1 Introduction
For the non-autonomous dynamics
$$\begin{eqnarray}x_{n+1}=A_{n}x_{n},\quad n\in \mathbb{N},\end{eqnarray}$$
defined by a sequence of matrices
$A_{n}$
, we consider the notion of a non-uniform exponential trichotomy for an arbitrary growth rate. In particular, the dynamics may have contracting, expanding and neutral directions in which the usual exponential behavior is replaced by one with respect to an arbitrary sequence
$\unicode[STIX]{x1D70C}(n)$
. This includes situations in which either all Lyapunov exponents are zero or all Lyapunov exponents are infinity, although still with well-separated contracting, expanding and neutral directions (we note that some of them may be absent).
The purpose of our work is two-fold.
-
(1) To use a regularity coefficient in order to show that the non-uniform exponential trichotomies occur naturally. Namely, using the information provided by asymptotic behavior of the dynamics and its adjoint, under mild assumptions we show that virtually all dynamics admit a non-uniform exponential trichotomy.
-
(2) To provide several alternative characterizations of the trichotomies for which the non-uniform part is arbitrarily small. This includes characterizations in terms of the growth rate of volumes and of the Lyapunov exponents of the dynamics and its adjoint.
In addition, we also obtain sharp lower and upper bounds for the regularity coefficient. This allows one to control the non-uniform part of an exponential trichotomy, that is, the part that deviates from the usual uniform exponential behavior along contracting, expanding and central directions. It turns out that the constants in the inequalities defining an exponential trichotomy can be expressed using simply the Lyapunov exponents of the original dynamics and of its adjoint dynamics. Some related results for the usual exponential growth rate
$\unicode[STIX]{x1D70C}(n)=n$
were obtained earlier in [Reference Barreira and Valls2].
Among the main technical difficulties caused by the presence of the non-uniform part in an exponential trichotomy is the fact that even though one can still establish the exponential stability of solutions along appropriate directions, the size of the neighborhood in which the initial condition must be chosen so that the corresponding solution satisfies a prescribed bound may depend on the initial time. In the uniform case the size of the neighborhood is independent of the initial time. We refer the reader to the books [Reference Barreira and Pesin1, Reference Barreira and Valls3] for details. It should be emphasized that the notion of a uniform exponential trichotomy is very stringent and so it is of considerable interest to look for more general types of hyperbolic behavior. This is precisely what happens with the notion of a non-uniform exponential trichotomy, particularly in the context of ergodic theory. Namely, from the point of view of ergodic theory almost all linear variational equations obtained from a measure-preserving flow admit a non-uniform exponential trichotomy up to an appropriate change of coordinates. On the other hand, there is also a large class of systems exhibiting a uniform exponential behavior (see for example the classical series of papers [Reference Sacker6–Reference Sacker and Sell10] and the books [Reference Hale4, Reference Henry5, Reference Sell and You11]).
2 Non-uniform exponential trichotomies
We first introduce the notion of a non-uniform exponential trichotomy for an arbitrary growth rate. Given a sequence of invertible
$k\times k$
matrices
$(A_{n})_{n\in \mathbb{N}}$
, let
$$\begin{eqnarray}{\mathcal{A}}(m,n)=\left\{\begin{array}{@{}ll@{}}A_{m-1}\cdots A_{n},\quad & m>n,\\ \text{Id},\quad & m=n.\end{array}\right.\end{eqnarray}$$
Now let
$\unicode[STIX]{x1D70C}:\mathbb{N}\rightarrow \mathbb{N}$
be an increasing function such that
$$\begin{eqnarray}\lim _{n\rightarrow +\infty }\frac{\unicode[STIX]{x1D70C}(n+1)}{\unicode[STIX]{x1D70C}(n)}=1.\end{eqnarray}$$
We say that the sequence
$(A_{n})_{n\in \mathbb{N}}$
admits a
$\unicode[STIX]{x1D70C}$
-non-uniform exponential trichotomy if there exist projections
$P_{n}$
,
$Q_{n}$
and
$R_{n}$
for
$n\in \mathbb{N}$
such that
$P_{n}+Q_{n}+R_{n}=\text{Id}$
and
$$\begin{eqnarray}P_{m}{\mathcal{A}}(m,n)={\mathcal{A}}(m,n)P_{n},\qquad Q_{m}{\mathcal{A}}(m,n)={\mathcal{A}}(m,n)Q_{n},\end{eqnarray}$$
$$\begin{eqnarray}R_{m}{\mathcal{A}}(m,n)={\mathcal{A}}(m,n)R_{n}\end{eqnarray}$$
for
$m,n\in \mathbb{N}$
and there exist constants
$$\begin{eqnarray}\displaystyle & -\infty \leqslant \text{}\underline{a}\leqslant \overline{a}<\text{}\underline{c}\leqslant \overline{c}<\text{}\underline{b}\leqslant \overline{b}\leqslant +\infty , & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & a,b,c\geqslant 0\quad \text{and}\quad D\geqslant 1 & \displaystyle\end{eqnarray}$$
such that
$$\begin{eqnarray}\begin{array}{@{}rcl@{}}\Vert {\mathcal{A}}(m,n)P_{n}\Vert \; & {\leqslant}\; & D\text{e}^{\overline{a}(\unicode[STIX]{x1D70C}(m)-\unicode[STIX]{x1D70C}(n))+a\unicode[STIX]{x1D70C}(n)},\\ \Vert {\mathcal{A}}(m,n)^{-1}P_{m}\Vert \; & {\leqslant}\; & D\text{e}^{-\text{}\underline{a}(\unicode[STIX]{x1D70C}(m)-\unicode[STIX]{x1D70C}(n))+a\unicode[STIX]{x1D70C}(m)},\\ \Vert {\mathcal{A}}(m,n)Q_{n}\Vert \; & {\leqslant}\; & D\text{e}^{\overline{b}(\unicode[STIX]{x1D70C}(m)-\unicode[STIX]{x1D70C}(n))+b\unicode[STIX]{x1D70C}(n)},\\ \Vert {\mathcal{A}}(m,n)^{-1}Q_{m}\Vert \; & {\leqslant}\; & D\text{e}^{-\text{}\underline{b}(\unicode[STIX]{x1D70C}(m)-\unicode[STIX]{x1D70C}(n))+b\unicode[STIX]{x1D70C}(m)},\\ \Vert {\mathcal{A}}(m,n)R_{n}\Vert \; & {\leqslant}\; & D\text{e}^{\overline{c}(\unicode[STIX]{x1D70C}(m)-\unicode[STIX]{x1D70C}(n))+c\unicode[STIX]{x1D70C}(n)},\\ \Vert {\mathcal{A}}(m,n)^{-1}R_{m}\Vert \; & {\leqslant}\; & D\text{e}^{-\text{}\underline{c}(\unicode[STIX]{x1D70C}(m)-\unicode[STIX]{x1D70C}(n))+c\unicode[STIX]{x1D70C}(m)}\end{array}\end{eqnarray}$$
for
$m\geqslant n$
. When
$R_{n}=0$
, that is,
$P_{n}+Q_{n}=\text{Id}$
for
$n\in \mathbb{N}$
, we say that the sequence
$(A_{n})_{n\in \mathbb{N}}$
admits a
$\unicode[STIX]{x1D70C}$
-non-uniform exponential dichotomy. In this case the identities in (2) correspond respectively to the invariance of the stable and unstable spaces
$$\begin{eqnarray}E_{n}=P_{n}(\mathbb{R}^{k})\quad \text{and}\quad F_{n}=Q_{n}(\mathbb{R}^{k}),\end{eqnarray}$$
that is,
$$\begin{eqnarray}{\mathcal{A}}(m,n)E_{n}=E_{m}\quad \text{and}\quad {\mathcal{A}}(m,n)F_{n}=F_{m}\end{eqnarray}$$
for
$m,n\in \mathbb{N}$
. Finally, when
$P_{n}=\text{Id}$
for
$n\in \mathbb{N}$
, we say that the sequence
$(A_{n})_{n\in \mathbb{N}}$
admits a
$\unicode[STIX]{x1D70C}$
-non-uniform exponential contraction.
The constants
$a$
,
$b$
and
$c$
in (4) measure the non-uniformity of the trichotomy. It turns out that these constants as well as those in (3) are related to the Lyapunov exponents (see §4).
For simplicity of the exposition, we shall only consider the particular case when each matrix
$A_{n}$
is in block form. Namely, we assume that there exists a decomposition
$$\begin{eqnarray}\mathbb{R}^{k}=E\oplus F\oplus G\end{eqnarray}$$
with respect to which each matrix
$A_{n}$
has the block form
$$\begin{eqnarray}\left(\begin{array}{@{}ccc@{}}B_{n} & 0 & 0\\ 0 & C_{n} & 0\\ 0 & 0 & D_{n}\end{array}\right).\end{eqnarray}$$
Given
$v=(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702},\unicode[STIX]{x1D708})\in E\times F\times G$
, we have
$$\begin{eqnarray}{\mathcal{A}}(m,n)v=({\mathcal{B}}(m,n)\unicode[STIX]{x1D709},{\mathcal{C}}(m,n)\unicode[STIX]{x1D702},{\mathcal{D}}(m,n)\unicode[STIX]{x1D708}),\end{eqnarray}$$
where
$$\begin{eqnarray}{\mathcal{B}}(m,n)=\left\{\begin{array}{@{}ll@{}}B_{m-1}\cdots B_{n},\quad & m>n,\\ \text{Id},\quad & m=n,\end{array}\right.\qquad {\mathcal{C}}(m,n)=\left\{\begin{array}{@{}ll@{}}C_{m-1}\cdots C_{n},\quad & m>n,\\ \text{Id},\quad & m=n\end{array}\right.\end{eqnarray}$$
and
$$\begin{eqnarray}{\mathcal{D}}(m,n)=\left\{\begin{array}{@{}ll@{}}D_{m-1}\cdots D_{n},\quad & m>n,\\ \text{Id},\quad & m=n.\end{array}\right.\end{eqnarray}$$
3 Lyapunov exponents
Let
$\unicode[STIX]{x1D70C}:\mathbb{N}\rightarrow \mathbb{N}$
be an increasing function and let
$(A_{n})_{n\in \mathbb{N}}$
be a sequence of invertible
$k\times k$
matrices such that
$$\begin{eqnarray}\limsup _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log ^{+}\max \{\Vert A_{n}\Vert ,\Vert A_{n}^{-1}\Vert \}=0,\end{eqnarray}$$
where
$\log ^{+}x=\max \{0,\log x\}$
. The
$\unicode[STIX]{x1D70C}$
-Lyapunov exponent (defined by the sequence
$(A_{n})_{n\in \mathbb{N}}$
) is the function
$\unicode[STIX]{x1D712}:\mathbb{R}^{k}\rightarrow \mathbb{R}\cup \{-\infty \}$
given by
$$\begin{eqnarray}\unicode[STIX]{x1D712}(v)=\limsup _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log \Vert {\mathcal{A}}(n,1)v\Vert .\end{eqnarray}$$
Condition (7) ensures in particular that
$\unicode[STIX]{x1D712}$
cannot take the value
$+\infty$
.
