ if $\eta(d\gamma)=1$. In the standard coordinates, the first
return map $T:I\to I$ is an IET.

For every saddle $\sigma\in\mathrm{Fix}(\psi_{\mathbb{R}})$ of multiplicity
$m=m_{\sigma}\geq 2$ let $(x,y)$ be a singular chart in a neighborhood
$U_{\sigma}$ of $\sigma$. Then the corresponding local Hamiltonian is of the
form $H(x,y)=\Im(x+iy)^{m}$. If the $\psi_{\mathbb{R}}$-invariant area form
$\omega=V(x,y)dx\wedge dy$, then the corresponding local Hamiltonian equation
in $U_{\sigma}$ is of the form

$\frac{dx}{dt}=\frac{\frac{\partial H}{\partial
y}(x,y)}{V(x,y)}=\frac{m\Re(x+iy)^{m-1}}{V(x,y)}\quad\text{and}\quad\frac{dy}{dt}=-\frac{\frac{\partial
H}{\partial x}(x,y)}{V(x,y)}=-\frac{m\Im(x+iy)^{m-1}}{V(x,y)},$

so

(8.1) $X(x,y)=X_{1}(x,y)+iX_{2}(x,y)=\frac{m\overline{(x+iy)^{m-1}}}{V(x,y)}$

and

$\eta=m\Im(x+iy)^{m-1}\,dx+m\Re(x+iy)^{m-1}\,dy.$

Therefore, a $C^{1}$-curve $\gamma:[a,b]\to U_{\sigma}$ is standard if and
only if

(8.2)
$\displaystyle\begin{split}1=\eta_{\gamma(t)}\gamma^{\prime}(t)&=m\Im(\gamma(t))^{m-1}\Re\gamma^{\prime}(t)+m\Re(\gamma(t))^{m-1}\Im\gamma^{\prime}(t)\\\
&=\Im\big{(}m(\gamma(t))^{m-1}\gamma^{\prime}(t)\big{)}=\Im\left(\frac{d}{dt}(\gamma(t))^{m}\right).\end{split}$

For every $f\in C^{m}(M)$ and any
$\alpha=(\alpha_{1},\alpha_{2})\in{\mathbb{Z}}_{\geq 0}^{2}$ with
$|\alpha|=\alpha_{1}+\alpha_{2}\leq m$ let
$\partial_{\sigma}^{\alpha}(f)=\frac{\partial^{|\alpha|}(f\cdot
V)}{\partial^{\alpha_{1}}x\partial^{\alpha_{2}}y}(0,0)$.

######  Lemma 8.1.

For every $f\in C^{m}(M)$ and any $\alpha\in{\mathbb{Z}}_{\geq 0}^{2}$ with
$|\alpha|\leq m-2$ we have

$\partial_{\sigma}^{\alpha}(f)=\partial_{\sigma}^{\alpha}(f\circ\psi_{t})\text{
for every }t\in{\mathbb{R}}.$

###### Proof.

First note that for every $(x,y)\in U_{\sigma}\cap\psi_{-t}(U_{\sigma})$ we
have

$\displaystyle\frac{d}{dt}((f\cdot V)\circ\psi_{t})(x,y)$
$\displaystyle=\frac{\frac{\partial(f\cdot V)}{\partial
x}(\psi_{t}(x,y))}{V(\psi_{t}(x,y))}(V\cdot X_{1})(\psi_{t}(x,y))$
$\displaystyle\quad+\frac{\frac{\partial(f\cdot V)}{\partial
y}(\psi_{t}(x,y))}{V(\psi_{t}(x,y))}(V\cdot X_{2})(\psi_{t}(x,y)).$

Therefore, by induction

$\displaystyle\frac{d}{dt}\frac{\partial^{|\alpha|}}{\partial^{\alpha_{1}}x\partial^{\alpha_{2}}y}((f\cdot
V)\circ\psi_{t})(x,y)=\frac{\partial^{|\alpha|}}{\partial^{\alpha_{1}}x\partial^{\alpha_{2}}y}\frac{d}{dt}((f\cdot
V)\circ\psi_{t})(x,y)$
$\displaystyle=\sum_{|\beta|\leq|\alpha|}W_{\beta,1}(t,x,y)\frac{\partial^{|\beta|}}{\partial^{\beta_{1}}x\partial^{\beta_{2}}y}(V\cdot
X_{1})(\psi_{t}(x,y))$
$\displaystyle\qquad+\sum_{|\beta|\leq|\alpha|}W_{\beta,2}(t,x,y)\frac{\partial^{|\beta|}}{\partial^{\beta_{1}}x\partial^{\beta_{2}}y}(V\cdot
X_{2})(\psi_{t}(x,y)).$

As $V\cdot X_{1}$ and $V\cdot X_{2}$ are homogenous polynomials of degree
$m-1$, we have

$\frac{\partial^{|\beta|}}{\partial^{\beta_{1}}x\partial^{\beta_{2}}y}(V\cdot
X_{1})(0,0)=\frac{\partial^{|\beta|}}{\partial^{\beta_{1}}x\partial^{\beta_{2}}y}(V\cdot
X_{2})(0,0)=0\text{ if }|\beta|\leq m-2.$

It follows that

$\frac{d}{dt}\frac{\partial^{|\alpha|}}{\partial^{\alpha_{1}}x\partial^{\alpha_{2}}y}((f\cdot
V)\circ\psi_{t})(0,0)=0\text{ for all }t\in{\mathbb{R}}\text{ and
}|\alpha|\leq m-2.$

Hence

$\partial_{\sigma}^{\alpha}(f\circ\psi_{t})=\frac{\partial^{|\alpha|}}{\partial^{\alpha_{1}}x\partial^{\alpha_{2}}y}((f\cdot
V)\circ\psi_{t})(0,0)=\frac{\partial^{|\alpha|}}{\partial^{\alpha_{1}}x\partial^{\alpha_{2}}y}(f\cdot
V)(0,0)=\partial_{\sigma}^{\alpha}(f).$

∎

Let $G_{0}:{\mathbb{C}}\to{\mathbb{C}}$ be the principal $m$-th root map, i.e.
$G_{0}(re^{is})=r^{1/m}e^{is/m}$ if $s\in[0,2\pi)$, and let
$\omega\in{\mathbb{C}}$ be the principal $2m$-th root of unity.

######  Definition 5.

For every $\varepsilon>0$ denote by $D_{\varepsilon}$ the pre-image of the
square $[-\varepsilon,\varepsilon]\times[-\varepsilon,\varepsilon]$ by the map
$z\mapsto z^{m}$. Given a neighborhood $U_{\sigma}$ of $\sigma$, choose
$\varepsilon>0$ such that $D_{\varepsilon}=D_{\sigma,\varepsilon}\subset
U_{\sigma}$.

Let us consider four curves that parametrize some incoming and outgoing
segments of the boundary of $D_{\varepsilon}$:
$\gamma_{+}^{in},\gamma_{+}^{out}:(0,\varepsilon)\to\partial D_{\varepsilon}$,
$\gamma_{-}^{in},\gamma_{-}^{out}:(-\varepsilon,0)\to\partial D_{\varepsilon}$
are given by

$\gamma_{\pm}^{in}(s)=G_{0}(-\varepsilon+is),\quad\gamma_{\pm}^{out}(s)=G_{0}(\varepsilon+is).$

For every interval $J\subset[0,2\pi)$ denote by $\mathcal{S}(J)$ the
corresponding angular sector $\\{z\in{\mathbb{C}}:\operatorname{Arg}(z)\in
J\\}$.

######  Lemma 8.2.

The following statements hold:

(i) The maps $\gamma_{\pm}^{in}$/$\gamma_{\pm}^{out}$ are standard
parametrizations of incoming/outgoing segments of
$D_{\varepsilon}\cap\mathcal{S}([0,2\pi/m))$ for the flow $\psi_{\mathbb{R}}$.

(ii) The orbit segments entering $D_{\varepsilon}$ at $\gamma_{\pm}^{in}(s)$
leave it at $\gamma_{\pm}^{out}(s)$.

Denote by $\tau(s)$ the time spent by this orbit in the set $D_{\varepsilon}$.
Then

(iii) for every $f\in C^{m}(M)$ we have

(8.3)
$\int_{0}^{\tau(s)}f(\psi_{t}(\gamma_{\pm}^{in}(s)))\,dt=\frac{1}{m^{2}}\int_{-\varepsilon}^{\varepsilon}\frac{(f\cdot
V)(G_{0}(u,s))}{(u^{2}+s^{2})^{\frac{m-1}{m}}}du.$

###### Proof.

As

$\Im(\gamma_{\pm}^{in}(s)^{m})=\Im(G_{0}(-\varepsilon+is)^{m})=s,\quad\Im(\gamma_{\pm}^{out}(s)^{m})=\Im(G_{0}(\varepsilon+is)^{m})=s,$

in view of (8.2), the parametrizations $\gamma_{\pm}^{in}$,
$\gamma_{\pm}^{out}$ are standard.

Since the map $z\mapsto z^{m}$ is a bijection between
$D_{\varepsilon}\cap\mathcal{S}([0,2\pi/m))$ and
$[-\varepsilon,\varepsilon]\times[-\varepsilon,\varepsilon]$, and $G_{0}$ is
its inverse, let us consider a local flow $\tilde{\psi}_{\mathbb{R}}$ on
$[-\varepsilon,\varepsilon]\times[-\varepsilon,\varepsilon]$ conjugated to the
flow $\psi_{\mathbb{R}}$ restricted to
$D_{\varepsilon}\cap\mathcal{S}([0,2\pi/m))$, i.e.
$\tilde{\psi}_{t}(z)=\psi_{t}(G_{0}(z))^{m}$. By (8.1),

$\displaystyle\frac{d}{dt}\tilde{\psi}_{t}(z)$
$\displaystyle=m\,\psi_{t}(G_{0}(z))^{m-1}\frac{d}{dt}\psi_{t}(G_{0}(z))=m\,\psi_{t}(G_{0}(z))^{m-1}X(\psi_{t}(G_{0}(z)))$
$\displaystyle=m^{2}\frac{|\psi_{t}(G_{0}(z))|^{2(m-1)}}{V(\psi_{t}(G_{0}(z)))}=m^{2}\frac{|\tilde{\psi}_{t}(z)|^{\frac{2(m-1)}{m}}}{V\circ
G_{0}(\tilde{\psi}_{t}(z))}.$

Hence

$\frac{d}{dt}\Re\tilde{\psi}_{t}(z)=m^{2}\frac{|\tilde{\psi}_{t}(z)|^{\frac{2(m-1)}{m}}}{V\circ
G_{0}(\tilde{\psi}_{t}(z))}>0\text{ and
}\frac{d}{dt}\Im\tilde{\psi}_{t}(z)=0.$

It follows that the interval
$\\{(-\varepsilon,s):s\in(-\varepsilon,\varepsilon)\\}$ is the incoming and
$\\{(\varepsilon,s):s\in(-\varepsilon,\varepsilon)\\}$ is the outgoing segment
of $[-\varepsilon,\varepsilon]\times[-\varepsilon,\varepsilon]$ for the local
flow $\tilde{\psi}_{\mathbb{R}}$. Moreover, the orbit segments entering
$[-\varepsilon,\varepsilon]\times[-\varepsilon,\varepsilon]$ at
$(-\varepsilon,s)$ leave it at $(\varepsilon,s)$. Passing via $G_{0}$ to the
flow $\psi_{\mathbb{R}}$, we obtain the first claim of the lemma.

Recall that $\tau(s)$ is the time spent by $\tilde{\psi}_{\mathbb{R}}$-orbit
starting at $(-\varepsilon,s)$ in the set
$[-\varepsilon,\varepsilon]\times[-\varepsilon,\varepsilon]$. Then

$\displaystyle\int_{0}^{\tau(s)}f(\psi_{t}(\gamma_{\pm}^{in}(s)))\,dt$
$\displaystyle=\int_{0}^{\tau(s)}f\circ
G_{0}\big{(}\tilde{\psi}_{t}(-\varepsilon,s)\big{)}\,dt$
$\displaystyle=\int_{0}^{\tau(s)}f\circ
G_{0}\big{(}\Re\tilde{\psi}_{t}(-\varepsilon,s),s\big{)}\,dt.$

Next we integrate by substituting $u(t)=\Re\tilde{\psi}_{t}(-\varepsilon,s)$.
As

$-\varepsilon=\Re\tilde{\psi}_{0}(-\varepsilon,s),\quad\varepsilon=\Re\tilde{\psi}_{\tau(s)}(-\varepsilon,s)$

and

$\displaystyle\frac{du}{dt}$
$\displaystyle=\frac{d}{dt}\Re\tilde{\psi}_{t}(-\varepsilon,s)=m^{2}\frac{|\tilde{\psi}_{t}(-\varepsilon,s)|^{\frac{2(m-1)}{m}}}{V\circ
G_{0}(\tilde{\psi}_{t}(-\varepsilon,s))}$
$\displaystyle=m^{2}\frac{((\Re\tilde{\psi}_{t}(-\varepsilon,s))^{2}+s^{2})^{\frac{m-1}{m}}}{V\circ
G_{0}(\Re\tilde{\psi}_{t}(-\varepsilon,s),s)}=m^{2}\frac{(u^{2}+s^{2})^{\frac{m-1}{m}}}{V\circ
G_{0}(u,s)},$

by change of variables, we have

$\displaystyle\int_{0}^{\tau(s)}f\circ
G_{0}\big{(}\Re\tilde{\psi}_{t}(-\varepsilon,s),s\big{)}\,dt$
$\displaystyle=\int_{-\varepsilon}^{\varepsilon}f\circ G_{0}(u,s)\frac{V\circ
G_{0}(u,s)}{m^{2}(u^{2}+s^{2})^{\frac{m-1}{m}}}du$
$\displaystyle=\frac{1}{m^{2}}\int_{-\varepsilon}^{\varepsilon}\frac{(f\cdot
V)(G_{0}(u,s))}{(u^{2}+s^{2})^{\frac{m-1}{m}}}du.$

This gives (8.3). ∎

######  Remark 8.3.

Lemma 8.2 describes incoming and outgoing segments on the boundary of
$D_{\varepsilon}$ but only in the angular sector $\mathcal{S}([0,2\pi/m))$.
The same arguments apply to the flow $\psi_{\mathbb{R}}$ restricted to
$\mathcal{S}([2\pi k/m,2\pi(k+1)/m))$ for $0\leq k<m$. As
$\omega\in{\mathbb{C}}$ is the principal $2m$-th root of unity, the
incoming/outgoing segments of $D_{\varepsilon}\cap\mathcal{S}([2\pi
k/m,2\pi(k+1)/m))$ are given by $\omega^{2k}\gamma_{\pm}^{in}$ and
$\omega^{2k}\gamma_{\pm}^{out}$ respectively. Moreover, if $\tau_{k}(s)$ is
the time spent by ${\psi}_{\mathbb{R}}$-orbit starting at
$\omega^{2k}\gamma_{\pm}^{in}(s)$ in the set $D_{\varepsilon}$, then

(8.4)
$\varphi_{f}^{\sigma,k}(s):=\int_{0}^{\tau_{k}(s)}f(\psi_{t}(\omega^{2k}\gamma_{\pm}^{in}(s)))\,dt=\frac{1}{m^{2}}\int_{-\varepsilon}^{\varepsilon}\frac{(f\cdot
V)(\omega^{2k}G_{0}(u,s))}{(u^{2}+s^{2})^{\frac{m-1}{m}}}du.$

Note that for $(u,s)\in{\mathbb{R}}^{2}_{\geq 0}$ we have

$G_{0}(-u,-s)=\omega G_{0}(u,s),\
G_{0}(u,-s)=\omega^{2}\overline{G_{0}}(u,s),\
G_{0}(-u,s)=\omega\overline{G_{0}}(u,s).$

It follows that for every $s\in(0,\varepsilon)$ we have

$\displaystyle m^{2}\varphi_{f}^{\sigma,k}(s)$
$\displaystyle=\int_{0}^{\varepsilon}\frac{(f\cdot
V)(\omega^{2k}G_{0}(u,s))}{(u^{2}+s^{2})^{\frac{m-1}{m}}}du+\int_{0}^{\varepsilon}\frac{(f\cdot
V)(\omega^{2k}G_{0}(-u,s))}{(u^{2}+s^{2})^{\frac{m-1}{m}}}du$
$\displaystyle=\int_{0}^{\varepsilon}\frac{(f\cdot
V)(\omega^{2k}G_{0}(u,s))}{(u^{2}+s^{2})^{\frac{m-1}{m}}}du+\int_{0}^{\varepsilon}\frac{(f\cdot
V)(\omega^{2k+1}\overline{G_{0}}(u,s))}{(u^{2}+s^{2})^{\frac{m-1}{m}}}du$

and

$\displaystyle m^{2}\varphi_{f}^{\sigma,k}(-s)$
$\displaystyle=\int_{0}^{\varepsilon}\frac{(f\cdot
V)(\omega^{2k}G_{0}(u,-s))}{(u^{2}+s^{2})^{\frac{m-1}{m}}}du+\int_{0}^{\varepsilon}\frac{(f\cdot
V)(\omega^{2k}G_{0}(-u,-s))}{(u^{2}+s^{2})^{\frac{m-1}{m}}}du$
$\displaystyle=\int_{0}^{\varepsilon}\frac{(f\cdot
V)(\omega^{2k+2}\overline{G_{0}}(u,s))}{(u^{2}+s^{2})^{\frac{m-1}{m}}}du+\int_{0}^{\varepsilon}\frac{(f\cdot
V)(\omega^{2k+1}{G_{0}}(u,s))}{(u^{2}+s^{2})^{\frac{m-1}{m}}}du.$

###  8.1. Singularities of $\varphi_{f}^{\sigma,k}$

The purpose of this section is to understand the type of singularity of
functions $\varphi_{f}^{\sigma,k}$. These functions are responsible for
reading the singularities of $\varphi_{f}$ and provide the tools to prove
Theorem 9.1 in §9.

