Abstract
In this paper we study ergodic optimization and multifractal behavior of Lyapunov exponents for matrix cocycles. We show that the restricted variational principle holds for generic cocycles [in the sense of (Bonatti and Viana in Ergod Theory Dyn Syst 24(5):1295–1330, 2004)] over mixing subshifts of finite type. We also show that the Lyapunov spectrum is equal to the closure of the set where the entropy spectrum is positive for such cocycles. Moreover, we show the continuity of the entropy spectrum at boundary of Lyapunov spectrum in the sense that \(h_{top}(E(\alpha _{t}))\ \rightarrow h_{top}(E(\beta ({\mathcal {A}}))\), where \(E(\alpha )=\{x\in X: \lim _{n\rightarrow \infty }\frac{1}{n}\log \Vert {\mathcal {A}}^{n}(x)\Vert =\alpha \}\), for such cocycles. We prove the continuity of the lower joint spectral radius for linear cocycles under the assumption that linear cocycles satisfy a cone condition.
Introduction and Statement of the Results
Let X be a compact metric space that is endowed with the metric d. We call (X, T) a topological dynamical system (TDS), if \(T: X\rightarrow X\) is a continuous map on the compact metric space X.
We denote by \({\mathcal {M}}(X,T)\) the space of all Tinvariant Borel probability measures on X. This space is a nonempty convex set and is compact with respect to the weak* topology. Moreover, we denote by \({\mathcal {E}}(X, T)\subset {\mathcal {M}}(X, T)\) the subset of ergodic measures, which are exactly the extremal points of \({\mathcal {M}}(X, T)\); see [35] for more information.
Let \(f:X\rightarrow {\mathbb {R}}\) be a continuous function. We denote by \(S_{n}f(x):=\sum _{k=0}^{n1}f(T^{k}(x))\) the Birkhoff sum, and we call
a Birkhoff average.
By Birkhoff theorem, for every \(\mu \in {\mathcal {M}}(X, T)\) and \(\mu \)almost every \(x\in X\), the Birkhoff average is welldefined. We denote by \(\beta (f)\) and \(\alpha (f)\) the supremum and infimum of the Birkhoff average over \(x\in X\), respectively; we call these numbers the maximal and minimal ergodic averages of f.
We say that \(\Phi :=\{\log \phi _{n}\}_{n=1}^{\infty }\) is a subadditive potential if each \(\phi _{n}\) is a continuous positivevalued function on X such that
Furthermore, \(\Phi =\{\log \phi _{n}\}_{n=1}^{\infty }\) is said to be an almost additive potential if there exists a constant \(C \ge 1\) such that for any \(m,n \in {\mathbb {N}}\), \(x\in X\), we have
We also say that \(\Phi =\{\log \phi _{n}\}_{n=1}^{\infty }\) is an additive potential if
in this case, \(\phi _{n}(x) = e^{S_n\log \phi _1(x)}\).
If \(\mu \) is an ergodic invariant probability measure, then the Birkhoff average converges to \(\int f d \mu \) for \(\mu \)almost all points, but there are plenty of ergodic invariant measures, for which the limit exists but converges to a different quantity. Moreover, there are plenty of points which are not generic points for any ergodic measure or even for which the Birkhoff average does not exist. So, one may ask about the size of the set of points
which we call \(\alpha \)level set of Birkhoff spectrum, for a given value \(\alpha \) from the set
which we call Birkhoff spectrum.
That size is usually calculated in terms of topological entropy (see Subsect. 3.3). Let \(Z\subset X\), we denote by \(h_{top}(T, Z)\) topological entropy of T restricted to Z or, simply, the topological entropy of Z, denote \(h_{top}(Z),\) when there is no confusion about T. In particular we write \(h_{top}(T)\) for \(h_{top}(T,X)\).