Proposition 1. The function
$\unicode[STIX]{x1D712}$
is a Lyapunov exponent, that is, the following properties hold:
-
(1)
$\unicode[STIX]{x1D712}(0)=-\infty$
; -
(2)
$\unicode[STIX]{x1D712}(cv)=\unicode[STIX]{x1D712}(v)$
for
$c\in \mathbb{R}\setminus \{0\}$
and
$v\in \mathbb{R}^{k}$
; -
(3)
$\unicode[STIX]{x1D712}(u+v)\leqslant \max \{\unicode[STIX]{x1D712}(u),\unicode[STIX]{x1D712}(v)\}$
for
$u,v\in \mathbb{R}^{k}$
.
Proof. The first two properties follow immediately from (8). For the last one, we note that given
$u,v\in \mathbb{R}^{k}$
and
$\unicode[STIX]{x1D700}>0$
, we have
$$\begin{eqnarray}\Vert {\mathcal{A}}(n,1)u\Vert \leqslant e^{(\unicode[STIX]{x1D712}(u)+\unicode[STIX]{x1D700})\unicode[STIX]{x1D70C}(n)}\Vert u\Vert \quad \text{and}\quad \Vert {\mathcal{A}}(n,1)v\Vert \leqslant e^{(\unicode[STIX]{x1D712}(v)+\unicode[STIX]{x1D700})\unicode[STIX]{x1D70C}(n)}\Vert v\Vert\end{eqnarray}$$
for all sufficiently large
$n$
. Therefore,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D712}(u+v) & {\leqslant} & \displaystyle \limsup _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log (\Vert {\mathcal{A}}(n,1)u\Vert +\Vert {\mathcal{A}}(n,1)v\Vert )\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \limsup _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log (\text{e}^{(\unicode[STIX]{x1D712}(u)+\unicode[STIX]{x1D700})\unicode[STIX]{x1D70C}(n)}\Vert u\Vert +\text{e}^{(\unicode[STIX]{x1D712}(v)+\unicode[STIX]{x1D700})\unicode[STIX]{x1D70C}(n)}\Vert v\Vert )\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \limsup _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log (2\text{e}^{(\max \{\unicode[STIX]{x1D712}(u),\unicode[STIX]{x1D712}(v)\}+\unicode[STIX]{x1D700})\unicode[STIX]{x1D70C}(n)}(\Vert u\Vert +\Vert v\Vert ))\nonumber\\ \displaystyle & = & \displaystyle \max \{\unicode[STIX]{x1D712}(u),\unicode[STIX]{x1D712}(v)\}+\unicode[STIX]{x1D700}\nonumber\end{eqnarray}$$
and it follows from the arbitrariness of
$\unicode[STIX]{x1D700}$
that
$$\begin{eqnarray}\unicode[STIX]{x1D712}(u+v)\leqslant \max \{\unicode[STIX]{x1D712}(u),\unicode[STIX]{x1D712}(v)\}.\end{eqnarray}$$
This completes the proof of the proposition. ◻
Since
$\unicode[STIX]{x1D712}$
is a Lyapunov exponent, by the abstract theory of Lyapunov exponents (see [Reference Barreira and Pesin1, §1.2]),
$\unicode[STIX]{x1D712}$
takes at most finitely many values on
$\mathbb{R}^{k}\setminus \{0\}$
, say
$\unicode[STIX]{x1D706}_{1}<\cdots <\unicode[STIX]{x1D706}_{r}$
for some integer
$r\leqslant k$
. We note that in general
$\unicode[STIX]{x1D706}_{1}$
may be
$-\infty$
. For
$i=1,\ldots ,r$
, let
$$\begin{eqnarray}V_{i}=\{v\in \mathbb{R}^{k}:\unicode[STIX]{x1D712}(v)\leqslant \unicode[STIX]{x1D706}_{i}\}.\end{eqnarray}$$
It follows from Proposition 1 that
$V_{i}$
is a vector space. The number
$$\begin{eqnarray}k_{i}=\dim V_{i}-\dim V_{i-1},\end{eqnarray}$$
with the convention that
$V_{0}=\{0\}$
, is called the multiplicity of the value
$\unicode[STIX]{x1D706}_{i}$
.
Now we introduce the notion of Lyapunov regularity. For this we consider the sequence of matrices
$B_{n}=(A_{n}^{\ast })^{-1}$
for
$n\in \mathbb{N}$
, where
$A^{\ast }$
denotes the transpose of
$A$
. Let
$$\begin{eqnarray}\tilde{{\mathcal{A}}}(m,n)=({\mathcal{A}}(m,n)^{\ast })^{-1}=\left\{\begin{array}{@{}ll@{}}(A_{m-1}^{\ast })^{-1}\cdots (A_{n}^{\ast })^{-1},\quad & m>n,\\ \text{Id},\quad & m=n.\end{array}\right.\end{eqnarray}$$
We also consider the function
$\tilde{\unicode[STIX]{x1D712}}:\mathbb{R}^{k}\rightarrow \mathbb{R}\cup \{-\infty \}$
defined by
$$\begin{eqnarray}\tilde{\unicode[STIX]{x1D712}}(w)=\limsup _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log \Vert \tilde{{\mathcal{A}}}(n,1)w\Vert .\end{eqnarray}$$
Condition (7) ensures that
$\tilde{\unicode[STIX]{x1D712}}$
cannot take the value
$+\infty$
. Repeating arguments in the proof of Proposition 1, one can show that
$\tilde{\unicode[STIX]{x1D712}}$
is a Lyapunov exponent. By the abstract theory of Lyapunov exponents,
$\tilde{\unicode[STIX]{x1D712}}$
takes at most finitely many values on
$\mathbb{R}^{k}\setminus \{0\}$
, say
$\tilde{\unicode[STIX]{x1D706}}_{s}<\cdots <\tilde{\unicode[STIX]{x1D706}}_{1}$
for some integer
$s\leqslant k$
. We note that in general
$\tilde{\unicode[STIX]{x1D706}}_{s}$
may be
$-\infty$
.
We define the regularity coefficient of (the pair of Lyapunov exponents)
$\unicode[STIX]{x1D712}$
and
$\tilde{\unicode[STIX]{x1D712}}$
by
$$\begin{eqnarray}\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D712},\tilde{\unicode[STIX]{x1D712}})=\min \max \{\unicode[STIX]{x1D712}(v_{i})+\tilde{\unicode[STIX]{x1D712}}(w_{i}):1\leqslant i\leqslant k\},\end{eqnarray}$$
where the minimum is taken over all dual bases
$v_{1},\ldots ,v_{k}$
and
$w_{1},\ldots ,w_{k}$
of
$\mathbb{R}^{k}$
, that is, all bases such that
$\langle v_{i},w_{\!j}\rangle =\unicode[STIX]{x1D6FF}_{ij}$
for
$i,j=1,\ldots ,k$
(where
$\unicode[STIX]{x1D6FF}_{ij}$
is the Kronecker symbol). Since
$$\begin{eqnarray}\displaystyle 1 & = & \displaystyle \langle v_{i},w_{i}\rangle =\langle {\mathcal{A}}(n,1)v_{i},({\mathcal{A}}(n,1)^{\ast })^{-1}w_{i}\rangle \nonumber\\ \displaystyle & {\leqslant} & \displaystyle \Vert {\mathcal{A}}(n,1)v_{i}\Vert \cdot \Vert \tilde{{\mathcal{A}}}(n,1)w_{i}\Vert ,\nonumber\end{eqnarray}$$
we have
$$\begin{eqnarray}\unicode[STIX]{x1D712}(v_{i})+\tilde{\unicode[STIX]{x1D712}}(w_{i})\geqslant 0\end{eqnarray}$$
for each
$i$
and hence
$\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D712},\tilde{\unicode[STIX]{x1D712}})\geqslant 0$
. We say that the sequence of matrices
$(A_{n})_{n\in \mathbb{N}}$
is regular if
$\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D712},\tilde{\unicode[STIX]{x1D712}})=0$
.
4 Existence of exponential trichotomies
In this section we show that the notion of a
$\unicode[STIX]{x1D70C}$
-non-uniform exponential trichotomy occurs naturally.
For a sequence of matrices
$(A_{n})_{n\in \mathbb{N}}$
with the block form in (6), we consider the Lyapunov exponents
$\unicode[STIX]{x1D712}|E$
,
$\unicode[STIX]{x1D712}|F$
and
$\unicode[STIX]{x1D712}|G$
associated to the sequences formed by the matrices
$B_{n}$
,
$C_{n}$
and
$D_{n}$
(see (6)) as well as the Lyapunov exponents
$\tilde{\unicode[STIX]{x1D712}}|E$
,
$\tilde{\unicode[STIX]{x1D712}}|F$
and
$\tilde{\unicode[STIX]{x1D712}}|G$
associated to the sequences formed by the matrices
$(B_{n}^{\ast })^{-1}$
,
$(C_{n}^{\ast })^{-1}$
and
$(D_{n}^{\ast })^{-1}$
. The corresponding regularity coefficients are
$$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{{\mathcal{B}}}=\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D712}|E,\tilde{\unicode[STIX]{x1D712}}|E),\qquad \unicode[STIX]{x1D6FE}_{{\mathcal{C}}}=\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D712}|F,\tilde{\unicode[STIX]{x1D712}}|F),\qquad \unicode[STIX]{x1D6FE}_{{\mathcal{D}}}=\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D712}|G,\tilde{\unicode[STIX]{x1D712}}|G).\end{eqnarray}$$
Theorem 1. Let
$(A_{n})_{n\in \mathbb{N}}$
be a sequence of matrices with the block form in (6). If
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D712}(v) & {<} & \displaystyle 0\quad \text{for }v\in E\setminus \{0\},\nonumber\\ \displaystyle \unicode[STIX]{x1D712}(v) & {>} & \displaystyle 0\quad \text{for }v\in F\setminus \{0\},\nonumber\\ \displaystyle \unicode[STIX]{x1D712}(v) & = & \displaystyle 0\quad \text{for }v\in G\setminus \{0\},\nonumber\end{eqnarray}$$
then, for each
$\unicode[STIX]{x1D700}>0$
, the sequence
$(A_{n})_{n\in \mathbb{N}}$
admits a
$\unicode[STIX]{x1D70C}$
-non-uniform exponential trichotomy with
$$\begin{eqnarray}\displaystyle & \text{}\underline{a}=\unicode[STIX]{x1D706}_{1}+\unicode[STIX]{x1D700},\qquad \overline{a}=\unicode[STIX]{x1D706}_{p-1}+\unicode[STIX]{x1D700},\qquad a=\unicode[STIX]{x1D6FE}_{{\mathcal{B}}}+2\unicode[STIX]{x1D700}, & \displaystyle \nonumber\\ \displaystyle & \text{}\underline{b}=\unicode[STIX]{x1D706}_{q+1}+\unicode[STIX]{x1D700},\qquad \overline{b}=\unicode[STIX]{x1D706}_{r}+\unicode[STIX]{x1D700},\qquad b=\unicode[STIX]{x1D6FE}_{{\mathcal{C}}}+2\unicode[STIX]{x1D700}, & \displaystyle \nonumber\\ \displaystyle & \text{}\underline{c}=\unicode[STIX]{x1D700},\qquad \overline{c}=\unicode[STIX]{x1D700},\qquad c=\unicode[STIX]{x1D6FE}_{{\mathcal{D}}}+2\unicode[STIX]{x1D700}, & \displaystyle \nonumber\end{eqnarray}$$
where
$\unicode[STIX]{x1D706}_{p-1}<0\leqslant \unicode[STIX]{x1D706}_{p}$
and
$\unicode[STIX]{x1D706}_{q}\leqslant 0<\unicode[STIX]{x1D706}_{q+1}$
.