For every $m\geq 2$ let $G:{\mathbb{R}}^{2}_{\geq 0}\to{\mathbb{C}}$ be a
continuous inverse of one of the maps $z\mapsto z^{m}$,
$z\mapsto\overline{z}^{m}$, $z\mapsto-z^{m}$ or $z\mapsto-\overline{z}^{m}$.
Then $G$ is homogenous of degree $1/m$ and analytic on
${\mathbb{R}}^{2}_{>0}$. If $G_{0}:{\mathbb{R}}^{2}_{\geq 0}\to{\mathbb{C}}$
is the principal $m$-th root and $\omega$ is the principal $2m$-th root of
unity, then $G$ is either $\omega^{l}G_{0}\text{ or
}\omega^{l}\overline{G}_{0}\text{ for some }0\leq l<2m.$

Let $f:{D}\to{\mathbb{R}}$ be a bounded Borel map where $D$ is the pre-image
of the square $[-1,1]\times[-1,1]$ by the map $z\mapsto z^{m}$. For every
$a\geq 1/2$ let us consider the map
$\varphi=\varphi_{f,a}:(0,1]\to{\mathbb{R}}$ given by

(8.5) $\varphi(s)=\int_{0}^{1}\frac{f(G(u,s))}{(u^{2}+s^{2})^{a}}\,du.$

######  Remark 8.4.

Note that for every $\varepsilon>0$ we have

$\displaystyle\varphi_{f,a,\varepsilon}(s)$
$\displaystyle=\int_{0}^{\varepsilon}\frac{f(G(u,s))}{(u^{2}+s^{2})^{a}}\,du=\frac{1}{\varepsilon}\int_{0}^{1}\frac{f(G(u/\varepsilon,s))}{((u/\varepsilon)^{2}+s^{2})^{a}}\,du$
$\displaystyle=\varepsilon^{2a-1}\int_{0}^{1}\frac{f(\varepsilon^{-1/m}G(u,\varepsilon
s))}{(u^{2}+(\varepsilon
s)^{2})^{a}}\,du=\varepsilon^{2a-1}\varphi_{f\circ\varepsilon^{-1/m},a}(\varepsilon
s).$

Therefore

$s^{2a}\varphi^{\prime}_{f,a,\varepsilon}(s)=(\varepsilon
s)^{2a}\varphi^{\prime}_{f\circ\varepsilon^{-1/m},a}(\varepsilon s).$

Notice that

$s^{2a-1}\int_{0}^{1}\frac{1}{(u^{2}+s^{2})^{a}}\,du=\int_{0}^{1}\frac{1}{((\frac{u}{s})^{2}+1)^{a}}\frac{du}{s}=\int_{0}^{1/s}\frac{dx}{(x^{2}+1)^{a}}.$

Let us recall the definitions of Gamma function $\Gamma(z)$ and Beta function
$B(x,y)$

$B(x,y):=\int_{0}^{1}t^{x-1}{(1-t)}^{y-1}dt,\quad\Gamma(z):=\int_{0}^{\infty}x^{z-1}e^{-x}dx,$

and let us denote

$\Gamma_{a}:=\int_{0}^{+\infty}\frac{dx}{(x^{2}+1)^{a}}=\frac{1}{2}B(\frac{1}{2},a-\frac{1}{2})=\frac{1}{2}\frac{\Gamma(\frac{1}{2})\Gamma(a-\frac{1}{2})}{\Gamma(a)}=\frac{\sqrt{\pi}}{2}\frac{\Gamma(a-\frac{1}{2})}{\Gamma(a)}.$

Then, for every $a>1/2$,

(8.6) $\displaystyle
s^{2a-1}\int_{0}^{1}\frac{1}{(u^{2}+s^{2})^{a}}\,du\leq\Gamma_{a}\text{ for
all }s\in(0,1],\text{ and}$ (8.7) $\displaystyle\lim_{s\to
0}s^{2a-1}\int_{0}^{1}\frac{1}{(u^{2}+s^{2})^{a}}\,du=\Gamma_{a}.$

If $a=1/2$ then

(8.8)
$\int_{0}^{1}\frac{1}{(u^{2}+s^{2})^{a}}\,du=\int_{0}^{1/s}\frac{dx}{\sqrt{x^{2}+1}}=\log\Big{(}\frac{1}{s}+\sqrt{\frac{1}{s^{2}}+1}\Big{)}\leq\log\frac{3}{s}\leq
2+|\log s|.$

In view of (8.6) and (8.8), for every $s\in(0,1]$,

(8.9) $\displaystyle\begin{aligned}
s^{2a-1}\varphi_{|f|,a}(s)&\leq\|f\|_{\sup}\Gamma_{a}\ &\text{ if }\ a\geq
1/2,\\\ \varphi_{|f|,a}(s)&\leq\|f\|_{\sup}(2+|\log s|)\ &\text{ if }\
a=1/2.\end{aligned}$

In the following lemmas, we find the upper bound of $\varphi^{\prime}$ by
$C^{k}$-norms of the function $f$ and the order of vanishing at the saddle.

######  Lemma 8.5.

Suppose that $f:D\to{\mathbb{R}}$ is a $C^{1}$-map. For every $1/2\leq a\leq
1$ we have

$|s^{2a}\varphi^{\prime}(s)|\leq 2\|f\|_{C^{1}}\Gamma_{a+\frac{m-1}{2m}}.$

Moreover,

$\lim_{s\to 0}s^{2a}\varphi^{\prime}(s)=-2af(0,0)\Gamma_{a+1}.$

###### Proof.

First note that $\varphi$ is a $C^{1}$-function on $(0,1]$ with

(8.10)
$\displaystyle\begin{split}\varphi^{\prime}(s)&=\int_{0}^{1}\frac{\frac{\partial
f}{\partial x}(G(u,s))\frac{\partial G_{1}}{\partial s}(u,s)+\frac{\partial
f}{\partial y}(G(u,s))\frac{\partial G_{2}}{\partial
s}(u,s)}{(u^{2}+s^{2})^{a}}\,du\\\
&\quad-2a\int_{0}^{1}\frac{sf(G(u,s))}{(u^{2}+s^{2})^{a+1}}\,du.\end{split}$

As $(G_{1}+iG_{2})^{m}=\pm u\pm is$, we have

(8.11) $m(G_{1}+iG_{2})^{m-1}\Big{(}\frac{\partial G_{1}}{\partial
s}+i\frac{\partial G_{2}}{\partial s}\Big{)}=\pm i.$

Hence

$\Big{|}\frac{\partial G_{1}}{\partial s}+i\frac{\partial G_{2}}{\partial
s}\Big{|}=\frac{1}{m|G_{1}+iG_{2}|^{m-1}}=\frac{1}{m(u^{2}+s^{2})^{\frac{m-1}{2m}}}.$

It follows that

(8.12) $\displaystyle\begin{split}\Big{|}&\int_{0}^{1}\frac{\frac{\partial
f}{\partial x}(G(u,s))\frac{\partial G_{1}}{\partial s}(u,s)+\frac{\partial
f}{\partial y}(G(u,s))\frac{\partial G_{2}}{\partial
s}(u,s)}{(u^{2}+s^{2})^{a}}\,du\Big{|}\\\
&\leq\frac{\|f^{\prime}\|_{C^{0}}}{m}\int_{0}^{1}\frac{1}{(u^{2}+s^{2})^{a+\frac{m-1}{2m}}}\,du\leq\frac{\|f^{\prime}\|_{C^{0}}}{m}\frac{\Gamma_{a+\frac{m-1}{2m}}}{s^{2a-\frac{1}{m}}}\leq\frac{\|f^{\prime}\|_{C^{0}}}{m}\frac{\Gamma_{a+\frac{m-1}{2m}}}{s^{2a}}\end{split}$

and

$\Big{|}\int_{0}^{1}\frac{sf(G(u,s))}{(u^{2}+s^{2})^{a+1}}\,du\Big{|}\leq\|f\|_{C^{0}}\int_{0}^{1}\frac{s}{(u^{2}+s^{2})^{a+1}}\,du\leq{\|f\|_{C^{0}}}\frac{\Gamma_{a+1}}{s^{2a}}.$

It follows that

$|\varphi^{\prime}(s)|\leq\Big{(}2a\|f\|_{C^{0}}\Gamma_{a+1}+\frac{\|f^{\prime}\|_{C^{0}}}{m}\Gamma_{a+\frac{m-1}{2m}}\Big{)}\frac{1}{s^{2a}}\leq
2\|f\|_{C^{1}}\Gamma_{a+\frac{m-1}{2m}}\frac{1}{s^{2a}}.$

Since $f$ is of class $C^{1}$, we have

$|f(G(u,s))-f(0,0)|\leq\|f\|_{C^{1}}\|G(u,s)\|\leq\|f\|_{C^{1}}(u^{2}+s^{2})^{\frac{1}{2m}}.$

Moreover, by (8.6),

$\int_{0}^{1}\frac{(u^{2}+s^{2})^{\frac{1}{2m}}}{(u^{2}+s^{2})^{a+1}}\,du\leq\frac{\Gamma_{a+1-\frac{1}{2m}}}{s^{2a+1-\frac{1}{m}}}.$

Therefore, in view of (8.10), (8.12), we have

$\displaystyle\Big{|}$ $\displaystyle
s^{2a}\varphi^{\prime}(s)+2af(0,0)s^{2a+1}\int_{0}^{1}\frac{du}{(u^{2}+s^{2})^{a+1}}\Big{|}$
$\displaystyle\leq s^{2a}\left|\int_{0}^{1}\frac{\frac{\partial f}{\partial
x}(G(u,s))\frac{\partial G_{1}}{\partial s}(u,s)+\frac{\partial f}{\partial
y}(G(u,s))\frac{\partial G_{2}}{\partial
s}(u,s)}{(u^{2}+s^{2})^{a}}\,du\right|$
$\displaystyle\quad+2as^{2a+1}\int_{0}^{1}\frac{|f(0,0)-f(G(u,s))|}{(u^{2}+s^{2})^{a+1}}du$
$\displaystyle=s^{2a}\|f^{\prime}\|_{C^{0}}\frac{\Gamma_{a+\frac{m-1}{2m}}}{s^{2a-\frac{1}{m}}}+2as^{2a+1}\|f\|_{C^{1}}\int_{0}^{1}\frac{(u^{2}+s^{2})^{\frac{1}{2m}}}{(u^{2}+s^{2})^{a+1}}\,du$
$\displaystyle\leq\|f\|_{C^{1}}\Gamma_{a+\frac{m-1}{2m}}s^{\frac{1}{m}}+2as^{2a+1}\|f\|_{C^{1}}\frac{\Gamma_{a+1-\frac{1}{2m}}}{s^{2a+1-\frac{1}{m}}}=O(s^{\frac{1}{m}}).$

Hence

$\lim_{s\to 0}s^{2a}\varphi^{\prime}(s)=-\lim_{s\to
0}2af(0,0)s^{2a+1}\int_{0}^{1}\frac{du}{(u^{2}+s^{2})^{a+1}}=-2af(0,0)\Gamma_{a+1}.$

∎

######  Lemma 8.6.

Assume that $f:D\to{\mathbb{R}}$ is a $C^{m}$-map, $1/2\leq a\leq 1$ and let
$k$ be a natural number such that $k\leq m(2a-1)$. Suppose that
$f^{(j)}(0,0)=0$ for $0\leq j<k$. Then

$|s^{2a-\frac{k}{m}}\varphi^{\prime}(s)|\leq\frac{\|f\|_{C^{k}}\Gamma_{1}}{(k-1)!}\text{
for all }s\in(0,1].$

Moreover, if $k<m(2a-1)$ then

$s^{2a-\frac{k}{m}-1}\varphi_{|f|,a}(s)\leq\frac{\|f\|_{C^{k}}\Gamma_{{a-\frac{k}{2m}}}}{k!}\text{
for all }s\in(0,1]$

and if $k=m(2a-1)$, then

$\varphi_{|f|,a}(s)\leq\frac{\|f\|_{C^{k}}}{k!}(2+|\log s|)\text{ for all
}s\in(0,1].$

###### Proof.

By assumption,

$\displaystyle|f(G(u,s))|$
$\displaystyle\leq\frac{\|f^{(k)}\|_{C^{0}}}{k!}\|G(u,s)\|^{k}\leq\frac{\|f^{(k)}\|_{C^{0}}}{k!}(u^{2}+s^{2})^{\frac{k}{2m}},$
$\displaystyle\left|\frac{\partial f}{\partial x}(G(u,s))\right|$
$\displaystyle\leq\frac{\|(\partial f/\partial
x)^{(k-1)}\|_{C^{0}}}{(k-1)!}\|G(u,s)\|^{k-1}\leq\frac{\|(\partial f/\partial
x)^{(k-1)}\|_{C^{0}}}{(k-1)!}(u^{2}+s^{2})^{\frac{k-1}{2m}},$
$\displaystyle\left|\frac{\partial f}{\partial y}(G(u,s))\right|$
$\displaystyle\leq\frac{\|(\partial f/\partial
y)^{(k-1)}\|_{C^{0}}}{(k-1)!}\|G(u,s)\|^{k-1}\leq\frac{\|(\partial f/\partial
y)^{(k-1)}\|_{C^{0}}}{(k-1)!}(u^{2}+s^{2})^{\frac{k-1}{2m}}.$

Therefore

$\varphi_{|f|,a}(s)\leq\frac{\|f^{(k)}\|_{C^{0}}}{k!}\int_{0}^{1}\frac{(u^{2}+s^{2})^{\frac{k}{2m}}}{(u^{2}+s^{2})^{a}}du=\frac{\|f^{(k)}\|_{C^{0}}}{k!}\int_{0}^{1}\frac{1}{(u^{2}+s^{2})^{a-\frac{k}{2m}}}du$

and, by (8.10),

$\displaystyle|\varphi^{\prime}(s)|$
$\displaystyle\leq\frac{\|f^{(k)}\|_{C^{0}}}{m(k-1)!}\int_{0}^{1}\frac{(u^{2}+s^{2})^{\frac{k-1}{2m}}}{(u^{2}+s^{2})^{\frac{m-1}{2m}}(u^{2}+s^{2})^{a}}du$
$\displaystyle\quad+2as\frac{\|f^{(k)}\|_{C^{0}}}{k!}\int_{0}^{1}\frac{(u^{2}+s^{2})^{\frac{k}{2m}}}{(u^{2}+s^{2})^{a+1}}du$
$\displaystyle\leq\frac{\|f^{(k)}\|_{C^{0}}}{m(k-1)!}\int_{0}^{1}\frac{1}{(u^{2}+s^{2})^{a+\frac{m-k}{2m}}}du+2as\frac{\|f^{(k)}\|_{C^{0}}}{k!}\int_{0}^{1}\frac{1}{(u^{2}+s^{2})^{a+1-\frac{k}{2m}}}du.$