We investigate the end points of Birkhoff spectrum, i.e., \(\alpha (f)\) and \(\beta (f)\). Since \(\alpha (f)=\beta (f)\), let us focus on the quantity \(\beta \). It can also be characterized as
By the compactness of \({\mathcal {M}}(X, T)\), there is at least one measure \(\mu \in {\mathcal {M}}(X, T)\) for which \(\beta (f)=\int f d\mu \); such measures are called Birkhoff maximizing measures.
It is wellknown (see, e.g. [25, 26, 43]) when (X, T) is a transitive subshift of finite type and f is an additive potential, then
and
where \(P_{f}(t)\) the topological pressure for a potential tf (see [45]).
In the almost additive potentials case, (1.2) was proven by Feng and Huang [29] under certain assumptions. In the subadditve potentials case, Feng and Huang [29] proved a similar result for \(t>0\) under the upper semi continuity entropy assumption.
A natural example of subadditive potentials is matrix cocycles. More precisely, given a continuous map \({\mathcal {A}}:X \rightarrow GL(k, {\mathbb {R}})\) taking values into the space \(k \times k\) invertible matrices. We consider the products
The pair \((T, {\mathcal {A}})\) is called a linear cocycle. That induces a skewproduct dynamics F on \(X\times {\mathbb {R}}^{k}\) by \((x, v)\mapsto X\times {\mathbb {R}}^{k}\), whose nth iterate is therefore
If T is invertible then so is F. Moreover, \(F^{n}(x)=(T^{n}(x), {\mathcal {A}}^{n}(x)v)\) for each \(n\ge 1\), where
In general, one could consider vector bundles over X instead of \(X \times {\mathbb {R}}^{d}\), and then consider bundle endomorphisms that fiber over \(T:X\rightarrow X.\)
A simple class of linear cocycles is locally constant cocycles which is defined as follows. Assume that \(X=\{1,...,q\}^{{\mathbb {Z}}}\) is a symbolic space. Suppose that \(T:X \rightarrow X\) is a shift map, i.e. \(T(x_{l})_{l}=(x_{l+1})_{l}\). Given a finite set of matrices \({\mathcal {A}} =\{A_{1},\ldots ,A_{q}\} \subset GL(k, {\mathbb {R}})\), we define the function \({\mathcal {A}}:X \rightarrow GL(k, {\mathbb {R}})\) by \({\mathcal {A}}(x)=A_{x_{0}}.\) In this case, we say that\((T,{\mathcal {A}})\) is a locally constant cocycle.
By Kingman’s subadditive ergodic theorem, for any \(\mu \in {\mathcal {M}}(X, T)\) and \(\mu \) almost every \(x\in X\) such that \(\log ^{+} \Vert {\mathcal {A}}\Vert \in L^{1}(\mu )\), the following limit, called the top Lyapunov exponent at x, exists:
where \(\Vert {\mathcal {A}}\Vert \) the Euclidean operator norm of a matrix \({\mathcal {A}}\), that is submultiplicative i.e.,
therefore, the potential \(\Phi _{{\mathcal {A}}}:=\{\log \Vert {\mathcal {A}}^{n}\Vert \}_{n=1}^{\infty }\) is subadditive.
Let us denote \(\chi (\mu , {\mathcal {A}})=\int \chi (. , {\mathcal {A}} )d\mu .\) If the measure \(\mu \) is ergodic, then \(\chi (x, {\mathcal {A}})=\chi (\mu , {\mathcal {A}})\) for \(\mu \)almost every \(x\in X.\)
Similarly to what we did for Birkhoff average (1.1), one can either maximize or minimize top Lyapunov exponent (1.3); the corresponding quantities will be denoted by \(\beta ({\mathcal {A}})\) and \(\alpha ({\mathcal {A}})\), respectively; we call these numbers the maximal and minimal Lyapunov exponents of \({\mathcal {A}}\). However, this time the maximization and the minimization problems are totally different. Even though \(\beta ({\mathcal {A}})\) is always attained by at least one measure (which is called a Lyapunov maximizing measure), that is not necessarily the case for the minimal Lyapunov exponent \(\alpha ({\mathcal {A}})\). In fact, in the locally constant cocycles case, Bochi and Morris [12] investigated the continuity properties of the minimal Lyapunov exponent. They showed that \(\alpha ({\mathcal {A}})\) is Lipschitz continuous at \({\mathcal {A}}\) under 1domination assumption. Breuillard and Sert [15] extended Bochi and Morris’s result to the joint spectrum under domination condition. In this case the \(\chi (\mu , {\mathcal {A}})\) depends continuously on the measure \(\mu \).