Proof. Given an invertible matrix
$H$
, let
$v_{1}(n),\ldots ,v_{k}(n)$
be the columns of
${\mathcal{B}}(n,1)H$
and let
$w_{1}(n),\ldots ,w_{k}(n)$
be the columns of
$\tilde{{\mathcal{B}}}(n,1)(H^{\ast })^{-1}$
. For
$i=1,\ldots ,k$
, let
$$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{i}=\unicode[STIX]{x1D712}(v_{i}(1))\quad \text{and}\quad \unicode[STIX]{x1D6FD}_{i}=\tilde{\unicode[STIX]{x1D712}}(w_{i}(1)).\end{eqnarray}$$
Given
$\unicode[STIX]{x1D700}>0$
, there exists
$D>0$
such that
$$\begin{eqnarray}\Vert v_{i}(n)\Vert \leqslant D\text{e}^{(\unicode[STIX]{x1D6FC}_{i}+\unicode[STIX]{x1D700})\unicode[STIX]{x1D70C}(n)}\quad \text{and}\quad \Vert w_{i}(n)\Vert \leqslant D\text{e}^{(\unicode[STIX]{x1D6FD}_{i}+\unicode[STIX]{x1D700})\unicode[STIX]{x1D70C}(n)}\end{eqnarray}$$
for
$i=1,\ldots ,k$
and
$n\in \mathbb{N}$
. On the other hand, it follows from the identity
$$\begin{eqnarray}(\tilde{{\mathcal{B}}}(n,1)(H^{\ast })^{-1})^{\ast }({\mathcal{B}}(n,1)H)=\text{Id}\end{eqnarray}$$
that
$\langle v_{i}(n),w_{\!j}(n)\rangle =\unicode[STIX]{x1D6FF}_{ij}$
for
$i,j=1,\ldots ,k$
and
$n\in \mathbb{N}$
. In particular, since the minimum in (11) takes only finitely many values, there exists a matrix
$H$
for which
$$\begin{eqnarray}\max \{\unicode[STIX]{x1D6FC}_{i}+\unicode[STIX]{x1D6FD}_{i}:i=1,\ldots ,k\}=\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D712},\tilde{\unicode[STIX]{x1D712}}).\end{eqnarray}$$
Now we notice that
$$\begin{eqnarray}{\mathcal{B}}(m,n)={\mathcal{B}}(m,1)\tilde{{\mathcal{B}}}(n,1)^{\ast }=({\mathcal{B}}(m,1)H)(\tilde{{\mathcal{B}}}(n,1)(H^{\ast })^{-1})^{\ast }.\end{eqnarray}$$
The entries
$b_{ij}(m,n)$
of the matrix
${\mathcal{B}}(m,n)$
are, thus,
$$\begin{eqnarray}b_{ij}(m,n)=\mathop{\sum }_{l=1}^{k}v_{il}(m)w_{jl}(n),\end{eqnarray}$$
where
$v_{il}(m)$
is the
$i$
th coordinate of
$v_{l}(m)$
and
$w_{jl}(n)$
is the
$j$
th coordinate of
$w_{l}(n)$
. By (13), we obtain
$$\begin{eqnarray}\displaystyle |b_{ij}(m,n)| & {\leqslant} & \displaystyle \mathop{\sum }_{l=1}^{k}\Vert v_{l}(m)\Vert \cdot \Vert w_{l}(n)\Vert \nonumber\\ \displaystyle & {\leqslant} & \displaystyle \mathop{\sum }_{l=1}^{k}D^{2}\text{e}^{(\unicode[STIX]{x1D6FC}_{l}+\unicode[STIX]{x1D700})\unicode[STIX]{x1D70C}(m)+(\unicode[STIX]{x1D6FD}_{l}+\unicode[STIX]{x1D700})\unicode[STIX]{x1D70C}(n)}\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{l=1}^{k}D^{2}\text{e}^{(\unicode[STIX]{x1D6FC}_{l}+\unicode[STIX]{x1D700})(\unicode[STIX]{x1D70C}(m)-\unicode[STIX]{x1D70C}(n))+(\unicode[STIX]{x1D6FC}_{l}+\unicode[STIX]{x1D6FD}_{l}+2\unicode[STIX]{x1D700})\unicode[STIX]{x1D70C}(n)}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle kD^{2}\text{e}^{(\unicode[STIX]{x1D706}_{p-1}+\unicode[STIX]{x1D700})(\unicode[STIX]{x1D70C}(m)-\unicode[STIX]{x1D70C}(n))+(\unicode[STIX]{x1D6FE}_{{\mathcal{B}}}+2\unicode[STIX]{x1D700})\unicode[STIX]{x1D70C}(n)}\nonumber\end{eqnarray}$$
for
$m\geqslant n$
. Therefore,
$$\begin{eqnarray}\Vert {\mathcal{B}}(m,n)\Vert \leqslant D^{\prime }\text{e}^{(\unicode[STIX]{x1D706}_{p-1}+\unicode[STIX]{x1D700})(\unicode[STIX]{x1D70C}(m)-\unicode[STIX]{x1D70C}(n))+(\unicode[STIX]{x1D6FE}_{{\mathcal{B}}}+2\unicode[STIX]{x1D700})\unicode[STIX]{x1D70C}(n)}\end{eqnarray}$$
for
$m\geqslant n$
, which yields the first inequality in (5) with
$\overline{a}=\unicode[STIX]{x1D706}_{p-1}+\unicode[STIX]{x1D700}$
and
$a=\unicode[STIX]{x1D6FE}_{{\mathcal{B}}}+2\unicode[STIX]{x1D700}$
. Similarly, the entries
$c_{ij}(m,n)=b_{ij}(n,m)$
of the matrix
$$\begin{eqnarray}{\mathcal{B}}(m,n)^{-1}={\mathcal{B}}(n,1)\tilde{{\mathcal{B}}}(m,1)^{\ast }=({\mathcal{B}}(n,1)H)(\tilde{{\mathcal{B}}}(m,1)(H^{\ast })^{-1})^{\ast }\end{eqnarray}$$
satisfy
$$\begin{eqnarray}\displaystyle |c_{ij}(m,n)| & {\leqslant} & \displaystyle \mathop{\sum }_{l=1}^{k}\Vert v_{l}(n)\Vert \cdot \Vert w_{l}(m)\Vert \nonumber\\ \displaystyle & {\leqslant} & \displaystyle \mathop{\sum }_{l=1}^{k}D^{2}\text{e}^{(\unicode[STIX]{x1D6FC}_{l}+\unicode[STIX]{x1D700})n+(\unicode[STIX]{x1D6FD}_{l}+\unicode[STIX]{x1D700})\unicode[STIX]{x1D70C}(m)}\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{l=1}^{k}D^{2}\text{e}^{-(\unicode[STIX]{x1D6FC}_{l}+\unicode[STIX]{x1D700})(\unicode[STIX]{x1D70C}(m)-\unicode[STIX]{x1D70C}(n))+(\unicode[STIX]{x1D6FC}_{l}+\unicode[STIX]{x1D6FD}_{l}+2\unicode[STIX]{x1D700})\unicode[STIX]{x1D70C}(m)}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle kD^{2}\text{e}^{-(\unicode[STIX]{x1D706}_{1}+\unicode[STIX]{x1D700})(\unicode[STIX]{x1D70C}(m)-\unicode[STIX]{x1D70C}(n))+(\unicode[STIX]{x1D6FE}_{{\mathcal{B}}}+2\unicode[STIX]{x1D700})\unicode[STIX]{x1D70C}(m)}\nonumber\end{eqnarray}$$
for
$m\geqslant n$
. This yields the second inequality in (5) with
$\text{}\underline{a}=\unicode[STIX]{x1D706}_{1}+\unicode[STIX]{x1D700}$
and again
$a=\unicode[STIX]{x1D6FE}_{{\mathcal{B}}}+2\unicode[STIX]{x1D700}$
. The four remaining inequalities in (5) can be obtained in a similar manner.◻
5 Bounds for the regularity coefficient
Theorem 1 shows that the constants
$a$
,
$b$
and
$c$
are related to the regularity coefficients
$\unicode[STIX]{x1D6FE}_{{\mathcal{B}}}$
,
$\unicode[STIX]{x1D6FE}_{{\mathcal{C}}}$
and
$\unicode[STIX]{x1D6FE}_{{\mathcal{D}}}$
. Therefore, it is of interest to obtain sharp bounds for
$\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D712},\tilde{\unicode[STIX]{x1D712}})$
in terms of the sequence
$(A_{n})_{n\in \mathbb{N}}$
.
We first obtain a lower bound for the regularity coefficient.
Theorem 2. We have
$$\begin{eqnarray}\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D712},\tilde{\unicode[STIX]{x1D712}})\geqslant \frac{1}{k}\limsup _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log a_{n}-\frac{1}{k}\liminf _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log a_{n},\end{eqnarray}$$
where
$a_{n}=|\text{det}\,{\mathcal{A}}(n,1)|$
.