As $k\leq m(2a-1)$, we have
$a+1-\frac{k}{2m}>a+\frac{m-k}{2m}=1+\frac{(2a-1)m-k}{2m}\geq 1$. By (8.6),
this gives

$\displaystyle|\varphi^{\prime}(s)|\leq\frac{\|f^{(k)}\|_{C^{0}}}{m(k-1)!}\frac{\Gamma_{a+\frac{m-k}{2m}}}{s^{2a-\frac{k}{m}}}+2as\frac{\|f^{(k)}\|_{C^{0}}}{k!}\frac{\Gamma_{a+1-\frac{k}{2m}}}{s^{2a+1-\frac{k}{m}}}\leq\frac{\|f^{(k)}\|_{C^{0}}}{(k-1)!}\frac{\Gamma_{1}}{s^{2a-\frac{k}{m}}}.$

If additionally $k<m(2a-1)$ then $a-\frac{k}{2m}>\frac{1}{2}$. By (8.6) again,

$\displaystyle\varphi_{|f|,a}(s)\leq\frac{\|f^{(k)}\|_{C^{0}}}{k!}\int_{0}^{1}\frac{1}{(u^{2}+s^{2})^{a-\frac{k}{2m}}}du\leq\frac{\|f^{(k)}\|_{C^{0}}}{k!}\frac{\Gamma_{a-\frac{k}{2m}}}{s^{2a-1-\frac{k}{m}}}.$

On the other hand, if $k=m(2a-1)$ then $a-\tfrac{k}{2m}=\tfrac{1}{2}$ and, by
(8.8),

$\displaystyle\varphi_{|f|,a}(s)\leq\frac{\|f^{(k)}\|_{C^{0}}}{k!}\int_{0}^{1}\frac{1}{(u^{2}+s^{2})^{a-\frac{k}{2m}}}du\leq\frac{\|f^{(k)}\|_{C^{0}}}{k!}(2+|\log
s|).$

∎

###  8.2. Quantities $C^{\pm}_{\alpha}(\varphi_{f})$

In this section, we develop some tools that help compute the non-vanishing
quantities $C^{\pm}_{\alpha}(\varphi_{f})$.

######  Definition 6.

For every real $\beta$ let $C_{\beta}({\mathbb{R}}_{>0}^{2})$ be the space of
continuous homogenous functions
$H:{\mathbb{R}}_{>0}\times{\mathbb{R}}_{>0}\to{\mathbb{C}}$ of degree $\beta$
such that $H(u,s)=O(\|(u,s)\|^{\beta})$.

For every real $a$ such that $2a-\beta>1$ and $H\in
C_{\beta}({\mathbb{R}}_{>0}^{2})$ let

$\Gamma_{a}(H):=\int_{0}^{+\infty}\frac{H(x,1)}{(x^{2}+1)^{a}}\,dx.$

As ${H(x,1)}/{(x^{2}+1)^{a}}=O(1/(1+x)^{2a-\beta})$, the quantity
$\Gamma_{a}(H)$ is well-defined and

(8.13)
$s^{2a-\beta-1}\int_{0}^{1}\frac{H(u,s)}{(u^{2}+s^{2})^{a}}\,du=\int_{0}^{1}\frac{H(u/s,1)}{((u/s)^{2}+1)^{a}}\,\frac{du}{s}=\int_{0}^{1/s}\frac{H(x,1)}{(x^{2}+1)^{a}}\,dx\to\Gamma_{a}(H).$

Note that if $H(u,s)=\tilde{H}(u,s)/\|(u,s)\|^{\max\\{-\beta,0\\}}$ and
$\tilde{H}:{\mathbb{R}}_{\geq 0}\times{\mathbb{R}}_{\geq 0}\to{\mathbb{C}}$ is
continuous homogenous of degree $\max\\{\beta,0\\}$ then $H\in
C_{\beta}({\mathbb{R}}_{>0}^{2})$.

######  Lemma 8.7.

Assume that $2a-\beta>1$, $H\in C_{\beta}({\mathbb{R}}_{>0}^{2})$ is of class
$C^{1}$ and $\tfrac{\partial H}{\partial y}\in
C_{\beta-1}({\mathbb{R}}_{>0}^{2})$. Then

(8.14) $\displaystyle\Gamma_{a}(\tfrac{\partial H}{\partial
y})=2a\Gamma_{a+1}(H)-(2a-1-\beta)\Gamma_{a}(H).$

###### Proof.

Note that for every $y>0$,

$\Gamma_{a}(H)=\int_{0}^{+\infty}\frac{H(x,1)}{(x^{2}+1)^{a}}\,dx=\int_{0}^{+\infty}\frac{H(x/y,1)}{(x^{2}/y^{2}+1)^{a}}\,\frac{dx}{y}=y^{2a-1-\beta}\int_{0}^{+\infty}\frac{H(x,y)}{(x^{2}+y^{2})^{a}}\,dx.$

As $H\in C_{\beta}({\mathbb{R}}_{>0}^{2})$ and $\tfrac{\partial H}{\partial
y}\in C_{\beta-1}({\mathbb{R}}_{>0}^{2})$, by differentiating with respect to
$y$, we get

$\displaystyle-(2a-1-\beta)y^{\beta-2a}\Gamma_{a}(H)$
$\displaystyle=\frac{d}{dy}y^{1+\beta-2a}\Gamma_{a}(H)=\frac{d}{dy}\int_{0}^{+\infty}\frac{H(x,y)}{(x^{2}+y^{2})^{a}}\,dx$
$\displaystyle=\int_{0}^{+\infty}\frac{\frac{\partial H}{\partial
y}(x,y)}{(x^{2}+y^{2})^{a}}\,dx-2ay\int_{0}^{+\infty}\frac{H(x,y)}{(x^{2}+y^{2})^{a+1}}\,dx.$

Taking $y=1,$ this yields (8.14). ∎

Recall that, by Lemma 8.5 for any $C^{1}$-map $f:D\to{\mathbb{R}}$ with
$f(0,0)\neq 0$ and for every $1/2\leq a\leq 1$ we have

$\lim_{s\to 0}s^{2a}\varphi^{\prime}(s)=-2af(0,0)\Gamma_{a+1}.$

In the next preliminary lemmas, we find precise asymptotics of
$\varphi^{\prime}$ at zero also when the function $f$ and some of its
derivatives vanish at the saddle. This plays a crucial role in calculating the
quantity $C^{\pm}_{\alpha}(\varphi_{f})$ explicitly in §9.

######  Lemma 8.8.

Let $k$ be an integer number such that $0\leq k\leq m(2a-1)$. Suppose that
$f:D\to{\mathbb{R}}$ is of class $C^{k+1}$ and $f^{(j)}(0,0)=0$ for $0\leq
j<k$. Then

(8.15) $\lim_{s\to
0}s^{2a-\frac{k}{m}}\varphi^{\prime}(s)=\sum_{j=0}^{k}\binom{k}{j}\frac{\partial^{k}f}{\partial
x^{j}\partial y^{k-j}}(0,0)\Gamma_{a}^{k,j}(G),$

where

$\displaystyle\Gamma_{a}^{k,j}(G)=-\frac{2a-1-\frac{k}{m}}{k}\Gamma_{a}(G_{1}^{j}G_{2}^{k-j})-2a\frac{k-1}{k}\Gamma_{a+1}(G_{1}^{j}G_{2}^{k-j})\text{
if }k\geq 1$

and $\Gamma_{a}^{0,0}(G)=-2a\Gamma_{a+1}$.

###### Proof.

If $k=0$, then (8.15) follows directly from Lemma 8.5.

Assume that $k\geq 1$. Then, by assumptions,

$\displaystyle f(G(u,s))$
$\displaystyle=\sum_{j=0}^{k}\binom{k}{j}\frac{\partial^{k}f}{\partial
x^{j}\partial y^{k-j}}(0,0)G_{1}^{j}(u,s)G_{2}^{k-j}(u,s)+O(\|G(u,s)\|^{k+1})$
$\displaystyle=\sum_{j=0}^{k}\binom{k}{j}\frac{\partial^{k}f}{\partial
x^{j}\partial
y^{k-j}}(0,0)G_{1}^{j}(u,s)G_{2}^{k-j}(u,s)+O((u^{2}+s^{2})^{\frac{k+1}{2m}}).$

Moreover,

$\displaystyle\frac{\partial f}{\partial x}(G(u,s))$
$\displaystyle=\sum_{j=0}^{k-1}\binom{k-1}{j}\frac{\partial^{k}f}{\partial
x^{j+1}\partial
y^{k-1-j}}(0,0)G_{1}^{j}(u,s)G_{2}^{k-1-j}(u,s)+O((u^{2}+s^{2})^{\frac{k}{2m}}),$
$\displaystyle\frac{\partial f}{\partial y}(G(u,s))$
$\displaystyle=\sum_{j=0}^{k-1}\binom{k-1}{j}\frac{\partial^{k}f}{\partial
x^{j}\partial
y^{k-j}}(0,0)G_{1}^{j}(u,s)G_{2}^{k-1-j}(u,s)+O((u^{2}+s^{2})^{\frac{k}{2m}}).$

As $\|G(u,s)\|=\|(u,s)\|^{\frac{1}{m}}=(u^{2}+s^{2})^{\frac{1}{2m}}$ and
$\|\frac{\partial G}{\partial
s}(u,s)\|=\frac{\|(u,s)\|^{-\frac{m-1}{m}}}{m}=\frac{(u^{2}+s^{2})^{-\frac{m-1}{2m}}}{m}$,
it follows that

(8.16)
$\displaystyle\begin{split}\frac{d}{ds}\frac{f(G(u,s))}{(u^{2}+s^{2})^{a}}&=\frac{\frac{\partial
f}{\partial x}(G(u,s))\frac{\partial G_{1}}{\partial s}(u,s)+\frac{\partial
f}{\partial y}(G(u,s))\frac{\partial G_{2}}{\partial
s}(u,s)}{(u^{2}+s^{2})^{a}}-2a\frac{sf(G(u,s))}{(u^{2}+s^{2})^{a+1}}\\\
&=\sum_{j=0}^{k}\binom{k}{j}\frac{\partial^{k}f}{\partial x^{j}\partial
y^{k-j}}(0,0)\frac{\big{(}\frac{j}{k}G_{1}^{j-1}G_{2}^{k-j}\frac{\partial
G_{1}}{\partial s}+\frac{k-j}{k}G_{1}^{j}G_{2}^{k-1-j}\frac{\partial
G_{2}}{\partial s}\big{)}(u,s)}{(u^{2}+s^{2})^{a}}\\\
&\quad-2as\sum_{j=0}^{k}\binom{k}{j}\frac{\partial^{k}f}{\partial
x^{j}\partial
y^{k-j}}(0,0)\frac{(G_{1}^{j}G_{2}^{k-j})(u,s)}{(u^{2}+s^{2})^{a+1}}\\\
&\quad+O\Big{(}\frac{1}{(u^{2}+s^{2})^{a+\frac{m-k-1}{2m}}}\Big{)}+O\Big{(}\frac{s}{(u^{2}+s^{2})^{a+1-\frac{k+1}{2m}}}\Big{)}.\end{split}$

Since $G_{1}^{j-1}G_{2}^{k-j}\frac{\partial G_{1}}{\partial s}$,
$G_{1}^{j}G_{2}^{k-j-1}\frac{\partial G_{2}}{\partial s}$ are homogenous of
degree $\frac{k-m}{m}<2a-1$ and $G_{1}^{j}G_{2}^{k-j}$ is homogenous of degree
$\frac{k}{m}<2(a+1)-1$, by (8.13) we have

(8.17) $\displaystyle\begin{split}\lim_{s\to
0}s^{2a-\frac{k}{m}}\int_{0}^{1}\frac{(G_{1}^{j-1}G_{2}^{k-j}\frac{\partial
G_{1}}{\partial
s})(u,s)}{(u^{2}+s^{2})^{a}}\,du=\Gamma_{a}(G_{1}^{j-1}G_{2}^{k-j}\tfrac{\partial
G_{1}}{\partial s}),\\\ \lim_{s\to
0}s^{2a-\frac{k}{m}}\int_{0}^{1}\frac{(G_{1}^{j}G_{2}^{k-j-1}\frac{\partial
G_{2}}{\partial
s})(u,s)}{(u^{2}+s^{2})^{a}}\,du=\Gamma_{a}(G_{1}^{j}G_{2}^{k-j-1}\tfrac{\partial
G_{2}}{\partial s}),\\\ \lim_{s\to
0}s^{2a-\frac{k}{m}}\int_{0}^{1}\frac{s(G_{1}^{j}G_{2}^{k-j})(u,s)}{(u^{2}+s^{2})^{a+1}}\,du=\Gamma_{a+1}(G_{1}^{j}G_{2}^{k-j}).\end{split}$

Furthermore,

$\displaystyle\lim_{s\to
0}s^{2a-\frac{k+1}{m}}\int_{0}^{1}\frac{1}{(u^{2}+s^{2})^{a+\frac{m-k-1}{2m}}}\,du=\Gamma_{a+\frac{m-k-1}{2m}},$
$\displaystyle\lim_{s\to
0}s^{2a-\frac{k+1}{m}}\int_{0}^{1}\frac{s}{(u^{2}+s^{2})^{a+1-\frac{k+1}{2m}}}=\Gamma_{a+1-\frac{k+1}{2m}}.$

In view of (8.16) and (8.17), this gives the statement (8.15) with

$\Gamma_{a}^{k,j}(G)=\frac{j}{k}\Gamma_{a}(G_{1}^{j-1}G_{2}^{k-j}\tfrac{\partial
G_{1}}{\partial
s})+\frac{k-j}{k}\Gamma_{a}(G_{1}^{j}G_{2}^{k-1-j}\tfrac{\partial
G_{2}}{\partial s})-2a\Gamma_{a+1}(G_{1}^{j}G_{2}^{k-j}).$

In view of (8.14) applied to $H=G_{1}^{j}G_{2}^{k-j}$ (which is homogenous of
degree $k/m$), we get

$\displaystyle\Gamma_{a}^{k,j}(G)$
$\displaystyle=\frac{1}{k}\Gamma_{a}\big{(}\tfrac{\partial}{\partial
s}(G_{1}^{j}G_{2}^{k-j})\big{)}-2a\Gamma_{a+1}(G_{1}^{j}G_{2}^{k-j})$
$\displaystyle=\frac{2a}{k}\Gamma_{a+1}(G_{1}^{j}G_{2}^{k-j})-\frac{2a-1-\frac{k}{m}}{k}\Gamma_{a}(G_{1}^{j}G_{2}^{k-j})-2a\Gamma_{a+1}(G_{1}^{j}G_{2}^{k-j})$
$\displaystyle=-\frac{2a-1-\frac{k}{m}}{k}\Gamma_{a}(G_{1}^{j}G_{2}^{k-j})-2a\frac{k-1}{k}\Gamma_{a+1}(G_{1}^{j}G_{2}^{k-j}).$

∎

Let $G_{0}:{\mathbb{R}}^{2}_{\geq 0}\to{\mathbb{C}}$ be the principal branch
of the $m$-th root and let $\omega$ and $\omega_{0}$ be the principal $2m$-th
and $4m$-th root of unity respectively.

######  Lemma 8.9.

Let $1/2<a\leq(m-1)/m$, $1\leq k<(2a-1)m$ and $a_{0},a_{1},\ldots,a_{k}$ are
real numbers not all equal to $0$. Then there exists $0\leq l<2m$ such that

$\sum_{j=0}^{k}a_{j}\big{(}\Gamma_{a}^{k,j}(\omega^{l}G_{0})+\Gamma_{a}^{k,j}(\omega^{l+1}\overline{G_{0}})\big{)}\neq
0.$

###### Proof.

Let $\mathfrak{G}:C_{k/m}({\mathbb{R}}_{>0}^{2})\to{\mathbb{C}}$ be the linear
operator given by

$\mathfrak{G}(H):=-\frac{2a-1-\frac{k}{m}}{k}\Gamma_{a}(H)-2a\frac{k-1}{k}\Gamma_{a+1}(H).$

Then $\overline{\mathfrak{G}(H)}=\mathfrak{G}(\overline{H})$ and
$\Gamma_{a}^{k,j}(G)=\mathfrak{G}(G_{1}^{j}G_{2}^{k-j})$.