Feng [26] proved (1.2) for continuous positive matrixvalued functions on the one side shift. He (see [27, 29]) also proved the first part (1.2) for locally constant cocycles under the irreducibility assumption.
The linear cocycles generated by the derivative of a diffeomorphism map or a smooth map \(T:X\rightarrow X\) on a closed Riemannian manifold X and a family of maps \({\mathcal {A}}(x):=D_{x}T: T_{x} X\rightarrow T_{T(x)}X\) are called derivative cocycles. Moreover, when \(T:X\rightarrow X\) is an Anosov diffeomorphism (or expanding map), Bowen [16] showed that there exists a symbolic coding of T by a subshift of finite type. Therefore, one can replace the derivative cocycle of a uniformly hyperbolic map by a linear cocycle over a subshift of finite type. In general, we know much more about locally constant cocycles than about the more general derivative cocycles.
In this paper, we are interested in linear cocycles \((T, {\mathcal {A}})\) generated by \(GL(k, {\mathbb {R}})\) valued functions \({\mathcal {A}}\) over two side subshifts of finite type \((\Sigma , T)\). We denote by \({\mathcal {L}}\) the set of admissible words of \(\Sigma \). For any \({\mathcal {A}}:\Sigma \rightarrow GL(k, {\mathbb {R}})\) and \(I\in {\mathcal {L}}\), we define
We define a positive continuous function \(\{\varphi _{{\mathcal {A}}, n}\}_{n\in {\mathbb {N}}}\) on \(\Sigma \) such that
We denote by \(\Phi _{{\mathcal {A}}}\) the subbadditive potential \(\{\log \varphi _{{\mathcal {A}}, n}\}_{n=1}^{\infty }\).
We say that \({\mathcal {A}}\) is quasimultiplicative if there exist \(C>0\) and \(m\in {\mathbb {N}}\) such that for every \(I, J \in {\mathcal {L}}\), there exists \(K\in {\mathcal {L}}\) with \(K\le m\) such that \(IKJ \in {\mathcal {L}}\) and
Bonatti and Viana [20] introduced the notion of typical cocycles among fiberbunched cocycles (see Definitions 2.4 and 2.5 for precise formulations). We assume that \(T:\Sigma \rightarrow \Sigma \) is a topologically mixing subshift of finite type. We denote by \(H^{r}(\Sigma , GL(k, {\mathbb {R}}))\) the space of all \(r\)Hölder continuous functions. We also denote by \(H_{b}^{r}(\Sigma , GL(k, {\mathbb {R}}))\) the space of all \(r\)Hölder continuous and fiber bunched functions, which says that the cocycles are nearly conformal. The set
is open in \(H_{b}^{r}(\Sigma , GL(k, {\mathbb {R}}))\), and also Bonatti and Viana [20] proved that \({\mathcal {W}}\) is dense in \(H_{b}^{r}(\Sigma , GL(k, {\mathbb {R}}))\) and that its complement has infinite codimension, i.e., it is contained in finite unions of closed submanifolds with arbitrary codimension.