Proof. Let
$v_{1},\ldots ,v_{k}$
be a basis of
$\mathbb{R}^{k}$
. Then
$$\begin{eqnarray}a_{n}=|\text{det}\,({\mathcal{A}}(n,1)H)|\leqslant \mathop{\prod }_{i=1}^{k}\Vert {\mathcal{A}}(n,1)v_{i}\Vert ,\end{eqnarray}$$
where
$H$
is the
$k\times k$
matrix whose columns are the vectors
$v_{1},\ldots ,v_{k}$
. Thus,
$$\begin{eqnarray}\limsup _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log a_{n}=\limsup _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log |\det ({\mathcal{A}}(n,1)H)|\leqslant \mathop{\sum }_{i=1}^{k}\unicode[STIX]{x1D712}(v_{i}).\end{eqnarray}$$
Similarly, if
$w_{1},\ldots ,w_{k}$
is a basis of
$\mathbb{R}^{k}$
, then (see (10))
$$\begin{eqnarray}\liminf _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log a_{n}=-\!\limsup _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log |\det \tilde{{\mathcal{A}}}(n,1)|\geqslant -\!\mathop{\sum }_{i=1}^{k}\tilde{\unicode[STIX]{x1D712}}(w_{i}).\end{eqnarray}$$
Therefore,
$$\begin{eqnarray}\limsup _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log a_{n}-\liminf _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log a_{n}\leqslant \mathop{\sum }_{i=1}^{k}(\unicode[STIX]{x1D712}(v_{i})+\tilde{\unicode[STIX]{x1D712}}(w_{i})).\end{eqnarray}$$
Now we assume that
$v_{1},\ldots ,v_{k}$
and
$w_{1},\ldots ,w_{k}$
are dual bases at which the minimum in (11) is attained, that is,
$$\begin{eqnarray}\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D712},\tilde{\unicode[STIX]{x1D712}})=\max \{\unicode[STIX]{x1D712}(v_{i})+\tilde{\unicode[STIX]{x1D712}}(w_{i}):1\leqslant i\leqslant k\}.\end{eqnarray}$$
Then
$$\begin{eqnarray}\mathop{\sum }_{i=1}^{k}(\unicode[STIX]{x1D712}(v_{i})+\tilde{\unicode[STIX]{x1D712}}(w_{i}))\leqslant k\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D712},\tilde{\unicode[STIX]{x1D712}}),\end{eqnarray}$$
In order to obtain an upper bound, we assume that each matrix
$A_{n}$
is upper triangular. Let
$a_{ij}(n)$
be the entries of
$A_{n}$
. For
$i=1,\ldots ,k$
, we define
$$\begin{eqnarray}\text{}\underline{\unicode[STIX]{x1D6FC}}_{i}=\liminf _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log \mathop{\prod }_{l=1}^{n}a_{ii}(l)\quad \text{and}\quad \overline{\unicode[STIX]{x1D6FC}}_{i}=\limsup _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log \mathop{\prod }_{l=1}^{n}a_{ii}(l).\end{eqnarray}$$
We obtain an upper bound for a large class of growth rates.
Theorem 3. If the matrices
$A_{n}$
are upper triangular for
$n\in \mathbb{N}$
, then
$$\begin{eqnarray}\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D712},\tilde{\unicode[STIX]{x1D712}})\leqslant \mathop{\sum }_{i=1}^{k}(\overline{\unicode[STIX]{x1D6FC}}_{i}-\text{}\underline{\unicode[STIX]{x1D6FC}}_{i}).\end{eqnarray}$$
Proof. For each
$i,j=1,\ldots ,k$
, let
$z_{ij}(n)$
be the entries of
${\mathcal{A}}(n+1,1)$
and let
$$\begin{eqnarray}\unicode[STIX]{x1D706}_{ij}=\limsup _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\text{log}\,|z_{ij}(n)|.\end{eqnarray}$$
Lemma 1. For
$i,j=1,\ldots ,k$
, we have
$\unicode[STIX]{x1D706}_{ii}=\overline{\unicode[STIX]{x1D6FC}}_{i}$
and
$$\begin{eqnarray}\unicode[STIX]{x1D706}_{ij}\leqslant \overline{\unicode[STIX]{x1D6FC}}_{\!j}+\mathop{\sum }_{m=i}^{j-1}(\overline{\unicode[STIX]{x1D6FC}}_{m}-\text{}\underline{\unicode[STIX]{x1D6FC}}_{m}),\quad i<j.\end{eqnarray}$$
Proof of the lemma.
Clearly,
$\unicode[STIX]{x1D706}_{ii}=\overline{\unicode[STIX]{x1D6FC}}_{i}$
for
$i=1,\ldots ,k$
. Now we consider the entries with
$i<j$
and we proceed by backwards induction on
$i$
. Assume that for a given
$i<k$
,
$$\begin{eqnarray}\unicode[STIX]{x1D706}_{lj}\leqslant \overline{\unicode[STIX]{x1D6FC}}_{\!j}+\mathop{\sum }_{m=l}^{j-1}(\overline{\unicode[STIX]{x1D6FC}}_{m}-\text{}\underline{\unicode[STIX]{x1D6FC}}_{m})\quad \text{whenever }i+1\leqslant l\leqslant j.\end{eqnarray}$$
We want to prove that
$$\begin{eqnarray}\unicode[STIX]{x1D706}_{ij}\leqslant \overline{\unicode[STIX]{x1D6FC}}_{\!j}+\mathop{\sum }_{m=i}^{j-1}(\overline{\unicode[STIX]{x1D6FC}}_{m}-\text{}\underline{\unicode[STIX]{x1D6FC}}_{m})\end{eqnarray}$$
for
$j\geqslant i+1$
. We consider two cases expressed in terms of the number
$$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{ij}=\overline{\unicode[STIX]{x1D6FC}}_{\!j}-\text{}\underline{\unicode[STIX]{x1D6FC}}_{i}+\mathop{\sum }_{m=i+1}^{j-1}(\overline{\unicode[STIX]{x1D6FC}}_{m}-\text{}\underline{\unicode[STIX]{x1D6FC}}_{m}).\end{eqnarray}$$
Case
$\unicode[STIX]{x1D6FD}_{ij}\geqslant 0$
The entries
$z_{ij}(n)$
of the matrix
${\mathcal{A}}(n+1,1)$
satisfy
$$\begin{eqnarray}z_{ij}(n)=\left\{\begin{array}{@{}ll@{}}0\quad & \text{if }i>j,\\ \displaystyle \mathop{\prod }_{l=1}^{n}a_{ii}(l)\quad & \text{if }i=j,\\ \displaystyle \mathop{\sum }_{l=i}^{j}a_{il}(n)z_{lj}(n-1)\quad & \text{if }i<j.\end{array}\right.\end{eqnarray}$$
Hence, for
$i<j$
, we obtain
$$\begin{eqnarray}\displaystyle z_{ij}(n) & = & \displaystyle \mathop{\sum }_{l=i+1}^{j}a_{il}(n)z_{lj}(n-1)+a_{ii}(n)z_{ij}(n-1)\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{l=i+1}^{j}a_{il}(n)z_{lj}(n-1)+a_{ii}(n)\mathop{\sum }_{l=i+1}^{j}a_{il}(n-1)z_{lj}(n-2)\nonumber\\ \displaystyle & & \displaystyle +\,a_{ii}(n)a_{ii}(n-1)z_{ij}(n-2).\nonumber\end{eqnarray}$$
Proceeding by induction yields that
$$\begin{eqnarray}z_{ij}(n)=\mathop{\sum }_{p=0}^{n-1}\mathop{\sum }_{l=i+1}^{j}a_{il}(n-p)z_{lj}(n-p-1)\mathop{\prod }_{m=n-p+1}^{n}a_{ii}(m),\end{eqnarray}$$
with the convention that
$\prod _{m=n+1}^{n}a_{ii}(m)=1$
.
Now we estimate
$\unicode[STIX]{x1D706}_{ij}$
. By (7), (1) and (19), for each
$\unicode[STIX]{x1D700}>0$
there exists
$D>0$
such that
$$\begin{eqnarray}|a_{il}(n)|\leqslant D\text{e}^{\unicode[STIX]{x1D700}\unicode[STIX]{x1D70C}(n)},\qquad D^{-1}\text{e}^{(\text{}\underline{\unicode[STIX]{x1D6FC}}_{i}-\unicode[STIX]{x1D700})\unicode[STIX]{x1D70C}(n)}\leqslant \mathop{\prod }_{l=1}^{n}|a_{ii}(l)|\leqslant D\text{e}^{(\overline{\unicode[STIX]{x1D6FC}}_{i}+\unicode[STIX]{x1D700})\unicode[STIX]{x1D70C}(n)}\end{eqnarray}$$
and
$$\begin{eqnarray}|z_{lj}(n)|\leqslant D\text{e}^{[\overline{\unicode[STIX]{x1D6FC}}_{\!j}+\mathop{\sum }_{m=l}^{j-1}(\overline{\unicode[STIX]{x1D6FC}}_{m}-\text{}\underline{\unicode[STIX]{x1D6FC}}_{m})+\unicode[STIX]{x1D700}]\unicode[STIX]{x1D70C}(n+1)}\end{eqnarray}$$
for
$n\geqslant 1$
and
$i+1\leqslant l\leqslant j$
. Hence, it follows from (22) that
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}_{ij} & {\leqslant} & \displaystyle \limsup _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log \biggl|\mathop{\sum }_{p=0}^{n-1}\mathop{\sum }_{l=i+1}^{j}a_{il}(n-p)z_{lj}(n-p-1)\mathop{\prod }_{m=n-p+1}^{n}a_{ii}(m)\biggr|\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \limsup _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log \mathop{\sum }_{p=0}^{n-1}\mathop{\sum }_{l=i+1}^{j}D^{2}\text{e}^{\unicode[STIX]{x1D700}\unicode[STIX]{x1D70C}(n-p)+[\overline{\unicode[STIX]{x1D6FC}}_{\!j}+\mathop{\sum }_{m=l}^{j-1}(\overline{\unicode[STIX]{x1D6FC}}_{m}-\text{}\underline{\unicode[STIX]{x1D6FC}}_{m})+\unicode[STIX]{x1D700}]\unicode[STIX]{x1D70C}(n-p)}\nonumber\\ \displaystyle & & \displaystyle \times \,\mathop{\prod }_{m=1}^{n}|a_{ii}(m)|D\text{e}^{(-\text{}\underline{\unicode[STIX]{x1D6FC}}_{i}+\unicode[STIX]{x1D700})\unicode[STIX]{x1D70C}(n-p)}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \overline{\unicode[STIX]{x1D6FC}}_{i}+\limsup _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log \biggl(kD^{3}\mathop{\sum }_{p=0}^{n-1}\text{e}^{(\unicode[STIX]{x1D6FD}_{ij}+3\unicode[STIX]{x1D700})\unicode[STIX]{x1D70C}(n-p)}\biggr).\end{eqnarray}$$
Since
$\unicode[STIX]{x1D70C}$
takes only integer values, we have
$$\begin{eqnarray}\unicode[STIX]{x1D70C}(k+m)\geqslant \unicode[STIX]{x1D70C}(k)+m.\end{eqnarray}$$
Hence, letting
$k=n-p$
and
$m=p$
yields that
$$\begin{eqnarray}\unicode[STIX]{x1D70C}(n-p)\leqslant \unicode[STIX]{x1D70C}(n-p+p)-p=\unicode[STIX]{x1D70C}(n)-p.\end{eqnarray}$$
Since
$\unicode[STIX]{x1D6FD}_{ij}\geqslant 0$
, we have
$\unicode[STIX]{x1D6FD}_{ij}+3\unicode[STIX]{x1D700}>0$
and it follows from (24) that
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}_{ij} & {\leqslant} & \displaystyle \overline{\unicode[STIX]{x1D6FC}}_{i}+\limsup _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log \biggl(kD^{3}\mathop{\sum }_{p=0}^{n-1}\text{e}^{(\unicode[STIX]{x1D6FD}_{ij}+3\unicode[STIX]{x1D700})(\unicode[STIX]{x1D70C}(n)-p)}\biggr)+3\unicode[STIX]{x1D700}\nonumber\\ \displaystyle & = & \displaystyle \overline{\unicode[STIX]{x1D6FC}}_{i}+\limsup _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log \biggl(\text{e}^{(\unicode[STIX]{x1D6FD}_{ij}+3\unicode[STIX]{x1D700})\unicode[STIX]{x1D70C}(n)}\frac{1-\text{e}^{-(\unicode[STIX]{x1D6FD}_{ij}+3\unicode[STIX]{x1D700})n}}{1-\text{e}^{-(\unicode[STIX]{x1D6FD}_{ij}+3\unicode[STIX]{x1D700})}}\biggr)\nonumber\\ \displaystyle & = & \displaystyle \overline{\unicode[STIX]{x1D6FC}}_{i}+\unicode[STIX]{x1D6FD}_{ij}+3\unicode[STIX]{x1D700}=\overline{\unicode[STIX]{x1D6FC}}_{\!j}+\mathop{\sum }_{m=i}^{j-1}(\overline{\unicode[STIX]{x1D6FC}}_{m}-\text{}\underline{\unicode[STIX]{x1D6FC}}_{m})+3\unicode[STIX]{x1D700}.\end{eqnarray}$$
Letting
$\unicode[STIX]{x1D700}\rightarrow 0$
yields inequality (20).