Suppose that, contrary to our claim,

(8.18)
$\sum_{j=0}^{k}a_{j}\big{(}\Gamma_{a}^{k,j}(\omega^{l}G_{0})+\Gamma_{a}^{k,j}(\omega^{l+1}\overline{G_{0}})\big{)}=0\text{
for every }0\leq l<2m.$

Denote by ${\mathbb{R}}_{k}[x,y]$ the linear space of homogenous polynomial of
degree $k$. The space ${\mathbb{R}}_{k}[x,y]$ coincides with the subspace
${\mathbb{C}}_{k,{\mathbb{R}}}[z,\overline{z}]$ of complex homogenous
polynomials ${\mathbb{C}}_{k}[z,\overline{z}]$ of the form
$\sum_{j=0}^{k}c_{j}z^{j}\overline{z}^{k-j}$ such that
$\overline{c}_{j}=c_{k-j}$ for $0\leq j\leq k$. For every
$P\in{\mathbb{R}}_{k}[x,y]$ denote by
$\widehat{P}\in{\mathbb{C}}_{k,{\mathbb{R}}}[z,\overline{z}]$ the unique
polynomial such that $\widehat{P}(z,\overline{z})=P(x,y)$.

As $Q(x,y)=\sum_{j=0}^{k}a_{j}x^{j}y^{k-j}\in{\mathbb{R}}_{k}[x,y]$ is non-
zero by assumptions, the corresponding polynomial
$\widehat{Q}(z,\overline{z})=\sum_{j=0}^{k}c_{j}z^{j}\overline{z}^{k-j}$ is
also non-zero. Note that

(8.19)
$\displaystyle\begin{split}\sum_{j=0}^{k}a_{j}\Gamma_{a}^{k,j}(G)&=\sum_{j=0}^{k}a_{j}\mathfrak{G}(G_{1}^{j}G_{2}^{k-j})={\mathfrak{G}}(Q(G_{1},G_{2}))\\\
&=\mathfrak{G}(\widehat{Q}(G,\overline{G}))=\sum_{j=0}^{k}c_{j}\mathfrak{G}(G^{j}\overline{G}^{k-j}).\end{split}$

As $k<m(2a-1)\leq m-2$, in view of (8.18) and (8.19), for every $0\leq l\leq
2k$ we have

$\displaystyle 0$
$\displaystyle=\sum_{j=0}^{k}c_{j}\Big{(}\mathfrak{G}\big{(}(\omega^{l}G_{0})^{j}(\overline{\omega^{l}G_{0}})^{k-j}\big{)}+\mathfrak{G}\big{(}(\omega^{l+1}\overline{G_{0}})^{j}(\overline{\omega^{l+1}\overline{G_{0}}})^{k-j}\big{)}\Big{)}$
$\displaystyle=\omega^{-kl}\sum_{j=0}^{k}\omega^{2jl}c_{j}\big{(}\mathfrak{G}(G_{0}^{j}\overline{G_{0}}^{k-j})+\omega^{2j-k}\mathfrak{G}(\overline{G_{0}}^{j}{G_{0}}^{k-j})\big{)}.$

Let us consider the matrix $\Omega_{k}=[\omega^{2lj}]_{0\leq l,j\leq k}\in
M_{(k+1)\times(k+1)}({\mathbb{C}})$. As $k<m$, by the Vandermonde determinant,

$\det\Omega_{k}=\prod_{0\leq i<j\leq k}(\omega^{2j}-\omega^{2i})\neq 0.$

This gives

$c_{j}\Big{(}\mathfrak{G}\big{(}G_{0}^{j}\overline{G_{0}}^{k-j}\big{)}+\omega^{2j-k}\overline{\mathfrak{G}\big{(}{G^{j}_{0}}\overline{G_{0}}^{k-j}\big{)}}\Big{)}=0\text{
for all }0\leq j\leq k.$

As $c_{k-j}=\overline{c_{j}}$ and $\widehat{Q}$ is non-zero, there exists
$0\leq j\leq k/2$ such that $c_{j}\neq 0$. Then

$\displaystyle 0$
$\displaystyle=\omega_{0}^{k-2j}\Big{(}\mathfrak{G}\big{(}G_{0}^{j}\overline{G_{0}}^{k-j}\big{)}+\omega_{0}^{2j-k}\overline{\mathfrak{G}\big{(}{G^{j}_{0}}\overline{G_{0}}^{k-j}\big{)}}\Big{)}$
$\displaystyle=\mathfrak{G}\big{(}(\omega^{-1}_{0}G_{0})^{j}\overline{(\omega^{-1}_{0}G_{0})}^{k-j}\big{)}+\overline{\mathfrak{G}\big{(}{(\omega^{-1}_{0}G_{0})}^{j}\overline{(\omega^{-1}_{0}G_{0})}^{k-j}\big{)}}.$

Hence

$0=\Re\mathfrak{G}\big{(}(\omega^{-1}_{0}G_{0})^{j}\overline{(\omega^{-1}_{0}G_{0})}^{k-j}\big{)}=\mathfrak{G}\big{(}|G_{0}|^{j}\Re((\omega_{0}\overline{G_{0}})^{k-2j})\big{)}.$

Since $G_{0}$ is the principal $m$-th root, for all $u,s>0$ we have
$\operatorname{Arg}G_{0}(u,s)\in(0,\tfrac{\pi}{2m})$. Hence

$\operatorname{Arg}(\omega_{0}\overline{G_{0}(u,s)})\in(0,\tfrac{\pi}{2m})\text{
and
}\operatorname{Arg}(\omega_{0}\overline{G_{0}(u,s)})^{k-2j}\in(0,(k-2j)\tfrac{\pi}{2m})\subset(0,\tfrac{\pi}{2}),$

so $\Re((\omega_{0}\overline{G_{0}}(u,s))^{k-2j})>0$. As $2a-1-\frac{k}{m}>0$,
by the definition of $\mathfrak{G}$ we have

$\mathfrak{G}\big{(}|G_{0}|^{j}\Re((\omega_{0}\overline{G_{0}})^{k-2j})\big{)}<0,$

which is a contradiction. ∎

##  9\. Global properties of the operator $f\mapsto\varphi_{f}$ and correcting
operators

In this section, we use the results of the previous section to prove an
extended version of Theorem 4.1, which is Theorem 9.1.

For every $\sigma\in\mathrm{Fix}(\psi_{\mathbb{R}})$, let
$G_{\sigma}:{\mathbb{C}}\to{\mathbb{C}}$ be the principal $m_{\sigma}$-root
map and let $\omega_{\sigma}$ be the principal $2m_{\sigma}$-root of unity.
For every $0\leq k\leq m_{\sigma}-2$, recall that

(9.1) $a(\sigma)=\frac{m_{\sigma}-2}{m_{\sigma}},\quad
b(\sigma,k)=\frac{m_{\sigma}-2-k}{m_{\sigma}}.$

Denote by $C^{m}_{\sigma,k}(M)$ the space of maps $f\in C^{m}(M)$, which
vanish on
$\bigcup_{\sigma^{\prime}\in\mathrm{Fix}(\psi_{\mathbb{R}})\setminus\\{\sigma\\}}U_{\sigma^{\prime}}$
and $f^{(j)}(\sigma)=0$ for all $0\leq j<k$.

######  Theorem 9.1.

The following statements hold:

(i) For every $f\in C^{m}(M)$ we have
$\varphi_{f}\in\operatorname{P_{a}G}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$
and
$\varphi_{|f|}\in\operatorname{\widehat{P}_{a}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$,
where $a=\frac{m-2}{m}$. Moreover, the operator

$f\in
C^{m}(M)\mapsto\varphi_{f}\in\operatorname{P_{a}G}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$

is bounded. More precisely, there exists $C>0$ such that
$\|\varphi_{f}\|_{a}\leq C\|f\|_{C^{1}}$ for every $f\in C^{m}(M)$.

(ii) For every $\sigma\in\mathrm{Fix}(\psi_{\mathbb{R}})\cap M^{\prime}$ and
$0\leq k\leq m_{\sigma}-2$, there exists $C_{\sigma,k}>0$ such that for every
$f\in C^{m}_{\sigma,k}(M)$ we have
$\varphi_{f}\in\operatorname{P_{b(\sigma,k)}G}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$,
$\|\varphi_{f}\|_{b(\sigma,k)}\leq C_{\sigma,k}\|f\|_{C^{k+1}}$ and
$\varphi_{|f|}\in\operatorname{\widehat{P}_{b(\sigma,k)}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$.

(iii) Moreover, if additionally $\psi_{\mathbb{R}}$ is minimal on $M$ (i.e.
$M^{\prime}=M$), then for every $f\in C^{m}_{\sigma,k}(M)$ and for every
$\alpha\in\mathcal{A}$ the quantity $C^{\pm}_{\alpha}(\varphi_{f})$ is zero or
is of the form

(9.2)
$-\frac{1}{m^{2}_{\sigma}}\sum_{j=0}^{k}\binom{k}{j}\partial_{\sigma}^{(j,k-j)}(f)\Big{(}\Gamma_{\frac{m_{\sigma}-1}{m_{\sigma}}}^{k,j}\big{(}\omega_{\sigma}^{l}G_{\sigma}\big{)}+\Gamma_{\frac{m_{\sigma}-1}{m_{\sigma}}}^{k,j}\big{(}\omega_{\sigma}^{l+1}\overline{G_{\sigma}}\big{)}\Big{)}$

for some $0\leq l<2m_{\sigma}$. On the other hand, for every $0\leq
l<2m_{\sigma}$, there exists $\alpha\in\mathcal{A}$ such that
$C^{\pm}_{\alpha}(\varphi_{f})$ is of the form (9.2).

###### Proof.

Without loss of generality we can assume that $\psi_{\mathbb{R}}$ is minimal
on $M$. The proof of (i) and (ii) in the general case proceeds in the same way
up to some complications in notation.

Choose $\varepsilon>0$ such that $D_{\sigma,\varepsilon}\subset U_{\sigma}$
for any $\sigma\in\mathrm{Fix}(\psi_{\mathbb{R}})$, where
$D_{\sigma,\varepsilon}$ is a closed neighborhood of $\sigma$ defined in §8.
Denote by $g:I\to{\mathbb{R}}_{>0}\cup\\{+\infty\\}$ the first return time
map. Since the flow $\psi_{\mathbb{R}}$ is smooth and $f$ is of class $C^{m}$,
both $g$ and $\varphi_{f}$ belong to
$C^{1}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$. Moreover, for every
$x\in\bigcup_{\alpha\in\mathcal{A}}IntI_{\alpha}$ we have

$|\varphi_{f}^{\prime}(x)|\leq|g^{\prime}(x)|\|f\|_{C^{0}}+\|f^{\prime}\|_{C^{0}}\int_{0}^{g(x)}\Big{\|}\frac{d\psi_{t}(x)}{dx}\Big{\|}dt.$

If additionally $\operatorname{dist}(x,End(T))\geq\varepsilon$ then
$|\varphi_{f}^{\prime}(x)|\leq C_{\varepsilon}\|f\|_{C_{1}}$, where

$C_{\varepsilon}:=\max\Big{\\{}|g^{\prime}(x)|+\int_{0}^{g(x)}\Big{\|}\frac{d\psi_{t}(x)}{dx}\Big{\|}dt:x\in
I,\operatorname{dist}(x,End(T))\geq\varepsilon\Big{\\}}<+\infty.$

Let $e\in End(T)$ and suppose that $e$ is the first backward intersection of a
separatrix incoming to a fixed point
$\sigma\in\mathrm{Fix}(\psi_{\mathbb{R}})$. For every
$x\in(e-\varepsilon,e)\cup(e,e+\varepsilon)$, let
$0<\tau_{1}(x)<\tau_{2}(x)<g(x)$ be the entrance ($\tau_{1}(x)$) and the exit
($\tau_{2}(x)$) time of the orbit segment $\\{\psi_{t}x:0\leq t\leq g(x)\\}$
to $D_{\sigma,\varepsilon}$. Let us consider
$\varphi_{f}^{1},\varphi_{f}^{2}:(e-\varepsilon,e)\cup(e,e+\varepsilon)\to{\mathbb{R}}$
given by

$\varphi_{f}^{1}(x)=\int_{\tau_{1}(x)}^{\tau_{2}(x)}f(\psi_{t}x)\,dt,\quad\varphi_{f}^{2}(x)=\int_{0}^{\tau_{1}(x)}f(\psi_{t}x)\,dt+\int_{\tau_{2}(x)}^{g(x)}f(\psi_{t}x)\,dt.$

Of course, $\varphi_{f}(x)=\varphi_{f}^{1}(x)+\varphi_{f}^{2}(x)$ for every
$x\in(e-\varepsilon,e)\cup(e,e+\varepsilon)$. In view of Lemma 8.2 and Remark
8.3, there exists $0\leq l<m_{\sigma}$ such that for every
$s\in(0,\varepsilon)$ we have

$\displaystyle m_{\sigma}^{2}\varphi_{f}^{1}(e+s)$
$\displaystyle=\int_{0}^{\varepsilon}\frac{(f\cdot
V)(\omega_{\sigma}^{2l}G_{\sigma}(u,s))}{(u^{2}+s^{2})^{\frac{m_{\sigma}-1}{m_{\sigma}}}}du+\int_{0}^{\varepsilon}\frac{(f\cdot
V)(\omega_{\sigma}^{2l+1}\overline{G_{\sigma}}(u,s))}{(u^{2}+s^{2})^{\frac{m_{\sigma}-1}{m_{\sigma}}}}du,$
$\displaystyle m_{\sigma}^{2}\varphi_{f}^{1}(e-s)$
$\displaystyle=\int_{0}^{\varepsilon}\frac{(f\cdot
V)(\omega_{\sigma}^{2l+1}{G_{\sigma}}(u,s))}{(u^{2}+s^{2})^{\frac{m_{\sigma}-1}{m_{\sigma}}}}du+\int_{0}^{\varepsilon}\frac{(f\cdot
V)(\omega_{\sigma}^{2l+2}\overline{G_{\sigma}}(u,s))}{(u^{2}+s^{2})^{\frac{m_{\sigma}-1}{m_{\sigma}}}}du.$

Note that $2\big{(}\tfrac{m_{\sigma}-1}{m_{\sigma}}\big{)}-1=a(\sigma)$. In
view of (8.9), Lemma 8.5 and Remark 8.4, it follows that for every
$x\in(e-\varepsilon,e)\cup(e,e+\varepsilon)$ we have

(9.3) $\displaystyle|x-e|^{a(\sigma)}\varphi_{|f|}^{1}(x)$
$\displaystyle\leq\Gamma_{\frac{m_{\sigma}-1}{m_{\sigma}}}\|f\cdot
V\|_{C^{0}}\text{ if }m_{\sigma}>2,$ (9.4) $\displaystyle\varphi_{|f|}^{1}(x)$
$\displaystyle\leq 2\|f\cdot V\|_{C^{0}}(2+|\log(\varepsilon s)|)\text{ if
}m_{\sigma}=2,$ (9.5)
$\displaystyle|x-e|^{a(\sigma)+1}|(\varphi_{f}^{1})^{\prime}(x)|$
$\displaystyle\leq\frac{4\varepsilon^{-1/m_{\sigma}}\Gamma_{\frac{3(m_{\sigma}-1)}{2m_{\sigma}}}}{m^{2}_{\sigma}}\|f\cdot
V\|_{C^{1}}\leq\frac{\varepsilon^{-1/m}}{m_{\sigma}^{2}}\Gamma_{3/4}\|V\|_{C^{1}}\|f\|_{C^{1}},$
(9.6) $\displaystyle\lim_{x\to
e^{\pm}}|x-e|^{a(\sigma)+1}(\varphi_{f}^{1})^{\prime}(x)$
$\displaystyle=\mp\frac{2(a(\sigma)+1)}{m_{\sigma}^{2}}f(\sigma)V(\sigma)\Gamma_{\frac{2m_{\sigma}-1}{m_{\sigma}}}.$