We denote \(E(\alpha )=E_{\Phi }(\alpha )\) when there is no confusion about \(\Phi .\)
Our main results are Theorems A, B, C and D formulated as follows:
Theorem A
Let \({\mathcal {A}}\in {\mathcal {W}}\). Then,
Furthermore, \(\alpha \mapsto h_{top}(E(\alpha ))\) is concave in \(\alpha \in \mathring{L}.\)
Theorem B
Suppose that \({\mathcal {A}}:\Sigma \rightarrow GL(k, {\mathbb {R}})\) belongs to typical functions \({\mathcal {W}}\). Then,
where \(\Omega :=\{\chi (\mu , {\mathcal {A}}) : \mu \in {\mathcal {M}}(\Sigma , T)\}.\)
Barreira and Gelfert [6] first obtained similar results for repellers of \(C^{1+\alpha }\) maps satisfying the cone condition (see Sect. 5) and bounded distortion. Feng and Huang [29, Theorem 4.8] improved the result to subadditive potentials for \(q \in {\mathbb {R}}_{+}\) (see (1.5)), and then Park [44, Corollary 6.6] used their result for cocycles that are quasimultiplicative. We extend the result for \(q \in {\mathbb {R}}\) (see (1.5)).
We show the continuity of the entropy spectrum of Lyapunov exponents for generic matrix cocycles.
Theorem C
Suppose \({\mathcal {A}}_{l}, {\mathcal {A}}\in {\mathcal {W}}\) with \({\mathcal {A}}_{l} \rightarrow {\mathcal {A}}\), and \(t_{l},t\in {\mathbb {R}}_{+}\) such that \(t_{l}\rightarrow t.\) Let \(\alpha _{t_{l}}=P_{\Phi _{{\mathcal {A}}_{l}}}^{'}(t_{l})\) and \(\alpha _{t}=P_{\Phi _{{\mathcal {A}}}}^{'}(t)\). Then
Moreover,
We also investigate the continuity of the minimal Lyapunov exponents for general cocycles. We prove the continuity of the minimal Lyapunov exponent under a cone condition. Moreover, our result implies the continuity of the minimal Lyapunov exponent under 1domination assumption.
Theorem D
Let (X, T) be a topologically mixing subshift of finite type. Suppose that \({\mathcal {A}}_{n}, {\mathcal {A}}:X \rightarrow GL(k, {\mathbb {R}})\) are matrix cocycles over (X, T), and \(\Phi _{{\mathcal {A}}}\) has bounded distortion. Assume that \((C_x)_{x\in X}\) is an invariant cone field on X.^{Footnote 1} Then \(\alpha ({\mathcal {A}}_{n}) \rightarrow \alpha ({\mathcal {A}})\) when \({\mathcal {A}}_{n}\rightarrow {\mathcal {A}}.\)
In this paper, we also obtain the high dimensional versions of Theorems A, B, and C.
The paper divides into two parts. The first part contains Sects. 2, 3 and 4, where we introduce necessary notions and results needed to state Theorems A, B, and C as well as the proofs of Theorems A, B, and C, and the second part contains Sect. 5, where we introduce necessary notions and results needed to state Theorem D and its proof.
Preliminaries
In this section, we recall some basic facts and definitions that we need to prove main theorems.
Symbolic Dynamics
In this section, we recall some definitions and basic facts related to subshift of finite type. For more information see [38].
Let \(Q=(q_{ij})\) be a \(k\times k\) with \(q_{ij}\in \{0, 1\}.\) The one side subshift of finite type associated to the matrix Q is a left shift map \(T:\Sigma _{Q}^{+}\rightarrow \Sigma _{Q}^{+}\) meaning that, \(T(x_{n})_{n\in {\mathbb {N}}_{0}}=(x_{n+1})_{n\in {\mathbb {N}}_{0}}\), where \(\Sigma _{Q}^{+}\) is the set of sequences
Similarly, one defines two sided subshift of finite type \(T:\Sigma _{Q}\rightarrow \Sigma _{Q}\), where
When the matrix Q has entries all equal to 1 we say this is the full shift. For simplicity, we denote that \(\Sigma _{Q}^{+}=\Sigma ^{+}\) and \(\Sigma _{Q}=\Sigma \).