Case
$\unicode[STIX]{x1D6FD}_{ij}<0$
Again by (21), we obtain
$$\begin{eqnarray}z_{ij}(n+1)a_{ii}(n+1)^{-1}=\mathop{\sum }_{l=i+1}^{j}a_{il}(n+1)z_{lj}(n)a_{ii}(n+1)^{-1}+z_{ij}(n).\end{eqnarray}$$
Writing
$$\begin{eqnarray}w_{ij}(t)=z_{ij}(t)\mathop{\prod }_{m=1}^{t}a_{ii}(m)^{-1},\quad t\geqslant n,\end{eqnarray}$$
it follows from (27) that
$$\begin{eqnarray}w_{ij}(n+1)=w_{ij}(n)+\mathop{\sum }_{l=i+1}^{j}a_{il}(n+1)z_{lj}(n)\mathop{\prod }_{m=1}^{n+1}a_{ii}(m)^{-1}.\end{eqnarray}$$
We will show that given
$\unicode[STIX]{x1D700}>0$
with
$\unicode[STIX]{x1D6FD}_{ij}+3\unicode[STIX]{x1D700}<0$
, there exists
$K>0$
such that
$$\begin{eqnarray}|w_{ij}(n)|\leqslant K\text{e}^{(\unicode[STIX]{x1D6FD}_{ij}+3\unicode[STIX]{x1D700})\unicode[STIX]{x1D70C}(n)},\quad n\geqslant 1.\end{eqnarray}$$
For this, we show that
$$\begin{eqnarray}w=(w(n))_{n\geqslant 1}=(w_{ij}(n))_{n\geqslant 1}\end{eqnarray}$$
is the unique fixed point of the operator
$S$
defined by
$$\begin{eqnarray}(Sw)(n)=w(n+1)-\mathop{\sum }_{l=i+1}^{j}a_{il}(n+1)z_{lj}(n)\mathop{\prod }_{m=1}^{n+1}a_{ii}(m)^{-1}\end{eqnarray}$$
in the space
$X$
of all sequences
$w$
of real numbers such that
$$\begin{eqnarray}\Vert w\Vert :=\sup _{n\geqslant 1}\frac{|w(n)|}{\text{e}^{(\unicode[STIX]{x1D6FD}_{ij}+3\unicode[STIX]{x1D700})\unicode[STIX]{x1D70C}(n)}}<\infty .\end{eqnarray}$$
We note that
$X$
is a Banach space when endowed with the norm
$\Vert \cdot \Vert$
and that
$$\begin{eqnarray}|w(n)|\leqslant \Vert w\Vert \text{e}^{(\unicode[STIX]{x1D6FD}_{ij}+3\unicode[STIX]{x1D700})\unicode[STIX]{x1D70C}(n)}\end{eqnarray}$$
for all
$n\geqslant 1$
. We have
$$\begin{eqnarray}\displaystyle |(Sw)(n)| & {\leqslant} & \displaystyle |w(n+1)|+\mathop{\sum }_{l=i+1}^{j}\biggl|a_{il}(n+1)z_{lj}(n)\mathop{\prod }_{m=1}^{n+1}a_{ii}(m)^{-1}\biggr|\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \Vert w\Vert \text{e}^{(\unicode[STIX]{x1D6FD}_{ij}+3\unicode[STIX]{x1D700})\unicode[STIX]{x1D70C}(n+1)}\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{l=i+1}^{j}D^{3}\text{e}^{\unicode[STIX]{x1D700}\unicode[STIX]{x1D70C}(n+1)}\text{e}^{[\overline{\unicode[STIX]{x1D6FC}}_{\!j}+\mathop{\sum }_{m=l}^{j-1}(\overline{\unicode[STIX]{x1D6FC}}_{m}-\text{}\underline{\unicode[STIX]{x1D6FC}}_{m})+\unicode[STIX]{x1D700}]\unicode[STIX]{x1D70C}(n+1)}\text{e}^{(-\text{}\underline{\unicode[STIX]{x1D6FC}}_{\!j}+\unicode[STIX]{x1D700})\unicode[STIX]{x1D70C}(n+1)}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \Vert w\Vert \text{e}^{(\unicode[STIX]{x1D6FD}_{ij}+3\unicode[STIX]{x1D700})\unicode[STIX]{x1D70C}(n+1)}+D^{3}k\text{e}^{(\unicode[STIX]{x1D6FD}_{ij}+3\unicode[STIX]{x1D700})\unicode[STIX]{x1D70C}(n+1)}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle (\Vert w\Vert +D^{3}k)\text{e}^{(\unicode[STIX]{x1D6FD}_{ij}+3\unicode[STIX]{x1D700})\unicode[STIX]{x1D70C}(n)}\nonumber\end{eqnarray}$$
and, thus,
$$\begin{eqnarray}\Vert Sw\Vert \leqslant \Vert w\Vert +D^{3}k<\infty .\end{eqnarray}$$
This shows that
$S(X)\subset X$
. Moreover, by (25),
$$\begin{eqnarray}\displaystyle |(Sw)(n)-(Su)(n)| & = & \displaystyle |w(n+1)-u(n+1)|\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \text{e}^{(\unicode[STIX]{x1D6FD}_{ij}+3\unicode[STIX]{x1D700})\unicode[STIX]{x1D70C}(n+1)}\Vert w-u\Vert \nonumber\\ \displaystyle & {\leqslant} & \displaystyle \text{e}^{(\unicode[STIX]{x1D6FD}_{ij}+3\unicode[STIX]{x1D700})(\unicode[STIX]{x1D70C}(n)+1)}\Vert w-u\Vert \nonumber\end{eqnarray}$$
and, thus,
$$\begin{eqnarray}\Vert Sw-Su\Vert \leqslant \text{e}^{\unicode[STIX]{x1D6FD}_{ij}+3\unicode[STIX]{x1D700}}\Vert w-u\Vert .\end{eqnarray}$$
Since
$\text{e}^{\unicode[STIX]{x1D6FD}_{ij}+3\unicode[STIX]{x1D700}}<1$
, the operator
$S$
is a contraction in the Banach space
$X$
. For the unique fixed point of
$S$
in
$X$
, it follows from (29) that given
$\unicode[STIX]{x1D700}>0$
sufficiently small, there exists
$K>0$
such that
$$\begin{eqnarray}|w_{ij}(n)|\leqslant K\text{e}^{(\unicode[STIX]{x1D6FD}_{ij}+3\unicode[STIX]{x1D700})\unicode[STIX]{x1D70C}(n)}\end{eqnarray}$$
for all
$n\geqslant 1$
. Hence, by (28), we obtain
$$\begin{eqnarray}\displaystyle |z_{ij}(n)| & {\leqslant} & \displaystyle K^{\prime }\text{e}^{(\unicode[STIX]{x1D6FD}_{ij}+\overline{\unicode[STIX]{x1D6FC}}_{i}+4\unicode[STIX]{x1D700})\unicode[STIX]{x1D70C}(n)}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle K\text{e}^{[\overline{\unicode[STIX]{x1D6FC}}_{\!j}+\mathop{\sum }_{m=i}^{j}(\overline{\unicode[STIX]{x1D6FC}}_{m}-\text{}\underline{\unicode[STIX]{x1D6FC}}_{m})+4\unicode[STIX]{x1D700}]\unicode[STIX]{x1D70C}(n)}\nonumber\end{eqnarray}$$
for some constant
$K^{\prime }>0$
and, thus,
$$\begin{eqnarray}\unicode[STIX]{x1D706}_{ij}=\limsup _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\text{log}\,|z_{i}(n)|\leqslant \overline{\unicode[STIX]{x1D6FC}}_{\!j}+\mathop{\sum }_{m=i}^{j}(\overline{\unicode[STIX]{x1D6FC}}_{m}-\text{}\underline{\unicode[STIX]{x1D6FC}}_{m})+4\unicode[STIX]{x1D700}.\end{eqnarray}$$
Letting
$\unicode[STIX]{x1D700}\rightarrow 0$
yields inequality (20).◻
Now we consider the matrices
$B_{n}=(A_{n}^{\ast })^{-1}$
. For each
$i,j=1,\ldots ,k$
, let
$w_{ij}(n)$
be the entries of
${\mathcal{B}}(n+1,1)$
and let
$$\begin{eqnarray}\unicode[STIX]{x1D707}_{ij}=\limsup _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log |w_{ij}(n)|.\end{eqnarray}$$
Lemma 2. For
$i,j=1,\ldots ,k$
, we have
$\unicode[STIX]{x1D707}_{jj}=-\text{}\underline{\unicode[STIX]{x1D6FC}}_{\!j}$
and
$$\begin{eqnarray}\unicode[STIX]{x1D707}_{ij}\leqslant -\text{}\underline{\unicode[STIX]{x1D6FC}}_{\!j}+\mathop{\sum }_{m=j+1}^{i}(\overline{\unicode[STIX]{x1D6FC}}_{m}-\text{}\underline{\unicode[STIX]{x1D6FC}}_{m}),\quad i>j.\end{eqnarray}$$
Proof of the lemma.
