##
**Characterization of regularity sets for families of sequences of matrices.**
*(English)*
Zbl 1487.37036

The authors consider one-parameter families \(A(\theta) = (A_{n}(\theta))_{n \in \mathbb{N}}\) of sequences of invertible \((q \times q)\)-matrices \(A_{n}(\theta)\) depending continuously on \(\theta \in \mathbb{R}\) for each \(n \in \mathbb{N}\). Their aim is to characterize the regularity sets of such families, namely the sets of parameters for which the dynamics of a sequence is Lyapunov regular.

The main novelty of the paper is to extend the known theory on the above problem to general sequences of matrices, in the sense that matrices (and their inverses) might be unbounded in norm. The unbounded case was not considered until now and it even seemed quite hard to obtain interesting results in this case.

The authors put emphasis on the fact that the notion of regularity is connected with that of hyperbolicity. The interconnection between regularity and hyperbolicity plays a crucial role in the theory of nonuniform hyperbolicity and smooth ergodic theory. In fact the notion of regularity quantifies the degree of nonuniform exponential stability of a linear dynamical system that persists under small nonlinear perturbations. The key point is to try to understand how regularity changes under such perturbations.

First they consider a one-parameter family \(A(\theta)\) of sequences of matrices and introduce the notion of Lyapunov exponent as the map \[ \lambda_{\theta}(v) = \limsup_{n \rightarrow \infty}\frac{1}{n} \log\|\mathcal{A}_{n}(\theta)v\|, \] where \(\lambda_{\theta}: \mathbb{R}^{q} \rightarrow \mathbb{R}\) and \(\mathcal{\theta} = A_{n-1}(\theta)\cdots A_{1}(\theta)\) for \(n > 1\) and the identity for \(n=1\).

To understand better the above formula, let \(k \in \{1, \dots, q\}\) be fixed and consider the corresponding Grassmannian \(\mathrm{Gr}_{k} (\mathbb{R}^{q})\), that is, the set of all \(k\)-dimensional subspaces of \(\mathbb{R}^{q}\). Let \[ \lambda_{k}'(\theta) = \inf_{L \in \mathrm{Gr}_{k}}(\mathbb{R}^{q}) \sup_{v \in L}\lambda_{\theta}(v). \] When \(\lambda_{\theta}\) takes only finite values on \(\mathbb{R}^{q} \setminus \{0\}\) the ordering is \(\lambda_{1} <\dots < \lambda_{p(\theta)}(\theta)\) for some integer \(p(\theta) \leq q\). There are corresponding linear subspaces \(E_{i}(\theta) = \{v \in \mathbb{R}^{q}: \lambda_{\theta}(v) \leq \lambda_{i}(\theta)\}\). In this case \(\lambda'_{1}(\theta) \leq \dots \leq \lambda'_{q}(\theta)\) are the values of \(\lambda_{\theta}\) on \(\mathbb{R}^{q}\) counted with their multiplicities.

Then \(A(\theta)\) is regular in Lyapunov sense if \(\lambda_{\theta}(v)\) is a real number and for all \(v \neq 0\) there holds \[ \liminf_{n \rightarrow \infty} \frac{1}{n} \log| \det \mathcal{A}_{n}(\theta)| = \sum_{k=1}^{q} \lambda'_{k}(\theta) .\tag{1} \]

When \(A(\theta)\) verifies the condition \[ \sup_{n}\|A_{n}(\theta)^{{\pm}1}\| < \infty \tag{2} \] then \(\lambda_{\theta}(v)\) is finite for any \(v \neq 0\) and as a consequence \(\lambda'_{k}(\theta)\) is also finite for \(k \in \{1,2,\dots,q\}\). Then \(A(\theta)\) satisfying \((2)\) is regular in Lyapunov sense if and only if \((1)\) holds. Formula \((1)\) is the classic notion of regularity and the notion of regularity used by the authors generalizes the classical one and it can be applied to arbitrary sequences of matrices.

The first step is to consider sequences \(A(\theta)\) for which \((2)\) does not hold. Now we recall the notion of Lyapunov coordinate change for general sequences \((U_{n})_{n \in \mathbb{N}}\). This can be performed when \[ \lim_{n \rightarrow \infty} \frac{1}{ n}\log\|U_{n}\| = \lim_{n \rightarrow \infty} \frac{1}{ n}\log\|U_{n}^{-1}\| = 0. \]

Such notion allows us to show when \(A(\theta)\) satisfying \((2)\) is regular: this happens if and only if there are a Lyapunov coordinate change \((U_{n})_{n}\) and nonzero constants \(c_{1}, c_{2}, \dots,c_{q}\) such that \[ U_{n+1}^{-1}A_{n}(\theta)U_{n} = \operatorname{diag}(c_{1}\dots c_{q}) \tag{3} \] for every \(n \in \mathbb{N}\).

It can be shown that property \((3)\) holds if and only if:

The type of problems considered in the paper were initially stated by Yu. Bogdanov [Dokl. Akad. Nauk SSSR 104, 813–814 (1955)], who asked if there exist regular equations of the form \(y'(t) = A(t)y(t)\) such that the regularity set of the family of equations \(y'(t) = \theta A(t)y(t)\) parametrized by \(\theta \in \mathbb{R}\) is not the whole line.

The most important part of the paper is devoted to provide a topological description of the regularity set associated with the sequence \(A(\theta)\) defined as the set of real parameters \(\theta\) for which the sequence is regular. The authors prove two interesting results:

Both results improve and extend previous results in the literature about this subject.

The main novelty of the paper is to extend the known theory on the above problem to general sequences of matrices, in the sense that matrices (and their inverses) might be unbounded in norm. The unbounded case was not considered until now and it even seemed quite hard to obtain interesting results in this case.