If additionally $f^{(j)}(\sigma)=0$ for all $0\leq j<k$ ($1\leq k\leq
m_{\sigma}-2$), then by Lemma 8.6, we have

(9.7)
$\displaystyle|x-e|^{a(\sigma)-\frac{k}{m_{\sigma}}+1}|(\varphi_{f}^{1})^{\prime}(x)|$
$\displaystyle\leq\frac{2\Gamma_{1}}{m_{\sigma}^{2}(k-1)!}\|f\cdot
V\|_{C^{k}}\leq\frac{2\Gamma_{1}}{m_{\sigma}^{2}}\|V\|_{C^{k}}\|f\|_{C^{k}},$
(9.8)
$\displaystyle|x-e|^{a(\sigma)-\frac{k}{m_{\sigma}}}\varphi_{|f|}^{1}(x)$
$\displaystyle\leq\frac{2\Gamma_{\frac{2m_{\sigma}-1-k}{2m_{\sigma}}}}{m_{\sigma}^{2}k!}\|f\cdot
V\|_{C^{k}},$

and, by Lemma 8.8,

(9.9) $\displaystyle\begin{split}\lim_{x\to
e^{+}}&(x-e)^{a(\sigma)-\frac{k}{m_{\sigma}}+1}(\varphi_{f}^{1})^{\prime}(x)\\\
&=\frac{1}{m^{2}_{\sigma}}\sum_{j=0}^{k}\binom{k}{j}\partial_{\sigma}^{(j,k-j)}(f)\Big{(}\Gamma_{\frac{m_{\sigma}-1}{m_{\sigma}}}^{k,j}\big{(}\omega_{\sigma}^{2l}G_{\sigma}\big{)}+\Gamma_{\frac{m_{\sigma}-1}{m_{\sigma}}}^{k,j}\big{(}\omega_{\sigma}^{2l+1}\overline{G_{\sigma}}\big{)}\Big{)},\end{split}$
(9.10) $\displaystyle\begin{split}\lim_{x\to
e^{-}}&(e-x)^{a(\sigma)-\frac{k}{m_{\sigma}}+1}(\varphi_{f}^{1})^{\prime}(x)\\\
&=-\frac{1}{m^{2}_{\sigma}}\sum_{j=0}^{k}\binom{k}{j}\partial_{\sigma}^{(j,k-j)}(f)\Big{(}\Gamma_{\frac{m_{\sigma}-1}{m_{\sigma}}}^{k,j}\big{(}\omega_{\sigma}^{2l+1}G_{\sigma}\big{)}+\Gamma_{\frac{m_{\sigma}-1}{m_{\sigma}}}^{k,j}\big{(}\omega_{\sigma}^{2l+2}\overline{G_{\sigma}}\big{)}\Big{)}.\end{split}$

Since $\tau^{1}$ and $g-\tau^{2}$ can be $C^{1}$-extended to the intervals
$[e-\varepsilon,e]$ and $[e,e+\varepsilon]$, for every
$x\in(e-\varepsilon,e)\cup(e,e+\varepsilon)$ we have
$|(\varphi_{f}^{2})^{\prime}(x)|\leq C_{\sigma,\varepsilon}\|f\|_{C^{1}}$,
where

$\displaystyle C_{\sigma,\varepsilon}:$
$\displaystyle=\max\Big{\\{}|\tau_{1}^{\prime}(x)|+\int_{0}^{\tau_{1}(x)}\Big{\|}\frac{d\psi_{t}(x)}{dx}\Big{\|}dt:0<|x-e|<\varepsilon\Big{\\}}$
$\displaystyle\quad+\max\Big{\\{}|(g-\tau_{2})^{\prime}(x)|+\int_{0}^{g(x)-\tau_{2}(x)}\Big{\|}\frac{d\psi_{-t}(Tx)}{dx}\Big{\|}dt:0<|x-e|<\varepsilon\Big{\\}}<+\infty.$

As $a(\sigma)+1=\frac{2(m_{\sigma}-1)}{m_{\sigma}}\leq\frac{2(m-1)}{m}=a+1$
for every $\sigma\in\mathrm{Fix}(\psi_{\mathbb{R}})$, in view of (9.3)-(9.6),
it follows that
$\varphi_{f}\in\operatorname{P_{a}G}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$
and
$\varphi_{|f|}\in\operatorname{\widehat{P}_{a}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$
for every $f\in C^{m}(M)$ and

$p_{a}(\varphi)\leq\Big{(}\sum_{\sigma\in\mathrm{Fix}(\psi_{\mathbb{R}})}\big{(}\varepsilon^{-1/m}\Gamma_{3/4}\|V\|_{C^{1}}+m_{\sigma}\varepsilon^{a+1}C_{\sigma,\varepsilon}\big{)}+|I|^{a+1}C_{\varepsilon}\Big{)}\|f\|_{C^{1}}.$

As $\|\varphi_{f}\|_{L^{1}}\leq\|f\|_{L^{1}}\leq\mu(M)\|f\|_{C^{0}}$, there
exists $C>0$ such that $\|\varphi_{f}\|_{a}\leq C\|f\|_{C^{1}}$ for every
$f\in C^{m}(M)$.

Since $a(\sigma)-\frac{k}{m_{\sigma}}=b(\sigma,k)$, applying similar arguments
for functions $f\in C^{m}_{\sigma,k}(M)$ and using (9.7)-(9.10) (instead of
(9.3)-(9.6)), we obtain
$\varphi_{f}\in\operatorname{P_{b(\sigma,k)}G}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$,
$\varphi_{|f|}\in\operatorname{\widehat{P}_{b(\sigma,k)}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$
and the existence of $C_{\sigma,k}>0$ such that
$\|\varphi_{f}\|_{b(\sigma,k)}\leq C_{\sigma,k}\|f\|_{C^{k}}$ for every $f\in
C^{m}_{\sigma,k}(M)$.

Moreover, (9.9) applied to $e=l_{\alpha}$ and (9.10) applied to $e=r_{\alpha}$
and combined with the inequality $|(\varphi_{f}^{2})^{\prime}(x)|\leq
C_{\sigma,\varepsilon}\|f\|_{C^{1}}$, yields either

$\displaystyle C^{+}_{\alpha}(\varphi_{f})$
$\displaystyle=-\frac{1}{m^{2}_{\sigma}}\sum_{j=0}^{k}\binom{k}{j}\partial_{\sigma}^{(j,k-j)}(f)\Big{(}\Gamma_{\frac{m_{\sigma}-1}{m_{\sigma}}}^{k,j}\big{(}\omega_{\sigma}^{2l}G_{\sigma}\big{)}+\Gamma_{\frac{m_{\sigma}-1}{m_{\sigma}}}^{k,j}\big{(}\omega_{\sigma}^{2l+1}\overline{G_{\sigma}}\big{)}\Big{)},$
$\displaystyle C^{-}_{\alpha}(\varphi_{f})$
$\displaystyle=-\frac{1}{m^{2}_{\sigma}}\sum_{j=0}^{k}\binom{k}{j}\partial_{\sigma}^{(j,k-j)}(f)\Big{(}\Gamma_{\frac{m_{\sigma}-1}{m_{\sigma}}}^{k,j}\big{(}\omega_{\sigma}^{2l+1}G_{\sigma}\big{)}+\Gamma_{\frac{m_{\sigma}-1}{m_{\sigma}}}^{k,j}\big{(}\omega_{\sigma}^{2l+2}\overline{G_{\sigma}}\big{)}\Big{)}$

or $C^{\pm}_{\alpha}(\varphi_{f})=0$ whenever the forward semi-orbit of $e$
returns to $I$ for the first time to one of its ends without visiting singular
points. The latter option appears exactly twice. On the other hand, since
every incoming separatix crosses $I$, it follows that every number of the form
(9.2) is obtained as $C^{\pm}_{\alpha}(\varphi_{f})$ for some $\alpha$. ∎

###  9.1. Correcting operators for observables

Let us consider the basis $h_{1},\ldots,h_{g}$ of $U_{g+1}\subset
H(\pi^{(0)})$, defined in Remark 3.1, such that
$\lim_{k\rightarrow\infty}\frac{1}{k}\left\|Q(k)h_{i}\right\|=\lambda_{i}$ for
$1\leq i\leq g$. Given $0\leq b<1$ we choose $2\leq j\leq g+1$ such that
$\lambda_{j}\leq\lambda_{1}b<\lambda_{j-1}$. Since $h_{1},\ldots,h_{j-1}$ is a
basis of $U_{j}$ (see also Remark 3.1) and the correction operator
$\mathfrak{h}_{j}:\operatorname{P_{b}G}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})\to
U_{j}$ (defined in the proof of Theorem 6.1) is bounded, for every $1\leq i<j$
there exists a bounded operator
$d_{b,i}:\operatorname{P_{b}G}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})\rightarrow{\mathbb{R}}$
such that

(9.11)
$\mathfrak{h}_{j}(\varphi)=\sum_{i=1}^{j-1}d_{b,i}(\varphi)h_{i}\quad\text{for
every
}\varphi\in\operatorname{P_{b}G}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha}).$

By Lemma 7.4 in [5], for every $h_{i}\in U_{g+1}\subset H(\pi^{(0)})$ ($1\leq
i\leq g$) there exists $f_{i}\in C^{\infty}(M)$ such that
$\varphi_{f_{i}}=h_{i}$ and $f_{i}$ vanishes on
$\bigcup_{\sigma\in\mathrm{Fix}(\psi_{\mathbb{R}})}U_{\sigma}$.

Finally, for every $\sigma\in\mathrm{Fix}(\psi_{\mathbb{R}})\cap M^{\prime}$
and any $0\leq k\leq m_{\sigma}-2$ we define a _correcting operator_
$\mathfrak{R}_{\sigma,k}:C^{m}_{\sigma,k}\to C^{m}_{\sigma,k}$ given by

(9.12)
$\mathfrak{R}_{\sigma,k}(\xi)=\xi-\sum_{i=1}^{j-1}d_{b(\sigma,k),i}(\varphi_{\xi})f_{i}.$

The correcting operator does not change the observable around all fixed
points, but it removes the influence of Lyapunov exponents of the K-Z cocycle
on the asymptotic of Birkhoff integrals.

######  Proposition 9.2.

Let $\sigma\in\mathrm{Fix}(\psi_{\mathbb{R}})\cap M^{\prime}$ and $0\leq k\leq
m_{\sigma}-2$. Then for every $\xi\in C^{m}_{\sigma,k}$ we have

$\displaystyle\limsup_{T\rightarrow\infty}\frac{\log{\big{|}\int_{0}^{T}\mathfrak{R}_{\sigma,k}(\xi)(\psi_{t}x)\,dt\big{|}}}{\log
T}\leq b(\sigma,k)\text{ for a.e. }x\in M^{\prime};$
$\displaystyle\limsup_{T\rightarrow\infty}\frac{\log{\big{|}\int_{0}^{T}\mathfrak{R}_{\sigma,k}(\xi)\circ\psi_{t}\,dt\big{|}}_{L^{1}(M^{\prime})}}{\log
T}\leq b(\sigma,k).$

###### Proof.

In view of Theorem 9.1,
$\varphi_{\mathfrak{R}_{\sigma,k}(\xi)}\in\operatorname{P_{b}G}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$
and
$\varphi_{|\mathfrak{R}_{\sigma,k}(\xi)|}\in{\operatorname{\widehat{P}_{b}}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$
for $b=b(\sigma,k)$. Therefore, by the definition of the correcting operator,

$\displaystyle\mathfrak{h}_{j}(\varphi_{\mathfrak{R}_{\sigma,k}(\xi)})$
$\displaystyle=\mathfrak{h}_{j}(\varphi_{\xi})-\sum_{i=1}^{j-1}{d_{b,i}(\varphi_{\xi})}\mathfrak{h}_{j}(\varphi_{f_{i}})=\mathfrak{h}_{j}(\varphi_{\xi})-\sum_{i=1}^{j-1}{d_{b,i}(\varphi_{\xi})}\mathfrak{h}_{j}(h_{i})$
$\displaystyle=\mathfrak{h}_{j}(\varphi_{\xi})-\sum_{i=1}^{j-1}d_{b,i}(\varphi_{\xi})h_{i}=0.$

Hence, by Corollary 6.2, for every $\tau>0$ we have
$\|\mathcal{M}^{(k)}(S(k)\varphi_{\mathfrak{R}_{\sigma,k}(\xi)})\|=O(e^{(\lambda_{1}b+\tau)k})$.
Finally, Theorems 7.8 and 7.10 applied to $g=\varphi_{1}$ ($a=(m-2)/2$) and
$f=\mathfrak{R}_{\sigma,k}(\xi)$, complete the proof. ∎

Note that, in view of (1.11), the inequalities are optimal whenever
$\psi_{\mathbb{R}}$ on $M$ is minimal. Hence, it follows that the correction
provided by the operator $\mathfrak{R}_{\sigma,k}$ is the most optimal one.

##  10\. Complete power deviation spectrum of Birkhoff integrals

In this section by combining previous results, we prove the full deviation
spectrum of Birkhoff integrals for locally Hamiltonian flows.

###### Proof of Theorem 1.1.

The proof splits into five parts.

_Part I: Deviations around fixed points._ For every
$\sigma\in\mathrm{Fix}(\psi_{\mathbb{R}})$ and $\alpha\in{\mathbb{Z}}_{\geq
0}\times{\mathbb{Z}}_{\geq 0}$ with $|\alpha|<m_{\sigma}-2$, choose a map
$\bar{\xi}_{\sigma}^{\alpha}\in C^{m}(M)$ supported on the neighborhood
$U_{\sigma}$ of the fixed point $\sigma$ so that
$\partial_{\sigma}^{\beta}(\bar{\xi}_{\sigma}^{\alpha})=\delta_{\alpha\beta}$
for all $\beta\in{\mathbb{Z}}_{\geq 0}\times{\mathbb{Z}}_{\geq 0}$ with
$|\beta|\leq m$, where $\delta_{\alpha\beta}$ is the Kronecker delta, i.e.
$\delta_{\alpha\beta}=1$ if $\alpha=\beta$ and $\delta_{\alpha\beta}=0$ if
$\alpha\neq\beta$. By definition, $\bar{\xi}_{\sigma}^{\alpha}\in
C_{\sigma,|\alpha|}^{m}(M)$. Let
$\xi_{\sigma}^{\alpha}:=\mathfrak{R}_{\sigma,|\alpha|}(\bar{\xi}_{\sigma}^{\alpha})\in
C_{\sigma,|\alpha|}^{m}(M)$ and
$c_{\sigma,\alpha}(T,x):=\int_{0}^{T}\xi_{\sigma}^{\alpha}(\psi_{t}(x))dt$.
Then, in view of Proposition 9.2 applied to $\xi=\bar{\xi}_{\sigma}^{\alpha}$,
for every $\sigma\in\mathrm{Fix}(\psi_{\mathbb{R}})\cap M^{\prime}$ and
$\alpha\in{\mathbb{Z}}_{\geq 0}\times{\mathbb{Z}}_{\geq 0}$ with
$|\alpha|<m_{\sigma}-2$ we have (1.5) and (1.6). Recall that, by Lemma 8.1,
the corresponding distribution $\partial_{\sigma}^{\alpha}$ is bounded and
$\psi_{\mathbb{R}}$-invariant.