Clearly,
$\unicode[STIX]{x1D707}_{jj}=-\text{}\underline{\unicode[STIX]{x1D6FC}}_{\!j}$
. For the entries with
$i>j$
, in a similar manner to that in the proof of Lemma 1, we assume that
$$\begin{eqnarray}\unicode[STIX]{x1D707}_{lj}\leqslant -\text{}\underline{\unicode[STIX]{x1D6FC}}_{\!j}+\mathop{\sum }_{m=j+1}^{l}(\overline{\unicode[STIX]{x1D6FC}}_{m}-\text{}\underline{\unicode[STIX]{x1D6FC}}_{m})\quad \text{whenever }j\leqslant l\leqslant i-1.\end{eqnarray}$$
We want to prove that
$$\begin{eqnarray}\unicode[STIX]{x1D707}_{ij}\leqslant -\text{}\underline{\unicode[STIX]{x1D6FC}}_{\!j}+\mathop{\sum }_{m=j+1}^{i}(\overline{\unicode[STIX]{x1D6FC}}_{m}-\text{}\underline{\unicode[STIX]{x1D6FC}}_{m}).\end{eqnarray}$$
Case
$\unicode[STIX]{x1D6FD}_{ji}\geqslant 0$
The entries
$w_{ij}(n)$
are given inductively by
$$\begin{eqnarray}w_{ij}(n)=\left\{\begin{array}{@{}ll@{}}0,\quad & i<j,\\ \displaystyle \mathop{\prod }_{l=1}^{n}a_{ii}(l)^{-1},\quad & i=j,\\ \displaystyle \mathop{\sum }_{l=j}^{i}a_{li}(n+1)w_{lj}(n+1),\quad & i>j.\end{array}\right.\end{eqnarray}$$
Hence, for
$i>j$
, we obtain
$$\begin{eqnarray}w_{ij}(n)=-\mathop{\sum }_{p=0}^{n-1}\mathop{\sum }_{l=j}^{i-1}a_{il}(n-p)w_{lj}(n-p)\mathop{\prod }_{m=n-p}^{n}a_{ii}(m)^{-1}\end{eqnarray}$$
(recall that
$w_{ij}(0)=0$
).
Now we estimate
$\unicode[STIX]{x1D707}_{ij}$
. By (7) and (31), for each
$\unicode[STIX]{x1D700}>0$
there exists
$D>0$
such that (23) holds as well as
$$\begin{eqnarray}|w_{lj}(n)|\leqslant D\text{e}^{[-\text{}\underline{\unicode[STIX]{x1D6FC}}_{\!j}+\mathop{\sum }_{m=j+1}^{l}(\overline{\unicode[STIX]{x1D6FC}}_{m}-\text{}\underline{\unicode[STIX]{x1D6FC}}_{m})+\unicode[STIX]{x1D700}]\unicode[STIX]{x1D70C}(n)}\end{eqnarray}$$
for
$n\geqslant 1$
and
$j\leqslant l\leqslant i-1$
. Hence, it follows from (33) that
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}_{ij} & {\leqslant} & \displaystyle \limsup _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log \biggl|\mathop{\sum }_{p=0}^{n-1}\mathop{\sum }_{l=j}^{i-1}a_{il}(n-p)w_{lj}(n-p)\mathop{\prod }_{m=n-p}^{n}a_{ii}(m)^{-1}\biggr|\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \limsup _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log \mathop{\sum }_{p=0}^{n-1}\mathop{\sum }_{l=j}^{i-1}D^{2}\text{e}^{\unicode[STIX]{x1D700}\unicode[STIX]{x1D70C}(n-p)+[-\text{}\underline{\unicode[STIX]{x1D6FC}}_{\!j}+\mathop{\sum }_{m=j+1}^{l}(\overline{\unicode[STIX]{x1D6FC}}_{m}-\text{}\underline{\unicode[STIX]{x1D6FC}}_{m})+\unicode[STIX]{x1D700}]\unicode[STIX]{x1D70C}(n-p)}\nonumber\\ \displaystyle & & \displaystyle \times \,\mathop{\prod }_{m=1}^{n}|a_{ii}(m)|^{-1}D\text{e}^{(\overline{\unicode[STIX]{x1D6FC}}_{i}+\unicode[STIX]{x1D700})\unicode[STIX]{x1D70C}(n-p)}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle -\text{}\underline{\unicode[STIX]{x1D6FC}}_{i}+\limsup _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log \biggl(kD^{3}\mathop{\sum }_{p=0}^{n-1}\text{e}^{(\unicode[STIX]{x1D6FD}_{ji}+3\unicode[STIX]{x1D700})\unicode[STIX]{x1D70C}(n-p)}\biggr)\nonumber\end{eqnarray}$$
and proceeding as in (26) yields that
$$\begin{eqnarray}\unicode[STIX]{x1D707}_{ij}\leqslant -\text{}\underline{\unicode[STIX]{x1D6FC}}_{\!j}+\unicode[STIX]{x1D6FD}_{ji}+3\unicode[STIX]{x1D700}=-\text{}\underline{\unicode[STIX]{x1D6FC}}_{\!j}+\mathop{\sum }_{m=j+1}^{i}(\overline{\unicode[STIX]{x1D6FC}}_{m}-\text{}\underline{\unicode[STIX]{x1D6FC}}_{m})+3\unicode[STIX]{x1D700}.\end{eqnarray}$$
Letting
$\unicode[STIX]{x1D700}\rightarrow 0$
yields inequality (30).
Case
$\unicode[STIX]{x1D6FD}_{ji}<0$
It follows from (32) that
$$\begin{eqnarray}\displaystyle & & \displaystyle w_{ij}(n+1)\mathop{\prod }_{m=1}^{n+1}a_{ii}(m)\nonumber\\ \displaystyle & & \displaystyle \quad =-\mathop{\sum }_{l=j}^{i-1}a_{li}(n+1)w_{lj}(n+1)\mathop{\prod }_{m=1}^{n}a_{ii}(m)+w_{ij}(n)\mathop{\prod }_{m=1}^{n}a_{ii}(m).\end{eqnarray}$$
Writing
$$\begin{eqnarray}u_{ij}(t)=w_{ij}(t)\mathop{\prod }_{m=1}^{t}a_{ii}(m),\quad t\geqslant n,\end{eqnarray}$$
identity (34) is equivalent to
$$\begin{eqnarray}u_{ij}(n+1)=u_{ij}(n)-\mathop{\sum }_{l=j}^{i-1}a_{li}(n+1)w_{lj}(n+1)\mathop{\prod }_{m=1}^{n}a_{ii}(m).\end{eqnarray}$$
We will show that given
$\unicode[STIX]{x1D700}>0$
with
$\unicode[STIX]{x1D6FD}_{ij}+3\unicode[STIX]{x1D700}<0$
, there exists
$K>0$
such that
$$\begin{eqnarray}|u_{ij}(n)|\leqslant K\text{e}^{(\unicode[STIX]{x1D6FD}_{ji}+3\unicode[STIX]{x1D700})\unicode[STIX]{x1D70C}(n)}.\end{eqnarray}$$
For this, we show that
$$\begin{eqnarray}u=(u(n))_{n\geqslant 1}=(u_{ij}(n))_{n\geqslant 1}\end{eqnarray}$$
is the unique fixed point of the operator
$T$
defined by
$$\begin{eqnarray}(Tu)(n)=u(n+1)+\mathop{\sum }_{l=j}^{i-1}a_{li}(n+1)w_{lj}(n+1)\mathop{\prod }_{m=1}^{n}a_{ii}(m)\end{eqnarray}$$
in the space
$Y$
of all sequences
$u$
of real numbers such that
$$\begin{eqnarray}\Vert u\Vert :=\sup _{n\geqslant 1}\frac{|u(n)|}{\text{e}^{(\unicode[STIX]{x1D6FD}_{ji}+3\unicode[STIX]{x1D700})\unicode[STIX]{x1D70C}(n)}}<\infty .\end{eqnarray}$$
We note that
$Y$
is a Banach space when endowed with the norm
$\Vert \cdot \Vert$
. Proceeding in a similar manner to that in the proof of Lemma 1, one can show that
$T$
is a contraction in
$Y$
. Together with (35), this implies that given
$\unicode[STIX]{x1D700}>0$
, there exists
$K>0$
such that
$$\begin{eqnarray}|w_{ij}(n)|\leqslant K\text{e}^{[-\text{}\underline{\unicode[STIX]{x1D6FC}}_{\!j}+\mathop{\sum }_{m=j+1}^{i}(\overline{\unicode[STIX]{x1D6FC}}_{m}-\text{}\underline{\unicode[STIX]{x1D6FC}}_{m})+4\unicode[STIX]{x1D700}]\unicode[STIX]{x1D70C}(n)}\end{eqnarray}$$
and, thus,
$$\begin{eqnarray}\unicode[STIX]{x1D707}_{ij}=\limsup _{n\rightarrow \infty }\frac{1}{n}\log |w_{ij}(n)|\leqslant -\text{}\underline{\unicode[STIX]{x1D6FC}}_{\!j}+\mathop{\sum }_{m=j+1}^{i}(\overline{\unicode[STIX]{x1D6FC}}_{m}-\text{}\underline{\unicode[STIX]{x1D6FC}}_{m})+4\unicode[STIX]{x1D700}.\end{eqnarray}$$
Letting
$\unicode[STIX]{x1D700}\rightarrow 0$
yields inequality (30).◻
We proceed with the proof of the theorem. Consider the columns
$$\begin{eqnarray}z_{\!j}(n)=(z_{1j}(n),\ldots ,z_{kj}(n))\quad \text{and}\quad w_{\!j}(n)=(w_{1j}(n),\ldots ,w_{kj}(n)),\end{eqnarray}$$
respectively, of the matrices
${\mathcal{A}}(n+1,1)$
and
${\mathcal{B}}(n+1,1)$
. If
$e_{1},\ldots ,e_{k}$
is the canonical basis of
$\mathbb{R}^{k}$
, then
$$\begin{eqnarray}\unicode[STIX]{x1D706}(e_{\!j})=\max \{\unicode[STIX]{x1D706}_{ij}:1\leqslant i\leqslant k\}\leqslant \overline{\unicode[STIX]{x1D6FC}}_{\!j}+\mathop{\sum }_{m=1}^{j-1}(\overline{\unicode[STIX]{x1D6FC}}_{m}-\text{}\underline{\unicode[STIX]{x1D6FC}}_{m})\end{eqnarray}$$
and
$$\begin{eqnarray}\unicode[STIX]{x1D707}(e_{\!j})=\max \{w_{ij}:1\leqslant i\leqslant k\}\leqslant -\text{}\underline{\unicode[STIX]{x1D6FC}}_{\!j}+\mathop{\sum }_{m=j+1}^{k}(\overline{\unicode[STIX]{x1D6FC}}_{m}-\text{}\underline{\unicode[STIX]{x1D6FC}}_{m}).\end{eqnarray}$$
Hence, for
$j=1,\ldots ,k$
, we have
$$\begin{eqnarray}\unicode[STIX]{x1D706}(e_{\!j})+\unicode[STIX]{x1D707}(e_{\!j})\leqslant \mathop{\sum }_{m=1}^{k}(\overline{\unicode[STIX]{x1D6FC}}_{m}-\text{}\underline{\unicode[STIX]{x1D6FC}}_{m}).\end{eqnarray}$$
Since the bases
$e_{1},\ldots ,e_{k}$
and
$e_{1},\ldots ,e_{k}$
are dual, inequality (18) follows readily from the definition of the regularity coefficient.◻
6 Characterizations of regularity
In this section we give several alternative characterizations of the regularity of a sequence of matrices. We first introduce another regularity coefficient. Consider the values