The authors put emphasis on the fact that the notion of regularity is connected with that of hyperbolicity. The interconnection between regularity and hyperbolicity plays a crucial role in the theory of nonuniform hyperbolicity and smooth ergodic theory. In fact the notion of regularity quantifies the degree of nonuniform exponential stability of a linear dynamical system that persists under small nonlinear perturbations. The key point is to try to understand how regularity changes under such perturbations.

First they consider a one-parameter family \(A(\theta)\) of sequences of matrices and introduce the notion of Lyapunov exponent as the map \[ \lambda_{\theta}(v) = \limsup_{n \rightarrow \infty}\frac{1}{n} \log\|\mathcal{A}_{n}(\theta)v\|, \] where \(\lambda_{\theta}: \mathbb{R}^{q} \rightarrow \mathbb{R}\) and \(\mathcal{\theta} = A_{n-1}(\theta)\cdots A_{1}(\theta)\) for \(n > 1\) and the identity for \(n=1\).

To understand better the above formula, let \(k \in \{1, \dots, q\}\) be fixed and consider the corresponding Grassmannian \(\mathrm{Gr}_{k} (\mathbb{R}^{q})\), that is, the set of all \(k\)-dimensional subspaces of \(\mathbb{R}^{q}\). Let \[ \lambda_{k}'(\theta) = \inf_{L \in \mathrm{Gr}_{k}}(\mathbb{R}^{q}) \sup_{v \in L}\lambda_{\theta}(v). \] When \(\lambda_{\theta}\) takes only finite values on \(\mathbb{R}^{q} \setminus \{0\}\) the ordering is \(\lambda_{1} <\dots < \lambda_{p(\theta)}(\theta)\) for some integer \(p(\theta) \leq q\). There are corresponding linear subspaces \(E_{i}(\theta) = \{v \in \mathbb{R}^{q}: \lambda_{\theta}(v) \leq \lambda_{i}(\theta)\}\). In this case \(\lambda'_{1}(\theta) \leq \dots \leq \lambda'_{q}(\theta)\) are the values of \(\lambda_{\theta}\) on \(\mathbb{R}^{q}\) counted with their multiplicities.

Then \(A(\theta)\) is regular in Lyapunov sense if \(\lambda_{\theta}(v)\) is a real number and for all \(v \neq 0\) there holds \[ \liminf_{n \rightarrow \infty} \frac{1}{n} \log| \det \mathcal{A}_{n}(\theta)| = \sum_{k=1}^{q} \lambda'_{k}(\theta) .\tag{1} \]

When \(A(\theta)\) verifies the condition \[ \sup_{n}\|A_{n}(\theta)^{{\pm}1}\| < \infty \tag{2} \] then \(\lambda_{\theta}(v)\) is finite for any \(v \neq 0\) and as a consequence \(\lambda'_{k}(\theta)\) is also finite for \(k \in \{1,2,\dots,q\}\). Then \(A(\theta)\) satisfying \((2)\) is regular in Lyapunov sense if and only if \((1)\) holds. Formula \((1)\) is the classic notion of regularity and the notion of regularity used by the authors generalizes the classical one and it can be applied to arbitrary sequences of matrices.

The first step is to consider sequences \(A(\theta)\) for which \((2)\) does not hold. Now we recall the notion of Lyapunov coordinate change for general sequences \((U_{n})_{n \in \mathbb{N}}\). This can be performed when \[ \lim_{n \rightarrow \infty} \frac{1}{ n}\log\|U_{n}\| = \lim_{n \rightarrow \infty} \frac{1}{ n}\log\|U_{n}^{-1}\| = 0. \]

Such notion allows us to show when \(A(\theta)\) satisfying \((2)\) is regular: this happens if and only if there are a Lyapunov coordinate change \((U_{n})_{n}\) and nonzero constants \(c_{1}, c_{2}, \dots,c_{q}\) such that \[ U_{n+1}^{-1}A_{n}(\theta)U_{n} = \operatorname{diag}(c_{1}\dots c_{q}) \tag{3} \] for every \(n \in \mathbb{N}\).

It can be shown that property \((3)\) holds if and only if:

- 1.
- For each \(v \in \mathbb{R}^{q} \setminus \{0\}\) one has \[ \lambda _{\theta}(v) = \lim_{n \rightarrow \infty}\log\|\mathcal{A}_{n}(\theta)v\| \in \mathbb{R};\]
- 2.
- Formula \((1)\) is satisfied.

The type of problems considered in the paper were initially stated by Yu. Bogdanov [Dokl. Akad. Nauk SSSR 104, 813–814 (1955)], who asked if there exist regular equations of the form \(y'(t) = A(t)y(t)\) such that the regularity set of the family of equations \(y'(t) = \theta A(t)y(t)\) parametrized by \(\theta \in \mathbb{R}\) is not the whole line.

The most important part of the paper is devoted to provide a topological description of the regularity set associated with the sequence \(A(\theta)\) defined as the set of real parameters \(\theta\) for which the sequence is regular. The authors prove two interesting results:

- 1.
- The regularity set of any one-parameter family A(\(\theta\)) is an \(F_{\sigma\delta}\)-set. It means that such set is a countable intersection of countable unions of closed sets;
- 2.
- Given an \(F_{\sigma\delta}\)-set \(M\) containing \(0\) it is possible construct a one-parameter family \(A(\theta)\) whose regularly set is exactly \(M\).

Both results improve and extend previous results in the literature about this subject.

Reviewer: Francisco Balibrea (Murcia)

### MSC:

37D25 | Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) |

37B55 | Topological dynamics of nonautonomous systems |

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\textit{L. Barreira} and \textit{C. Valls}, Topol. Methods Nonlinear Anal. 57, No. 2, 675--693 (2021; Zbl 1487.37036)

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