_Part II: Construction of the remainder._ Let us consider $f_{r}\in C^{m}(M)$
given by

(10.1)
$f=\sum_{\sigma\in\mathrm{Fix}(\psi_{\mathbb{R}})}\sum_{\begin{subarray}{c}\alpha\in{\mathbb{Z}}^{2}_{\geq
0}\\\
|\alpha|<m_{\sigma}-2\end{subarray}}\partial^{\alpha}_{\sigma}(f)\xi_{\sigma}^{\alpha}+f_{r}.$

In view of Theorem 9.1, we have
$\varphi_{f_{r}}\in\operatorname{P_{0}G}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$.
Indeed, let
$\\{\chi_{\sigma}:\sigma\in\mathrm{Fix}(\psi_{\mathbb{R}})\\}\subset C^{m}(M)$
be a partition of unity such that for any pair of fixed points
$(\sigma,\sigma^{\prime})$ we have
$\chi_{\sigma}(x)=\delta_{\sigma\sigma^{\prime}}$ for all $x\in
U_{\sigma^{\prime}}$. Then

$\displaystyle
f_{r}=\sum_{\sigma\in\mathrm{Fix}(\psi_{\mathbb{R}})}\Big{(}f\cdot\chi_{\sigma}-\sum_{|\alpha|<m_{\sigma}-2}\partial^{\alpha}_{\sigma}(f)\xi_{\sigma}^{\alpha}\Big{)}$

and
$f_{\sigma}:=f\cdot\chi_{\sigma}-\sum_{|\alpha|<m_{\sigma}-2}\partial^{\alpha}_{\sigma}(f)\xi_{\sigma}^{\alpha}$
vanishes on
$\bigcup_{\sigma^{\prime}\in\mathrm{Fix}(\psi_{\mathbb{R}})\setminus\\{\sigma\\}}U_{\sigma^{\prime}}$
and for every $\beta\in{\mathbb{Z}}^{2}_{\geq 0}$ with $|\beta|<m_{\sigma}-2$
we have

$\partial_{\sigma}^{\beta}(f_{\sigma})=\partial_{\sigma}^{\beta}(f\cdot\chi_{\sigma})-\sum_{|\alpha|<m_{\sigma}-2}\partial^{\alpha}_{\sigma}(f)\partial_{\sigma}^{\beta}(\xi_{\sigma}^{\alpha})=\partial_{\sigma}^{\beta}(f)-\sum_{|\alpha|<m_{\sigma}-2}\delta_{\alpha\beta}\partial^{\alpha}_{\sigma}(f)=0.$

Therefore, $(f_{\sigma})^{(l)}(\sigma)=0$ for all $0\leq l<m_{\sigma}-2$. As
$f_{\sigma}\in C^{m}_{\sigma,m_{\sigma}-2}$, in view of Theorem 9.1, it
follows that for every $\sigma\in\mathrm{Fix}(\psi_{\mathbb{R}})\cap
M^{\prime}$ we have
$\varphi_{f_{\sigma}}\in\operatorname{P_{0}G}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$
and

$\|\varphi_{f_{\sigma}}\|_{0}\leq C\|f_{\sigma}\|_{C^{m_{\sigma}-1}}\leq
C\|f\|_{C^{m_{\sigma}-1}}\|\chi_{\sigma}\|_{C^{m_{\sigma}-1}}+C\sum_{|\alpha|<m_{\sigma}-2}|\partial^{\alpha}_{\sigma}(f)|\|\xi_{\sigma}^{\alpha}\|_{C^{m_{\sigma}-1}}.$

By definition, for every $\sigma\in\mathrm{Fix}(\psi_{\mathbb{R}})$ and
$\alpha\in{\mathbb{Z}}_{\geq 0}^{2}$ there exists $C_{\sigma,\alpha}>0$ such
that $|\partial^{\alpha}_{\sigma}(f)|\leq
C_{\sigma,\alpha}\|f\|_{C^{|\alpha|}}$ for every $f\in C^{m}(M)$. It follows
that there exists another $C>0$ such that $\|\varphi_{f_{r}}\|_{0}\leq
C\|f\|_{C^{m-1}}$ for every $f\in C^{m}(M)$.

_Part III: Deviation of the remainder $f_{r}$._ Applying Theorem 6.1 to $a=0$,
we have a bounded (correction) operator
$\mathfrak{h}_{g+1}:\operatorname{P_{0}G}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})\rightarrow
U_{g+1}\subset H(\pi^{(0)})$ such that ${\mathfrak{h}_{g+1}}(h)=h$ for every
$h\in U_{g+1}$. Let us consider bounded operators
${d_{i}}:\operatorname{P_{0}G}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})\rightarrow{\mathbb{R}}$
for $1\leq i\leq g$ such that

(10.2)
$\mathfrak{h}_{g+1}(\varphi)=\sum_{i=1}^{g}{d_{i}}(\varphi)h_{i}\quad\text{for
every
}\varphi\in\operatorname{P_{0}G}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha}).$

Let $D_{i}:C^{m}(M)\rightarrow{\mathbb{R}}$, $1\leq i\leq g$ be operators
given by

$D_{i}({f})=d_{i}(\varphi_{f_{r}}),\text{ for }f\in C^{m}(M).$

Since $C^{m}(M)\ni
f\mapsto\varphi_{f_{r}}\in\operatorname{P_{0}G}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$
and
${d_{i}}:\operatorname{P_{0}G}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})\rightarrow{\mathbb{R}}$
are bounded linear operator, the operators $D_{i}$ are also bounded.

Recall that we have $f_{i}\in C^{\infty}(M)$ such that $\varphi_{f_{i}}=h_{i}$
for $1\leq i\leq g$. Let us consider $f_{e}\in C^{m}(M)$ given by

(10.3) $f_{r}=\sum_{i=1}^{g}D_{i}(f)f_{i}+f_{e}.$

For every $1\leq i\leq g$ let $u_{i}(T,x):=\int_{0}^{T}f_{i}(\psi_{t}(x))dt$.
As $\lim_{k\rightarrow\infty}\frac{1}{k}\left\|Q(k)h_{i}\right\|=\lambda_{i}$
for $1\leq i\leq g$, in view of Proposition 7.12, we have (1.7) and (1.8) with
$\nu_{i}=\lambda_{i}/\lambda_{1}$.

By the definition of $f_{e}$, we have

$\displaystyle\varphi_{f_{e}}=\varphi_{f_{r}}-\sum_{i=1}^{g}D_{i}(f)\varphi_{f_{i}}=\varphi_{f_{r}}-\sum_{i=1}^{g}D_{i}(f)h_{i}.$

As
$\varphi_{f_{r}}\in\operatorname{P_{0}G}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$,
we have
$\varphi_{f_{e}}\in\operatorname{P_{0}G}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$
and

$\displaystyle\mathfrak{h}_{g+1}(\varphi_{f_{e}})$
$\displaystyle=\mathfrak{h}_{g+1}(\varphi_{f_{r}})-\sum_{i=1}^{g}D_{i}(f)\mathfrak{h}_{g+1}(h_{i})=\mathfrak{h}_{g+1}(\varphi_{f_{r}})-\sum_{i=1}^{g}{d_{i}}(\varphi_{f_{r}})h_{i}=0.$

By Corollary 6.2, it follows that

$\|\mathcal{M}^{(k)}(S(k)\varphi_{f_{e}})\|=O(e^{\tau k})\text{ for every
}\tau>0.$

Let $err(f,T,x)=\int_{0}^{T}f_{e}(\psi_{t}(x))dt$. If $f_{e}\neq 0$ then, in
view of Proposition 7.13 and Remark 7.14, this gives (1.9) and (1.10).

By (10.1) and (10.3), we have

(10.4)
$f=\sum_{\sigma\in\mathrm{Fix}(\psi_{\mathbb{R}})}\sum_{\begin{subarray}{c}\alpha\in{\mathbb{Z}}^{2}_{\geq
0}\\\
|\alpha|<m_{\sigma}-2\end{subarray}}\partial^{\alpha}_{\sigma}(f)\xi_{\sigma}^{\alpha}+\sum_{i=1}^{g}D_{i}(f)f_{i}+f_{e}.$

Passing to the Birkhoff integrals, we obtain (1.4).

_Part IV: Invariance of distributions._ We need to show that the distributions
$D_{i}$ for $1\leq i\leq g$ are $\psi_{\mathbb{R}}$-invariant. By (10.4), for
every $s\in{\mathbb{R}}$ we have

$f\circ\psi_{s}=\sum_{\sigma\in\mathrm{Fix}(\psi_{\mathbb{R}})}\sum_{\begin{subarray}{c}\alpha\in{\mathbb{Z}}^{2}_{\geq
0}\\\
|\alpha|<m_{\sigma}-2\end{subarray}}\partial^{\alpha}_{\sigma}(f\circ\psi_{s})\xi_{\sigma}^{\alpha}+\sum_{i=1}^{g}D_{i}(f\circ\psi_{s})f_{i}+(f\circ\psi_{s})_{e}.$

Since
$\partial^{\alpha}_{\sigma}(f\circ\psi_{s})=\partial^{\alpha}_{\sigma}(f)$
(see Lemma 8.1), it follows that

$\bar{f}:=\sum_{i=1}^{g}D_{i}(f\circ\psi_{s}-f)f_{i}=f\circ\psi_{s}-f+f_{e}-(f\circ\psi_{s})_{e}.$

Note that for any $T>0$ we have

$\left|\int_{0}^{T}(f\circ\psi_{s}-f)(\psi_{t}x)\,dt\right|\leq\int_{0}^{s}|f(\psi_{t}x)|\,dt+\int_{T}^{T+s}|f(\psi_{t}x)|\,dt\leq
2s\|f\|_{C^{0}}.$

In view of (1.9), it follows that

(10.5)
$\limsup_{T\to+\infty}\frac{\log\left|\int_{0}^{T}\bar{f}(\psi_{t}x)\,dt\right|}{\log
T}\leq 0\text{ for a.e. }x\in M^{\prime}.$

On the other hand,

$\varphi_{\bar{f}}=\sum_{i=1}^{g}D_{i}(f\circ\psi_{s}-f)\varphi_{f_{i}}=\sum_{i=1}^{g}D_{i}(f\circ\psi_{s}-f)h_{i}\in
U_{g+1}.$

Suppose that, contrary to our claim, $D_{i}(f\circ\psi_{s}-f)\neq 0$ for some
$1\leq i\leq g$. As $h_{1},\ldots,h_{g}$ are linearly independent,

$h:=\varphi_{\bar{f}}=\sum_{i=1}^{g}D_{i}(f\circ\psi_{s}-f)h_{i}\neq 0.$

In view of (3.4), it follows that

$\lambda(h):=\lim_{k\to\infty}\frac{\log\|Q(k)h\|}{k}\geq\lambda_{g}>0.$

By Proposition 7.12, this yields

$\limsup_{T\to+\infty}\frac{\log\left|\int_{0}^{T}\bar{f}(\psi_{t}x)\,dt\right|}{\log
T}=\frac{\lambda(h)}{\lambda_{1}}>0\text{ for a.e. }x\in M^{\prime},$

contrary to (10.5). Consequently, $D_{i}(f\circ\psi_{s})=D_{i}(f)$ for all
$1\leq i\leq g$ and $s\in{\mathbb{R}}$.

_Part V: Lower bounds._ Suppose that $M^{\prime}=M$. Let us consider any $f\in
C^{m}_{\sigma,l}$ with $f^{(l)}(\sigma)\neq 0$ for $0\leq l<m_{\sigma}-2$ and
$\sigma\in\mathrm{Fix}(\psi_{\mathbb{R}})$. Then
$\varphi=\varphi_{f}\in{\operatorname{P_{b}G}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$
with $b=b(\sigma,l)>0$. The purpose of this part is to show (1.11).

In view of Theorem 9.1 (the last sentence) combined with Lemma 8.9 applied to
$a=\frac{m_{\sigma}-1}{m_{\sigma}}$ and $a_{i}=\partial^{(i,l-i)}_{\sigma}(f)$
for $0\leq i\leq l$ (at least one of them is non-zero, since
$f^{(l)}(\sigma)\neq 0$), there exists $\alpha\in\mathcal{A}$ such that
$C^{+}_{\alpha}(\varphi_{f})\neq 0$ or $C^{-}_{\alpha}(\varphi_{f})\neq 0$. We
focus only on the case $C^{+}_{\alpha}(\varphi_{f})\neq 0$. In the latter
case, the proof runs similarly.

For every $k\geq 1$ let us consider the interval
$\big{(}l^{(k)}_{\alpha},l^{(k)}_{\alpha}+\varepsilon|I^{(k)}_{\alpha}|\big{]}$
with

$\varepsilon:=\left(\frac{|C^{+}_{\alpha}|}{2^{\frac{4+4b}{b}}\kappa^{1+b}d\zeta(1+b)p_{b}(\varphi)}\right)^{1/(1+b)}.$

As $d,\kappa,\zeta(b+1)\geq 1$, by (4.1), we have $16\varepsilon^{b}\leq 1$.
In view of Proposition 5.5, for $k$ large enough ($|I^{(k)}|\leq\delta$) and
for every
$x\in\big{(}l^{(k)}_{\alpha},l^{(k)}_{\alpha}+\varepsilon|I^{(k)}_{\alpha}|\big{]}$,
we have

$\big{|}(x-l^{(k)}_{\alpha})^{1+b}(S(k)\varphi)^{\prime}(x)\big{|}\geq\frac{|C^{+}_{\alpha}|}{2}-2^{2+b}\kappa^{1+b}d\zeta(1+b)p_{b}(\varphi)\left(\frac{\varepsilon|I^{(k)}_{\alpha}|}{|I^{(k)}_{\alpha}|}\right)^{1+b}\geq\frac{|C^{+}_{\alpha}|}{4}>0.$

In view of Lemma 4.7, there exists an interval
$\widehat{J}\subset\big{(}l^{(k)}_{\alpha},l^{(k)}_{\alpha}+\varepsilon|I_{\alpha}^{(k)}|\big{]}$
such that

(10.6) $|\widehat{J}|\geq\frac{\varepsilon|I^{(k)}_{\alpha}|}{4}\text{ and }\
|(S(k)\varphi)(x)|\geq\frac{|C^{+}_{\alpha}|}{4(\varepsilon|I^{(k)}_{\alpha}|)^{b}}\geq\frac{|C^{+}_{\alpha}|}{|I^{(k)}_{\alpha}|^{b}}\text{
for }x\in\widehat{J}.$

Finally we can choose an interval $J^{(k)}\subset\widehat{J}$ such that

(10.7)
$\displaystyle|J^{(k)}|\geq\varepsilon|I^{(k)}_{\alpha}|/(2^{4}\kappa);$
(10.8)
$\displaystyle\operatorname{dist}(J^{(k)},End(T^{(k)}))\geq\varepsilon|I^{(k)}_{\alpha}|/2^{4};$
(10.9)
$\displaystyle\operatorname{dist}((T^{(k)})^{-1}J^{(k)},End(T^{(k)}))\geq\varepsilon|I^{(k)}_{\alpha}|/(2^{4}\kappa).$

Let us consider the set

$\displaystyle B_{k}:$
$\displaystyle=\\{T^{g}_{t}(x,0):x\in(T^{(k)})^{-1}J^{(k)},0\leq
t<(S(k)g)(x)\\}$ $\displaystyle=\\{T^{g}_{t}(x,0):x\in
J^{(k)},-(S(k)g)((T^{(k)})^{-1}x)\leq t<0\\}.$

As $g\geq\underline{g}>0$, by (10.6), (10.7) and (3.12), we have

$Leb(B_{k})=\int_{(T^{(k)})^{-1}J^{(k)}}S(k)g(x)\,dx\geq\underline{g}|J^{(k)}|\min_{\beta\in\mathcal{A}}Q_{\beta}(k)\geq\frac{\delta\varepsilon\underline{g}}{2^{4}\kappa^{2}}|I|.$

For every $(x^{\prime},r^{\prime})=T^{g}_{r}(x,0)\in B_{k}$ let

$\displaystyle\tau_{k}^{0}$
$\displaystyle=\tau_{k}^{0}(x^{\prime},r^{\prime}):=(S(k)g)(x)-r\text{ and }$
$\displaystyle\tau_{k}^{1}$
$\displaystyle=\tau_{k}^{1}(x^{\prime},r^{\prime}):=(S(k)g)(x)+(S(k)g)(T^{(k)}x)-r.$

Then $T^{g}_{\tau_{k}^{0}}(x^{\prime},r^{\prime})=(T^{(k)}x,0)$ and

$\displaystyle\int_{0}^{\tau_{k}^{1}}f(T^{g}_{t}(x^{\prime},r^{\prime}))\,dt-\int_{0}^{\tau_{k}^{0}}f(T^{g}_{t}(x^{\prime},r^{\prime}))\,dt$
$\displaystyle=\int_{0}^{(S(k)g)(T^{(k)}x)}f(T^{g}_{t}(T^{(k)}x,0))\,dt$
$\displaystyle=(S(k)\varphi_{f})(T^{(k)}x).$