$$\begin{eqnarray}\unicode[STIX]{x1D706}_{1}^{\prime }\leqslant \cdots \leqslant \unicode[STIX]{x1D706}_{k}^{\prime }\quad \text{and}\quad \unicode[STIX]{x1D707}_{1}^{\prime }\geqslant \cdots \geqslant \unicode[STIX]{x1D707}_{k}^{\prime },\end{eqnarray}$$
respectively, of the Lyapunov exponents
$\unicode[STIX]{x1D712}$
and
$\tilde{\unicode[STIX]{x1D712}}$
on
$\mathbb{R}^{k}\setminus \{0\}$
counted with their multiplicities. We define the Perron coefficient of the pair of Lyapunov exponents
$\unicode[STIX]{x1D712}$
and
$\tilde{\unicode[STIX]{x1D712}}$
by
$$\begin{eqnarray}\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D712},\tilde{\unicode[STIX]{x1D712}})=\max \{\unicode[STIX]{x1D706}_{i}^{\prime }+\unicode[STIX]{x1D707}_{i}^{\prime }:1\leqslant i\leqslant k\}.\end{eqnarray}$$
In view of (12), it follows from [Reference Barreira and Pesin1, Theorem 1.2.6] that
$$\begin{eqnarray}0\leqslant \unicode[STIX]{x1D70B}(\unicode[STIX]{x1D712},\tilde{\unicode[STIX]{x1D712}})\leqslant \unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D712},\tilde{\unicode[STIX]{x1D712}})\leqslant k\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D712},\tilde{\unicode[STIX]{x1D712}}).\end{eqnarray}$$
We say that a basis
$v_{1},\ldots ,v_{k}$
of
$\mathbb{R}^{k}$
is normal for the family of spaces
$V_{i}$
in (9) if for each
$i=1,\ldots ,r$
there exists a basis of
$V_{i}$
composed of vectors in
$\{v_{1},\ldots ,v_{k}\}$
. Similarly, we say that a basis
$w_{1},\ldots ,w_{k}$
of
$\mathbb{R}^{k}$
is normal for the family of spaces
$$\begin{eqnarray}W_{i}=\{w\in \mathbb{R}^{k}:\tilde{\unicode[STIX]{x1D712}}(w)\leqslant \tilde{\unicode[STIX]{x1D706}}_{i}\}\end{eqnarray}$$
if for each
$i=1,\ldots ,s$
there exists a basis of
$W_{i}$
composed of vectors in
$\{w_{1},\ldots ,w_{k}\}$
. One can easily verify that there exist normal bases
$v_{1},\ldots ,v_{k}$
and
$w_{1},\ldots ,w_{k}$
(respectively for the families of spaces
$V_{i}$
and
$W_{i}$
) that are dual.
Finally, given vectors
$v_{1},\ldots ,v_{m}\in \mathbb{R}^{k}$
, we denote by
$\unicode[STIX]{x1D6E4}(v_{1},\ldots ,v_{m})$
the
$m$
-volume determined by these vectors. It is equal to
$|\text{det}\,K|^{1/2}$
, where
$K$
is the
$m\times m$
matrix with entries
$k_{ij}=\langle v_{i},v_{\!j}\rangle$
, where
$\langle \cdot \,,\cdot \rangle$
is the standard inner product.
The following result gives several alternative characterizations of the regularity of a sequence of matrices.
Theorem 4. The following properties are equivalent:
-
(1)
$\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D712},\tilde{\unicode[STIX]{x1D712}})=0$
; -
(2)
$\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D712},\tilde{\unicode[STIX]{x1D712}})=0$
; -
(3) for any dual normal bases
$v_{1},\ldots ,v_{k}$
and
$w_{1},\ldots ,w_{k}$
of
$\mathbb{R}^{k}$
, we have (38)
$$\begin{eqnarray}\unicode[STIX]{x1D712}(v_{i})+\tilde{\unicode[STIX]{x1D712}}(w_{i})=0\quad \text{for }i=1,\ldots ,k;\end{eqnarray}$$
-
(4)
$\unicode[STIX]{x1D706}_{i}^{\prime }+\unicode[STIX]{x1D707}_{i}^{\prime }=0$
for
$i=1,\ldots ,k$
; -
(5)
(39)
$$\begin{eqnarray}\lim _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log |\det {\mathcal{A}}(n,1)|=\mathop{\sum }_{i=1}^{r}(\dim V_{i}-\dim V_{i-1})\unicode[STIX]{x1D706}_{i};\end{eqnarray}$$
-
(6) for any normal orthonormal basis
$v_{1},\ldots ,v_{k}$
of
$\mathbb{R}^{k}$
and
$m\leqslant k$
, the limit exists.
$$\begin{eqnarray}\lim _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log \unicode[STIX]{x1D6E4}({\mathcal{A}}(n,1)v_{1},\ldots ,{\mathcal{A}}(n,1)v_{m})\end{eqnarray}$$
Proof. The equivalence of the first two properties is immediate from (37).
Now we establish the equivalence of properties 2–4. Let
$v_{1},\ldots ,v_{k}$
and
$w_{1},\ldots ,w_{k}$
be dual normal bases. By (12), we have
$\unicode[STIX]{x1D712}(v_{i})+\tilde{\unicode[STIX]{x1D712}}(w_{i})\geqslant 0$
for all
$i$
. Therefore,
$$\begin{eqnarray}\displaystyle 0 & {\leqslant} & \displaystyle \unicode[STIX]{x1D706}(v_{i})+\tilde{\unicode[STIX]{x1D712}}(w_{i})\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \mathop{\sum }_{i=1}^{k}(\unicode[STIX]{x1D712}(v_{i})+\tilde{\unicode[STIX]{x1D712}}(w_{i}))\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{i=1}^{k}(\unicode[STIX]{x1D706}_{i}^{\prime }+\unicode[STIX]{x1D707}_{i}^{\prime })\leqslant k\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D712},\tilde{\unicode[STIX]{x1D712}})\end{eqnarray}$$
(the equality follows from the fact that the bases are normal). So, if property 2 holds, then (38) also holds.
Moreover, if property 3 holds, then
$\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D712},\tilde{\unicode[STIX]{x1D712}})=0$
and, thus, by (37), we have
$\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D712},\tilde{\unicode[STIX]{x1D712}})=0$
. Hence, by (40),
$$\begin{eqnarray}\mathop{\sum }_{i=1}^{k}(\unicode[STIX]{x1D706}_{i}^{\prime }+\unicode[STIX]{x1D707}_{i}^{\prime })=0.\end{eqnarray}$$
On the other hand, again since
$\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D712},\tilde{\unicode[STIX]{x1D712}})=0$
, it follows from the definition in (36) that
$\unicode[STIX]{x1D706}_{i}^{\prime }+\unicode[STIX]{x1D707}_{i}^{\prime }\leqslant 0$
for
$i=1,\ldots ,k$
and thus
$\unicode[STIX]{x1D706}_{i}^{\prime }+\unicode[STIX]{x1D707}_{i}^{\prime }=0$
for every
$i$
, that is, property 4 holds. Finally, property 4 clearly implies property 2.
In order to prove the equivalence to the remaining properties, we first establish an auxiliary result.
Lemma 3. If
$v_{1},\ldots ,v_{k}$
is a normal orthonormal basis of
$\mathbb{R}^{k}$
, then there exists a sequence
$(U_{n})_{n\in \mathbb{N}}$
of orthogonal matrices such that:
-
(1)
$C_{n}=U_{n+1}^{\ast }A_{n}U_{n}$
is upper triangular for each
$n\in \mathbb{N}$
; -
(2) the canonical basis
$e_{1},\ldots ,e_{n}$
of
$\mathbb{R}^{k}$
is normal for the family of spaces associated to the Lyapunov exponent defined by the sequence
$(C_{n})_{n\in \mathbb{N}}$
; -
(3)
$U_{1}e_{i}=v_{i}$
for
$i=1,\ldots ,k$
.
Proof of the lemma.
For each
$n\in \mathbb{N}$
and
$i=1,\ldots ,k$
, let
$v_{i}(n)={\mathcal{A}}(n,1)v_{i}$
. Since the matrices
$A_{n}$
are invertible, the vectors
$v_{1}(n),\ldots ,v_{k}(n)$
form a basis of
$\mathbb{R}^{k}$
for each
$n$
. Applying the Gram–Schmidt process, we obtain an orthonormal basis
$u_{1}(n),\ldots ,u_{k}(n)$
of
$\mathbb{R}^{k}$
. That is,
$$\begin{eqnarray}\langle u_{i}(n),u_{\!j}(n)\rangle =\unicode[STIX]{x1D6FF}_{ij}\quad \text{for }i,j=1,\ldots ,k.\end{eqnarray}$$
Now we consider the matrices
$U_{n}$
and
$V_{n}$
whose columns are respectively
$u_{1}(n),\ldots ,u_{k}(n)$
and
$v_{1}(n),\ldots ,v_{k}(n)$
. Since each vector
$u_{i}(n)$
is obtained in the Gram–Schmidt process as a linear combination of
$v_{1}(n),\ldots ,v_{i}(n)$
, with non-vanishing coefficient in
$v_{i}(n)$
, the vector
$v_{i}(n)$
is also a linear combination of
$u_{1}(n),\ldots ,u_{i}(n)$
, with non-vanishing coefficient in
$u_{i}(n)$
. Thus, there exist numbers
$d_{ij}(n)$
for
$i,j=1,\ldots ,k$
with
$i\leqslant j$
such that
$d_{jj}(n)\neq 0$
and
$$\begin{eqnarray}v_{\!j}(n)=\mathop{\sum }_{i=1}^{k}d_{ij}(n)u_{i}(n)\end{eqnarray}$$
for each
$j$
. Then the matrix
$D_{n}$
with entries
$$\begin{eqnarray}(D_{n})_{ij}=\left\{\begin{array}{@{}ll@{}}d_{ij}(n),\quad & i\leqslant j,\\ 0,\quad & i>j\end{array}\right.\end{eqnarray}$$
is upper triangular and
$U_{n}D_{n}=V_{n}$
, that is,
$D_{n}=U_{n}^{-1}V_{n}$
.
Now we consider the matrices
$$\begin{eqnarray}C_{n}=U_{n+1}^{\ast }A_{n}U_{n}=U_{n+1}^{-1}A_{n}U_{n}\end{eqnarray}$$
(notice that in view of (41), each matrix
$U_{n}$
is orthogonal). We have
$$\begin{eqnarray}{\mathcal{C}}(n,1)=U_{n}^{-1}{\mathcal{A}}(n,1)U_{1}\end{eqnarray}$$
and so
$$\begin{eqnarray}\displaystyle \langle e_{i},{\mathcal{C}}(n,1)e_{\!j}\rangle & = & \displaystyle \langle e_{i},U_{n}^{-1}{\mathcal{A}}(n,1)U_{1}e_{\!j}\rangle \nonumber\\ \displaystyle & = & \displaystyle \langle U_{n}e_{i},{\mathcal{A}}(n,1)v_{\!j}\rangle ,\nonumber\end{eqnarray}$$
since
$U_{1}$
is obtained by applying the Gram–Schmidt process to the basis
$v_{1},\ldots ,v_{k}$
that is already orthonormal. In particular, we have
$U_{1}e_{\!j}=v_{\!j}$
, thus yielding the third property in the lemma. Therefore,
$$\begin{eqnarray}\langle e_{i},{\mathcal{C}}(n,1)e_{\!j}\rangle =\langle u_{i}(n),v_{\!j}(n)\rangle =0\end{eqnarray}$$
for
$i>j$
, since
$v_{\!j}(n)$
is a linear combination of
$u_{1}(n),\ldots ,u_{\!j}(n)$
and the vectors
$u_{1}(n),\ldots ,u_{k}(n)$
are pairwise orthogonal. This shows that the matrix
${\mathcal{C}}(n,1)$
is upper triangular and so the same happens to
$$\begin{eqnarray}C_{n}={\mathcal{C}}(n+1,1){\mathcal{C}}(n,1)^{-1},\end{eqnarray}$$
thus yielding the first property in the lemma.
For the second property, we note that
$$\begin{eqnarray}{\mathcal{C}}(n,1)=U_{n}^{-1}{\mathcal{A}}(n,1)U_{1}.\end{eqnarray}$$
Since the matrix
$U_{n}$
is orthogonal, the Lyapunov exponents
$\unicode[STIX]{x1D712}$
and
$\tilde{\unicode[STIX]{x1D712}}$
defined, respectively, by the sequences
$(A_{n})_{n\in \mathbb{N}}$
and
$(C_{n})_{n\in \mathbb{N}}$
satisfy
$\tilde{\unicode[STIX]{x1D712}}(v)=\unicode[STIX]{x1D712}(U_{1}v)$
. In particular, they take exactly the same values with the same multiplicities (because
$U_{1}$
is invertible). Hence, since
$v_{1},\ldots ,v_{k}$
is a normal basis for the spaces associated to
$\unicode[STIX]{x1D712}$
, the vectors
$U_{1}^{-1}v_{i}=e_{i}$
, for
$i=1,\ldots ,k$
, form a normal basis for the spaces associated to
$\tilde{\unicode[STIX]{x1D712}}$
.◻
In what follows, we show that properties
$2$
,
$5$
and
$6$
are equivalent. Assume first that
$\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D712},\tilde{\unicode[STIX]{x1D712}})=0$
. By Theorem 2, we have
$$\begin{eqnarray}\liminf _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log |\det {\mathcal{A}}(n,1)|=\limsup _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log |\det {\mathcal{A}}(n,1)|\end{eqnarray}$$
and so the limit in the left-hand side of (39) exists. Given dual normal bases, it follows from property (38) that
$$\begin{eqnarray}\mathop{\sum }_{i=1}^{k}\unicode[STIX]{x1D712}(v_{i})=-\mathop{\sum }_{i=1}^{k}\tilde{\unicode[STIX]{x1D712}}(w_{i}).\end{eqnarray}$$
Hence, it follows readily from (15) and (16) that identity (39) holds.
Now we assume that property 5 holds. For the upper triangular matrices
$C_{n}$
constructed in Lemma 3, we have
$$\begin{eqnarray}{\mathcal{C}}(n,1)e_{i}=U_{n}^{-1}{\mathcal{A}}(n,1)U_{1}e_{i}=U_{n}^{-1}{\mathcal{A}}(n,1)v_{i}\end{eqnarray}$$
(see (42)) and, so,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}({\mathcal{A}}(n,1)v_{1},\ldots ,{\mathcal{A}}(n,1)v_{m}) & =\unicode[STIX]{x1D6E4}({\mathcal{C}}(n,1)e_{1},\ldots ,{\mathcal{C}}(n,1)e_{m})=\mathop{\prod }_{i=1}^{m}|c_{i}(n)| & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
for
$m\leqslant k$
, where
$$\begin{eqnarray}c_{i}(n)=\langle {\mathcal{C}}(n,1)e_{i},e_{i}\rangle .\end{eqnarray}$$
In particular, taking
$m=k$
, we obtain
$$\begin{eqnarray}\displaystyle \lim _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log |\det {\mathcal{A}}(n,1)| & = & \displaystyle \lim _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log \unicode[STIX]{x1D6E4}({\mathcal{A}}(n,1)v_{1},\ldots ,{\mathcal{A}}(n,1)v_{k})\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \mathop{\sum }_{i=1}^{k}\limsup _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log |c_{i}(n)|\nonumber\end{eqnarray}$$
(the limit exists by property 5). On the other hand, by (43),
$$\begin{eqnarray}\mathop{\sum }_{i=1}^{k}\unicode[STIX]{x1D712}(v_{i})=\mathop{\sum }_{i=1}^{k}\limsup _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\text{log}\,\Vert {\mathcal{A}}(n,1)v_{i}\Vert \geqslant \mathop{\sum }_{i=1}^{k}\limsup _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\text{log}\,|c_{i}(n)|.\end{eqnarray}$$
By property 5, we obtain
$$\begin{eqnarray}\displaystyle \lim _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log \unicode[STIX]{x1D6E4}({\mathcal{A}}(n,1)v_{1},\ldots ,{\mathcal{A}}(n,1)v_{k}) & = & \displaystyle \mathop{\sum }_{i=1}^{k}\unicode[STIX]{x1D712}(v_{i})\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{i=1}\limsup _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\text{log}\,|c_{i}(n)|.\nonumber\end{eqnarray}$$
On the other hand, by (44),
$$\begin{eqnarray}\lim _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log \unicode[STIX]{x1D6E4}({\mathcal{A}}(n,1)v_{1},\ldots ,{\mathcal{A}}(n,1)v_{k})=\lim _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\mathop{\sum }_{i=1}^{k}\text{log}\,|c_{i}(n)|\end{eqnarray}$$
and, so,
$$\begin{eqnarray}\lim _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\mathop{\sum }_{i=1}^{k}\text{log}\,|c_{i}(n)|=\mathop{\sum }_{i=1}^{k}\limsup _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\text{log}\,|c_{i}(n)|.\end{eqnarray}$$
It follows from this identity that each
$\limsup$
in (46) is a limit. Indeed, if
$$\begin{eqnarray}\text{}\underline{c}_{i}:=\liminf _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\text{log}\,|c_{i}(n)|<\limsup _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\text{log}\,|c_{i}(n)|=:\overline{c}_{i}\end{eqnarray}$$
for some
$i=j$
, then choosing a sequence
$k_{n}$
such that
$$\begin{eqnarray}\lim _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(k_{n})}\text{log}\,|c_{\!j}(k_{n})|=\text{}\underline{c}_{\!j},\end{eqnarray}$$
we obtain
$$\begin{eqnarray}\displaystyle \lim _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\mathop{\sum }_{i=1}^{k}\text{log}\,|c_{i}(n)| & = & \displaystyle \lim _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(k_{n})}\mathop{\sum }_{i=1}^{k}\text{log}\,|c_{i}(k_{n})|\nonumber\\ \displaystyle & = & \displaystyle \text{}\underline{c}_{\!j}+\lim _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(k_{n})}\mathop{\sum }_{i\neq j}\text{log}\,|c_{i}(k_{n})|<\overline{c}_{\!j}+\mathop{\sum }_{i\neq j}\overline{c}_{i}\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{i=1}^{k}\overline{c}_{i},\nonumber\end{eqnarray}$$
which contradicts (46). Therefore, by (44), we obtain
$$\begin{eqnarray}\lim _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log \unicode[STIX]{x1D6E4}({\mathcal{A}}(n,1)v_{1},\ldots ,{\mathcal{A}}(n,1)v_{m})=\mathop{\sum }_{i=1}^{m}\lim _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\text{log}\,|c_{i}(n)|.\end{eqnarray}$$
This establishes property 6.
Finally, we assume that property 6 holds. Consider again the upper triangular matrices
$C_{n}$
constructed in Lemma 3. By (43), since
$U_{n}$
is orthogonal, it follows from property 6 and (44) that
$$\begin{eqnarray}\displaystyle & & \displaystyle \lim _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log \frac{\unicode[STIX]{x1D6E4}({\mathcal{A}}(n,1)v_{1},\ldots ,{\mathcal{A}}(n,1)v_{m})}{\unicode[STIX]{x1D6E4}({\mathcal{A}}(n,1)v_{1},\ldots ,{\mathcal{A}}(n,1)v_{m-1})}\nonumber\\ \displaystyle & & \displaystyle \quad =\lim _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\log \frac{\unicode[STIX]{x1D6E4}({\mathcal{C}}(n,1)e_{1},\ldots ,{\mathcal{C}}(n,1)e_{m})}{\unicode[STIX]{x1D6E4}({\mathcal{C}}(n,1)e_{1},\ldots ,{\mathcal{C}}(n,1)e_{m-1})}\nonumber\\ \displaystyle & & \displaystyle \quad =\lim _{n\rightarrow +\infty }\frac{1}{\unicode[STIX]{x1D70C}(n)}\text{log}\,|c_{m}(n)|,\nonumber\end{eqnarray}$$
with
$c_{m}(n)$
as in (45). Hence, by Theorem 3, the regularity coefficient of the pair of Lyapunov exponents defined by the matrices
$C_{n}$
and
$(C_{n}^{\ast })^{-1}$
vanishes. But, since the Lyapunov exponents defined by the matrices
$A_{n}$
and
$C_{n}$
as well as the Lyapunov exponents defined by the matrices
$(A_{n}^{\ast })^{-1}$
and
$(C_{n}^{\ast })^{-1}$
take exactly the same values with the same multiplicities, the regularity coefficient
$\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D712},\tilde{\unicode[STIX]{x1D712}})$
also vanishes. This completes the proof of the theorem.◻