As $x\in(T^{(k)})^{-1}J^{(k)}$, by (10.6), we have
$|(S(k)\varphi_{f})(T^{(k)}x)|\geq{|C^{+}_{\alpha}|}/|I^{(k)}_{\alpha}|^{b}$.
It follows that

(10.10)
$\max\Big{\\{}\Big{|}\int_{0}^{\tau_{k}^{1}}f(T^{g}_{t}(x^{\prime},r^{\prime}))\,dt\Big{|},\Big{|}\int_{0}^{\tau_{k}^{0}}f(T^{g}_{t}(x^{\prime},r^{\prime}))\,dt\Big{|}\Big{\\}}\geq\frac{|C^{+}_{\alpha}|}{2|I^{(k)}_{\alpha}|^{b}}.$

Choose $\tau_{k}=\tau_{k}(x^{\prime},r^{\prime})$ among $\tau_{k}^{0}$ and
$\tau_{k}^{1}$ such that

$\left|\int_{0}^{\tau_{k}}f(T^{g}_{t}(x^{\prime},r^{\prime}))\,dt\right|=\max\Big{\\{}\Big{|}\int_{0}^{\tau_{k}^{1}}f(T^{g}_{t}(x^{\prime},r^{\prime}))\,dt\Big{|},\Big{|}\int_{0}^{\tau_{k}^{0}}f(T^{g}_{t}(x^{\prime},r^{\prime}))\,dt\Big{|}\Big{\\}}.$

As $x\in(T^{(k)})^{-1}J^{(k)}$, in view of (10.8), (10.9) and (4.14), we have

$\displaystyle|(S(k)g)((T^{(k)})^{-1}x)|$
$\displaystyle\leq\|\mathcal{M}^{(k)}(S(k)g)\|+p_{a}(S(k)g)O(|I^{(k)}|^{-a})$
$\displaystyle|(S(k)g)(x)|$
$\displaystyle\leq\|\mathcal{M}^{(k)}(S(k)g)\|+p_{a}(S(k)g)O(|I^{(k)}|^{-a}).$

Moreover, by (6.6), (5.1) and (5.10), we have

$\|\mathcal{M}^{(k)}(S(k)g)\|\leq\frac{2\kappa}{|I^{(k)}|}\|S(k)g\|_{L^{1}(I^{(k)})}\leq\frac{2\kappa\|g\|_{L^{1}(I^{(0)})}}{|I^{(k)}|},\quad
p_{a}(S(k)g)\leq O(p_{a}(g)).$

Therefore,

$\displaystyle|(S(k)g)((T^{(k)})^{-1}x)|\leq
O(|I^{(k)}|^{-1}),\quad|(S(k)g)(x)|\leq O(|I^{(k)}|^{-1}).$

Hence there exists $C>0$ such that for every $k\geq 1$ and
$(x^{\prime},r^{\prime})\in B_{k}$ we have
$\tau_{k}(x^{\prime},r^{\prime})\leq C|I^{(k)}|^{-1}$. In view of (10.10), it
follows that for every $(x^{\prime},r^{\prime})\in B_{k}$ we have

$\frac{\log|\int_{0}^{\tau_{k}}f(T^{g}_{t}(x^{\prime},r^{\prime}))\,dt|}{\log\tau_{k}}\geq\frac{\log(|C^{+}_{\alpha}||I^{(k)}|^{-b}/2)}{\log(C|I^{(k)}|^{-1})}.$

Since $(B_{k})_{k\geq 1}$ is a sequence of asymptotically invariant sets (i.e.
for every $t\in{\mathbb{R}}$ we have $Leb(B_{k}\triangle T^{g}_{t}B_{k})\to 0$
as $k\to\infty$) and their measures are separated from zero, by the ergodicity
of the flow, a.e. $(x,r)\in I^{g}$ belongs to $B_{k}$ for infinitely many $k$.
It follows that for a.e. $(x,r)\in I^{g}$ we have

$\displaystyle\limsup_{T\to+\infty}\frac{\log|\int_{0}^{T}f(T^{g}_{t}(x,r))\,dt|}{\log
T}$
$\displaystyle\geq\limsup_{k\to+\infty}\frac{\log|\int_{0}^{\tau_{k}}f(T^{g}_{t}(x,r))\,dt|}{\log\tau_{k}}$
$\displaystyle\geq\lim_{k\to+\infty}\frac{\log(|C^{+}_{\alpha}||I^{(k)}|^{-b}/2)}{\log(C|I^{(k)}|^{-1})}=b.$

Finally, (1.12) follows directly from (1.5) and (1.11), since
$c_{\sigma,\alpha}(T,x)=\int_{0}^{T}\xi_{\sigma}^{\alpha}(\psi_{t}(x))dt$ and
$\xi_{\sigma}^{\alpha}\in C_{\sigma,|\alpha|}^{m}(M)$ with
$\partial_{\sigma}^{\alpha}(\xi_{\sigma}^{\alpha})=1$. ∎

## Acknowledgements

M.K. would like to thank the Center of Excellence “Dynamics, mathematical
analysis and artificial intelligence” at the Nicolaus Copernicus University in
Toruń for hospitality during his post-doc grant. Research was partially
supported by the Narodowe Centrum Nauki Grant 2017/27/B/ST1/00078.

##  Appendix A Proof of Theorem 3.2

We review the natural extension of the Rauzy-Veech induction and prove the
full measure of IETs satisfying LABEL:FDC.

###  A.1. Extension of the Rauzy-Veech induction

Let $\mathcal{G}\subset\mathcal{S}^{0}_{\mathcal{A}}$ be a Rauzy class and set
$\Delta^{\mathcal{A}}:=\\{\lambda\in{\mathbb{R}}_{>0}^{\mathcal{A}}:|\lambda|=1\\}.$
Let
${\mathcal{R}}:\mathcal{G}\times{\mathbb{R}}_{>0}^{\mathcal{A}}\rightarrow\mathcal{G}\times{\mathbb{R}}_{>0}^{\mathcal{A}}$
be the standard Rauzy-Veech map defined in §2.4 by

$\mathcal{R}(\pi,\lambda)=(\widetilde{\pi},\widetilde{\lambda}),\text{ where
}\widetilde{\lambda}=A^{-1}(\pi,\lambda)\text{ and $\widetilde{\pi}$ is given
by \eqref{def:pi}.}$

Then we define (normalized) Rauzy-Veech renormalization

$\widetilde{\mathcal{R}}:\mathcal{G}\times\Delta^{\mathcal{A}}\rightarrow\mathcal{G}\times\Delta^{\mathcal{A}},\quad\widetilde{\mathcal{R}}(\pi,\lambda)=(\tilde{\pi},\tilde{\lambda}/|\tilde{\lambda}|).$

By Veech [27], there exists an $\widetilde{\mathcal{R}}$-invariant ergodic
recurrent measure $\mu_{\mathcal{G}}$ which is equivalent to the product of
the counting measure on $\mathcal{G}$ and the Lebesgue measure on
$\Delta^{\mathcal{A}}$. For every $\pi\in\mathcal{S}^{0}_{\mathcal{A}}$, let

$\Theta_{\pi}:=\Big{\\{}\tau\in{\mathbb{R}}^{\mathcal{A}}:\sum_{\pi_{0}(\alpha)\leq
k}\tau_{\alpha}>0,\sum_{\pi_{1}(\alpha)\leq k}\tau_{\alpha}<0\text{ for }1\leq
k\leq d\Big{\\}}$

and let

$X(\mathcal{G}):=\bigcup_{\pi\in\mathcal{G}}\left\\{(\pi,\lambda,\tau)\in\\{\pi\\}\times\Delta^{\mathcal{A}}\times\Theta_{\pi}:\langle\lambda,\Omega_{\pi}\tau\rangle=1\right\\}.$

Then the natural (invertible) extension of $\widetilde{\mathcal{R}}$ is of the
form

$\widehat{\mathcal{R}}:X(\mathcal{G})\to
X(\mathcal{G}),\quad\widehat{\mathcal{R}}(\pi,\lambda,\tau)=\left(\tilde{\pi},\frac{A^{-1}(\pi,\lambda)\lambda}{|A^{-1}(\pi,\lambda)\lambda|},|A^{-1}(\pi,\lambda)\lambda|A^{-1}(\pi,\lambda)\tau\right).$

The natural extension, constructed by Veech in [27], of the measure
$\mu_{\mathcal{G}}$ on $X(\mathcal{G})$ is denoted by
$\widehat{\mu}_{\mathcal{G}}$. Then $\widehat{\mu}_{\mathcal{G}}$ is
$\widehat{\mathcal{R}}$-invariant and $\widehat{\mathcal{R}}$ is ergodic and
recurrent with respect to $\widehat{\mu}_{\mathcal{G}}$.

We extend the map $A:\mathcal{G}\times\Delta^{\mathcal{A}}\rightarrow
SL_{\mathcal{A}}({\mathbb{Z}})$ defined in §2.4 to
$\widehat{A}:X(\mathcal{G})\rightarrow SL_{\mathcal{A}}({\mathbb{Z}})$ given
by $\widehat{A}(\pi,\lambda,\tau)={A}(\pi,\tau)$. Let us consider the extended
cocycle $\widehat{A}:{\mathbb{Z}}\times X(\mathcal{G})\rightarrow
SL_{\mathcal{A}}({\mathbb{Z}})$

$\widehat{A}^{(n)}(\pi,\lambda,\tau)=\begin{cases}\widehat{A}(\pi,\lambda,\tau)\cdot\widehat{A}(\widehat{\mathcal{R}}(\pi,\lambda,\tau))\cdots\widehat{A}(\widehat{\mathcal{R}}^{n-1}(\pi,\lambda,\tau))&\text{
if }n\geq 0\\\
\widehat{A}(\widehat{\mathcal{R}}^{-1}(\pi,\lambda,\tau))\cdot\widehat{A}(\widehat{\mathcal{R}}^{-2}(\pi,\lambda,\tau))\cdots\widehat{A}(\widehat{\mathcal{R}}^{-n}(\pi,\lambda,\tau))&\text{
if }n<0.\end{cases}$

Then

(A.1) $\widehat{A}^{(n)}(\pi,\lambda,\tau)={A}^{(n)}(\pi,\tau)\text{ if }n\geq
0.$

Let $Y\subset X(\mathcal{G})$ be a subset such that
$0<\widehat{\mu}_{\mathcal{G}}(Y)<\infty$. For a.e $(\pi,\lambda,\tau)\in Y$,
let $r(\pi,\lambda,\tau)\geq 1$ by the first return time of
$(\pi,\lambda,\tau)$ for the map $\widehat{\mathcal{R}}$. Denote by
$\widehat{\mathcal{R}}_{Y}:Y\rightarrow Y$ the induced map and by
$\widehat{{A}}_{Y}:Y\rightarrow SL_{\mathcal{A}}({\mathbb{Z}})$ the induced
cocycle

$\widehat{\mathcal{R}}_{Y}(\pi,\lambda,\tau)=\widehat{\mathcal{R}}^{r(\pi,\lambda,\tau)}(\pi,\lambda,\tau),\quad\widehat{{A}}_{Y}(\pi,\lambda,\tau)=\widehat{{A}}^{(r(\pi,\lambda,\tau))}(\pi,\lambda,\tau)$

for a.e $(\pi,\lambda,\tau)\in Y$. Let $\widehat{\mu}_{Y}$ be the restriction
of $\widehat{\mu}_{\mathcal{G}}$ to $Y$.

###  A.2. Oseledets splitting

Assume that $\log\|\widehat{A}_{Y}\|$ and $\log\|\widehat{A}^{-1}_{Y}\|$ are
$\widehat{\mu}_{Y}$-integrable. By the Oseledets multiplicative theorem, the
symplecticity of $\widehat{A}_{Y}$ (see [31]) and the simplicity of spectrum
(see [1]), there exist Lyapunov exponents
$\lambda_{1}>\ldots>\lambda_{g}>\lambda_{g+1}=0$ such that for
$\widehat{\mu}_{Y}$-a.e. $(\pi,\lambda,\tau)\in Y$ we have a splitting

${\mathbb{R}}^{\mathcal{A}}=\bigoplus_{1\leq i\leq
g+1}F_{i}(\pi,\lambda,\tau)\oplus\bigoplus_{1\leq i\leq
g}F_{-i}(\pi,\lambda,\tau),$

for which

(A.2)
$\displaystyle\lim_{n\to\pm\infty}\frac{1}{n}\log\|\widehat{A}^{(n)}_{Y}(\pi,\lambda,\tau)^{t}\upharpoonright_{F_{i}(\pi,\lambda,\tau)}\|=\lambda_{i}\
\text{ if }1\leq i\leq g+1$ (A.3)
$\displaystyle\lim_{n\to\pm\infty}\frac{1}{n}\log\|\widehat{A}^{(n)}_{Y}(\pi,\lambda,\tau)^{t}\upharpoonright_{F_{-i}(\pi,\lambda,\tau)}\|=-\lambda_{i}\
\text{ if }1\leq i\leq g$ (A.4)
$\displaystyle\widehat{A}^{(n)}_{Y}(\pi,\lambda,\tau)^{t}F_{i}(\pi,\lambda,\tau)=F_{i}(\widehat{\mathcal{R}}^{n}_{Y}(\pi,\lambda,\tau))$

for all $i\in\\{-g,\ldots,-1,1,\ldots,g+1\\}$ and $n\in{\mathbb{Z}}$ and

$\dim F_{\pm i}(\pi,\lambda,\tau)=1\quad\text{for}\quad i=1,\ldots,g.$

Moreover, for every partition $\\{I_{1},I_{2}\\}$ of the set
$\\{-g,\ldots,-1,1,\ldots,g+1\\}$ we have

(A.5)
$\lim_{n\to\pm\infty}\frac{1}{n}\log\Big{|}\sin\angle\Big{(}\bigoplus_{i\in
I_{1}}F_{i}(\pi,\lambda,\tau),\bigoplus_{i\in
I_{2}}F_{i}(\pi,\lambda,\tau)\Big{)}\Big{|}=0,$

and

(A.6) $H(\pi):=\bigoplus_{i\neq g+1}F_{i}(\pi,\lambda,\tau).$

For every $1\leq j\leq g+1$ let

$\displaystyle E_{j}(\pi,\lambda,\tau)$ $\displaystyle:=\bigoplus_{j\leq i\leq
g+1}F_{i}(\pi,\lambda,\tau)\oplus\bigoplus_{1\leq i\leq
g}F_{-i}(\pi,\lambda,\tau)$ $\displaystyle U_{j}(\pi,\lambda,\tau)$
$\displaystyle:=\bigoplus_{1\leq i<j}F_{i}(\pi,\lambda,\tau)\subset H(\pi).$

Then

$E_{j}(\pi,\lambda,\tau)\oplus
U_{j}(\pi,\lambda,\tau)={\mathbb{R}}^{\mathcal{A}}.$

By (A.4), for every $n\in{\mathbb{Z}}$ we have

$\displaystyle\widehat{A}^{(n)}_{Y}(\pi,\lambda,\tau)^{t}E_{j}(\pi,\lambda,\tau)=E_{j}(\widehat{\mathcal{R}}^{n}_{Y}(\pi,\lambda,\tau)),\
\widehat{A}^{(n)}_{Y}(\pi,\lambda,\tau)^{t}U_{j}(\pi,\lambda,\tau)=U_{j}(\widehat{\mathcal{R}}^{n}_{Y}(\pi,\lambda,\tau)).$

In view of (A.2), (A.3) and (A.5), for every $1\leq j\leq g+1$ we have

(A.7)
$\displaystyle\lim_{n\to+\infty}\frac{1}{n}\log\|\widehat{A}^{(n)}_{Y}(\pi,\lambda,\tau)^{t}\upharpoonright_{E_{j}(\pi,\lambda,\tau)}\|=\lambda_{j}$
(A.8)
$\displaystyle\lim_{n\to+\infty}\frac{1}{n}\log\|\widehat{A}^{(-n)}_{Y}(\pi,\lambda,\tau)^{t}\upharpoonright_{U_{j}(\pi,\lambda,\tau)}\|=-\lambda_{j-1}\text{
if }j\geq 2$ (A.9)
$\displaystyle\lim_{n\to\pm\infty}\frac{1}{n}\log\left|\sin\angle\left(E_{j}(\widehat{\mathcal{R}}^{n}_{Y}(\pi,\lambda,\tau)),U_{j}(\widehat{\mathcal{R}}^{n}_{Y}(\pi,\lambda,\tau))\right)\right|=0\text{
if }j\geq 2.$

###  A.3. Proof of Theorem 3.2

The arguments used in the proof runs similarly to that used to prove Theorem
3.8 in [8]. We will omit some repetitive arguments.

###### Proof of Theorem 3.2.

Let us consider a subset $Y\subset X(\mathcal{G})$ which satisfies the
assumptions below:

  * $(i)$

the projection $\underline{Y}$ of $Y$ on
$\mathcal{G}\times\Lambda^{\mathcal{A}}$ is precompact with respect to the
Hilbert metric;

  * $(ii)$

there exists $0<\delta<1$ such that for every $(\pi,\lambda,\tau)\in Y$ we
have

$\min\Big{\\{}\Big{\\{}\sum_{\pi_{0}(\alpha)\leq k}\tau_{\alpha}:1\leq
k<d\Big{\\}}\cup\\{(\Omega_{\pi}(\tau))_{\alpha}:\alpha\in\mathcal{A}\\}\Big{\\}}>\delta\max\\{(\Omega_{\pi}(\tau))_{\alpha}:\alpha\in\mathcal{A}\\};$

  * $(iii)$

$\widehat{\mu}_{Y}$ is finite;

  * $(iv)$

the functions $\log\|\widehat{A}_{Y}\|$ and $\log\|\widehat{A}_{Y}^{-1}\|$ are
$\widehat{\mu}_{Y}$-integrable.

_Acceleration._ In view of (A.8) and (A.7), for every $\tau>0$ the maps

$\displaystyle Y\ni(\pi,\lambda,\tau)\mapsto\sup_{n\geq
0}e^{-(\lambda_{j}-\tau)n}\|\widehat{A}^{(n)}_{Y}(\pi,\lambda,\tau)^{t}\upharpoonright_{E_{j}(\pi,\lambda,\tau)}\|\in{\mathbb{R}}\text{
for }1\leq j\leq g+1,$ $\displaystyle Y\ni(\pi,\lambda,\tau)\mapsto\sup_{n\geq
0}e^{(\lambda_{j-1}+\tau)n}\|\widehat{A}^{(-n)}_{Y}(\pi,\lambda,\tau)^{t}\upharpoonright_{U_{j}(\pi,\lambda,\tau)}\|\in{\mathbb{R}}\text{
for }2\leq j\leq g+1$

are a.e. defined and measurable. Therefore, there exists a closed subset
$K\subset Y$ with $\widehat{\mu}_{Y}(K)/\widehat{\mu}_{Y}(Y)>1-\tau/2$ and a
constant $C>0$ such that if $(\pi,\lambda,\tau)\in K$ then for every $n\geq 0$
we have

(A.10)
$\displaystyle\|\widehat{A}^{(n)}_{Y}(\pi,\lambda,\tau)^{t}\upharpoonright_{E_{j}(\pi,\lambda,\tau)}\|\leq
Ce^{(\lambda_{j}+\tau)n}\text{ for }1\leq j\leq g+1,$ (A.11)
$\displaystyle\|(\widehat{A}^{(n)}_{Y}(\mathcal{R}_{Y}^{-n}(\pi,\lambda,\tau))^{t}\upharpoonright_{U_{j}(\pi,\lambda,\tau)})^{-1}\|\leq
Ce^{(-\lambda_{j-1}+\tau)n}\text{ for }2\leq j\leq g+1.$

Let $\widehat{\mathcal{R}}_{K}:K\to K$ be the induced map and let
$\widehat{A}_{K}:K\to SL_{\mathcal{A}}({\mathbb{Z}})$ be the induced cocycle,
i.e.

$\widehat{\mathcal{R}}_{K}(\pi,\lambda,\tau)=\widehat{\mathcal{R}}^{r_{K}(\pi,\lambda,\tau)}_{Y}(\pi,\lambda,\tau),$

where $r_{K}(\pi,\lambda,\tau)\geq 1$ is the first return time of
$(\pi,\lambda,\tau)\in K$ to $K$ for the map $\widehat{\mathcal{R}}_{Y}$ and

$\widehat{A}_{K}^{(n)}=\widehat{A}_{Y}^{(r_{K}^{(n)})}\text{ for every }n\geq
0,$

where $r_{K}^{(n)}:=\sum_{0\leq i<n}r_{K}\circ\widehat{\mathcal{R}}_{K}^{i}$
for every $n\geq 0$. Then

(A.12)
$\frac{r_{K}^{(n)}(\pi,\lambda,\tau)}{n}\to\frac{\widehat{\mu}_{Y}(Y)}{\widehat{\mu}_{Y}(K)}\in(1,1+\tau)\text{
for a.e. }(\pi,\lambda,\tau)\in K.$

In view of (A.10) and (A.11), for every $(\pi,\lambda,\tau)\in K$,

(A.13)
$\displaystyle\|\widehat{A}^{(n)}_{K}(\pi,\lambda,\tau)^{t}\upharpoonright_{E_{j}(\pi,\lambda,\tau)}\|\leq
Ce^{(\lambda_{j}+\tau)r_{K}^{(n)}(\pi,\lambda,\tau)}\text{ for }1\leq j\leq
g+1,$ (A.14)
$\displaystyle\|(\widehat{A}^{(n)}_{K}(\mathcal{R}_{K}^{-n}(\pi,\lambda,\tau))^{t}\upharpoonright_{U_{j}(\pi,\lambda,\tau)})^{-1}\|\leq
Ce^{(-\lambda_{j-1}+\tau)r_{K}^{(n)}(\mathcal{R}_{K}^{-n}(\pi,\lambda,\tau))}$

for $2\leq j\leq g+1$. Moreover, for a.e. $(\pi,\lambda,\tau)\in K$,

(A.15)
$\lim_{n\to+\infty}\frac{1}{n}\log\|\widehat{A}^{(n)}_{K}(\pi,\lambda,\tau)\|=\lambda_{1}\frac{\widehat{\mu}_{Y}(Y)}{\widehat{\mu}_{Y}(K)}\in(\lambda_{1},\lambda_{1}(1+\tau)).$

Since the maps $\log\|\widehat{A}_{K}\|$ and $\log\|\widehat{A}_{K}^{-1}\|$
are $\widehat{\mu}_{K}$-integrable, for a.e. $(\pi,\lambda,\tau)\in K$,

(A.16)
$\lim_{n\to+\infty}\frac{1}{n}\log\|\widehat{A}_{K}(\widehat{\mathcal{R}}_{K}^{n}(\pi,\lambda,\tau))\|=0.$

By the ergodicity of $\widehat{\mathcal{R}}:X(\mathcal{G})\to X(\mathcal{G})$,
for a.e. $(\pi,\lambda,\tau)\in X(\mathcal{G})$

(A.17) $\displaystyle\begin{split}&\text{there exists }n_{1}\geq 0\text{ such
that }\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau)\in K\\\ &\text{and
$\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau)$ satisfies
\eqref{eq:laangle}, \eqref{eq:retK}, \eqref{eq:maxlap} and
\eqref{eq:zzero}}.\end{split}$

By Fubini argument, there exists a measurable subset
$\Xi\subset\mathcal{G}\times\Lambda^{\mathcal{A}}$ such that
$\mu_{\mathcal{G}}(\mathcal{G}\times\Lambda^{\mathcal{A}}\setminus\Xi)=0$ and
for every $(\pi,\lambda)\in\Xi$, there exists $\tau\in\Theta_{\pi}$ such that
$(\pi,\lambda,\tau)\in X(\mathcal{G})$ satisfies (A.17).

_Full measure._ We now show that every $(\pi,\lambda)\in\Xi$ satisfies the
LABEL:FDC.

Suppose that $(\pi,\lambda)\in\Xi$ and $(\pi,\lambda,\tau)\in X(\mathcal{G})$
satisfies (A.17). Then the accelerating sequence $(n_{k})_{k\geq 0}$ required
by Definition 2 is defined as follows:

  * •

$n_{0}=0$;

  * •

for $k\geq 1$ we take $n_{k}$ so that
$\widehat{\mathcal{R}}^{n_{k}}(\pi,\lambda,\tau)=\widehat{\mathcal{R}}_{K}^{k-1}\widehat{\mathcal{R}}^{n_{1}(\pi,\lambda,\tau)}(\pi,\lambda,\tau)$.

Since $(\pi,\lambda,\tau)$ is Oseledets generic, $(\pi,\lambda)$ is also
Oseledets generic with the Oseledets filtration

$\displaystyle\\{0\\}=E_{0}(\pi,\lambda)\subset
E_{-1}(\pi,\lambda)\subset\ldots\subset E_{-g}(\pi,\lambda)\subset
E_{cs}(\pi,\lambda)$ $\displaystyle=E_{g+1}(\pi,\lambda)\subset
E_{g}(\pi,\lambda)\subset\ldots\subset E_{1}(\pi,\lambda)=\Gamma$

given by

$E_{j}(\pi,\lambda):=E_{j}(\pi,\lambda,\tau)\text{ for
}j=-1,-2,\ldots,-g,g+1,g,\ldots,2,1.$

We can define a complementary filtration $\\{0\\}=U_{1}\subset
U_{2}\subset\ldots\subset U_{g}\subset U_{g+1}$ by

$U_{j}=U_{j}(\pi,\lambda,\tau)\text{ for }1\leq j\leq g+1.$

Then for every $k\geq 1$,

$E_{j}^{(k)}=E_{j}(\widehat{\mathcal{R}}_{K}^{k-1}(\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau)))\text{
and
}U_{j}^{(k)}=U_{j}(\widehat{\mathcal{R}}_{K}^{k-1}(\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau))).$

By the definition of $Q$ and (A.1),
$Q(k,l)=\widehat{A}_{K}^{(l-k)}(\widehat{\mathcal{R}}_{K}^{k-1}(\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau)))^{t}$
for $1\leq k\leq l$, so

$\displaystyle\|Q|_{E_{j}}(k,l)\|$
$\displaystyle=\big{\|}\widehat{A}_{K}^{(l-k)}(\widehat{\mathcal{R}}_{K}^{k-1}(\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau)))^{t}\upharpoonright_{E_{j}(\widehat{\mathcal{R}}_{K}^{k-1}(\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau)))}\big{\|}$
$\displaystyle\|Q|_{U_{j}}(k,l)^{-1}\|$
$\displaystyle=\big{\|}(\widehat{A}_{K}^{(l-k)}(\widehat{\mathcal{R}}_{K}^{k-1}(\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau)))^{t}\upharpoonright_{U_{j}(\widehat{\mathcal{R}}_{K}^{k-1}(\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau)))})^{-1}\big{\|}$
$\displaystyle=\big{\|}(\widehat{A}_{K}^{(l-k)}(\widehat{\mathcal{R}}_{K}^{-(l-k)}(\widehat{\mathcal{R}}_{K}^{l-1}\circ\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau)))^{t}\upharpoonright_{U_{j}(\widehat{\mathcal{R}}_{K}^{k-1}(\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau)))})^{-1}\big{\|}.$

Since
$\widehat{\mathcal{R}}_{K}^{k-1}\circ\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau),\widehat{\mathcal{R}}_{K}^{l-1}\circ\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau)\in
K$ for every $1\leq k\leq l$, by (A.13) and (A.14), we have

$\displaystyle\|Q|_{E_{j}}(k,l)\|\leq
Ce^{(\lambda_{j}+\tau)r_{K}^{(l-k)}(\widehat{\mathcal{R}}_{K}^{k-1}\circ\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau))}\text{
for }1\leq j\leq g+1,$ $\displaystyle\|Q|_{U_{j}}(k,l)^{-1}\|\leq
Ce^{(-\lambda_{j-1}+\tau)r_{K}^{(l-k)}(\widehat{\mathcal{R}}_{K}^{k-1}\circ\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau))}\text{
for }2\leq j\leq g+1.$

Let us consider a sequence $(r_{n})_{n\geq 0}$ given by $r_{0}=1$ and for
$n\geq 1$,

$r_{n}=r_{K}(\widehat{\mathcal{R}}_{K}^{n-1}\circ\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau))).$

Then for all $1\leq k\leq l$ we have
$r(k,l)=r_{K}^{(l-k)}(\widehat{\mathcal{R}}_{K}^{k-1}\circ\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau))$,
so

$\|Q|_{E_{j}}(k,l)\|\leq Ce^{(\lambda_{j}+\tau)r(k,l)}\text{ and
}\|Q|_{U_{j}}(k,l)^{-1}\|\leq Ce^{(-\lambda_{j-1}+\tau)r(k,l)}.$

Both inequalities extend to the case where $k=0$. Then the constant $C$ must
be multiplied additionally by
$\max\\{\|Q(0,1)\|,e^{\lambda_{1}}\|Q(0,1)^{-1}\|\\}$. This gives (3.6) and
(3.7).

_Return time estimate._ Since
$\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau)\in K$ satisfies (A.12),
(A.15) and (A.16), we also have

(A.18)
$\displaystyle\frac{r(0,n)}{n}=\frac{1+r_{K}^{(n-1)}(\mathcal{R}^{n_{1}}(\pi,\lambda,\tau))}{n}\to\frac{\widehat{\mu}_{Y}(Y)}{\widehat{\mu}_{Y}(K)}\in(1,1+\tau),$
$\displaystyle\frac{1}{n}\log\|Z(n+1)\|=\frac{1}{n}\log\|\widehat{A}_{K}(\widehat{\mathcal{R}}_{K}^{n-1}\circ\mathcal{R}^{n_{1}}(\pi,\lambda,\tau))\|\to
0,$
$\displaystyle\frac{1}{n}\log\|Q(1,n)\|=\frac{1}{n}\log\|\widehat{A}^{(n-1)}_{K}(\mathcal{R}^{n_{1}}(\pi,\lambda,\tau))\|\to\lambda_{1}\frac{\widehat{\mu}_{Y}(Y)}{\widehat{\mu}_{Y}(K)}\in(\lambda_{1},\lambda_{1}(1+\tau)).$

As $\|Q(0,1)^{-1}\|^{-1}\|Q(1,n)\|\leq\|Q(0,n)\|\leq\|Q(0,1)\|\|Q(1,n)\|$,
this gives

(A.19)
$\frac{1}{n}\log\|Q(n)\|\to\lambda_{1}\frac{\widehat{\mu}_{Y}(Y)}{\widehat{\mu}_{Y}(K)}\in(\lambda_{1},\lambda_{1}(1+\tau)).$

The above convergences lead directly to (3.5), (3.8) and (3.9).

_Angle estimate._ Since,

$\displaystyle\frac{\log\left|\sin\angle\left(E_{j}^{(k)},U_{j}^{(k)}\right)\right|}{\log\|Q(k)\|}=$
$\displaystyle\frac{\log\left|\sin\angle\left(E_{j}(\widehat{\mathcal{R}}_{K}^{k-1}(\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau))),U_{j}(\widehat{\mathcal{R}}_{K}^{k-1}(\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau)))\right)\right|}{r_{K}^{(k-1)}(\mathcal{R}^{n_{1}}(\pi,\lambda,\tau))}$
$\displaystyle\cdot\frac{r_{K}^{(k-1)}(\mathcal{R}^{n_{1}}(\pi,\lambda,\tau))}{k}\frac{k}{\log\|Q(k)\|},$

in view of (A.9), (A.18) and (A.19),

$\lim_{k\to\infty}\frac{\log\left|\sin\angle\left(E_{j}^{(k)},U_{j}^{(k)}\right)\right|}{\log\|Q(k)\|}=\frac{0}{\lambda_{1}}=0>-\tau.$

This leads to (3.11).

_Rokhlin tower condition._ Since
$\widehat{\mathcal{R}}^{n_{k}}(\pi,\lambda,\tau)\in Y$ and the set $Y\subset
X(\mathcal{G})$ is chosen to satisfy the conditions $(i)$ and $(ii)$, by the
proof of Lemma 3.6 in [8], $(n_{k})_{k\geq 0}$ is a Rokhlin-balanced
accelerating sequence. Thus $(n_{k})_{k\geq 0}$ satisfies (3.10) and (RT). ∎

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