We say that \(i_{0}...i_{k1}\) is an admissible word if \(Q_{i_{n},i_{n+1}}= 1\) for all \(0\le n \le k2\). We denote by \({\mathcal {L}}\) the collection of admissible words. We denote by I the length of \(I\in {\mathcal {L}}.\) Denote by \({\mathcal {L}}(n)\) the set of admissible words of length n. That is, a word \(i_{0},..,i_{n1}\) with \(i_{j}\in \{1,...,k\}\) such that \(Q_{x_{i}, x_{i+1}}=1\). One can define nth level cylinder [I] as follows:
for any \(i_{0}...i_{n1}\in {\mathcal {L}}(n).\)
Observe that the partition of \(\Sigma _Q\) (or \(\Sigma _Q^+\)) into first level cylinders is generating, for this reason the partition into first level cylinders is the partition canonically used in symbolic dynamics to calculate the metric entropy.
Definition 2.1
The matrix Q is called primitive when there exists \(n>0\) such that all the entries of \(Q^n\) are positive.
It is wellknown that a subshift of a finite type associated with a primitive matrix Q is topologically mixing. That is, for every open nonempty \(U, V \subset \Sigma \), there is N such that for every \(n\ge N\), \(T^{n}(U)\cap V \ne \emptyset .\) We say that T is topological transitive if there is a point with dense orbit.
We fix \(\omega \in (0,1)\) and consider the space \(\Sigma \) is endowed with the metric \(d_{\omega }\) which is defined as follows: For \(x=(x_{i})_{i\in {\mathbb {Z}}}, y=(y_{i})_{i\in {\mathbb {Z}}} \in \Sigma \), we have
where k is the largest integer such that \(x_{i}=y_{i}\) for all \(i <k.\) We denote \(d:=d_{\omega }\) when there is no confusion about \(\omega .\)
In the twosided dynamics, we define the local stable set
and the local unstable set
Furthermore, the global stable and unstable manifolds of \(x \in \Sigma \) are
If \(\Sigma \) is equipped by the metric \(d_{\omega }\) (see (2.1)), then the two side subshift of finite type \(T:\Sigma \rightarrow \Sigma \) becomes a hyperbolic homeomorphism; see [3, Subsect. 2.3].
Multilinear Algebra
We recall some basic facts about the exterior algebra. We use it for studying the singular value function.
We denote by \(\sigma _{1},...,\sigma _{k}\) the singular values of the matrix A, which are the square roots of the eigenvalues of the positive semi definite matrix \(A^{*}A\) listed in decreasing order according to multiplicity.
\(\{e_{1},..,e_{k}\}\) is the standard orthogonal basis of \({\mathbb {R}}^{k}\) and define
for all \(l\in \{1,...,k\}\) with the convention that \(\wedge ^{0} {\mathbb {R}}^{k}={\mathbb {R}}\). It is called the lth exterior power of \({\mathbb {R}}^{k}\). That is a \(\left( {\begin{array}{c}k\\ l\end{array}}\right) \)dimensional \({\mathbb {R}}\)vector space spanned by decomposable vectors \(v_1 \wedge \ldots \wedge v_k\) with the usual identifications.
We are interested in the group \(k\times k\) invertible matrices of real numbers \(GL(k, {\mathbb {R}})\) that can be seen as a subset of \({\mathbb {R}}^{k^{2}}\). This space has a topology induced from \({\mathbb {R}}^{k^{2}}\). For \(A\in GL(k, {\mathbb {R}})\), we define an invertible linear map \(A^{\wedge l} : \wedge ^{l} {\mathbb {R}}^{k} \rightarrow \wedge ^{l} {\mathbb {R}}^{k}\) as follows
\(A^{\wedge l}\) can be represented by a \(\left( {\begin{array}{c}k\\ l\end{array}}\right) \times \left( {\begin{array}{c}k\\ l\end{array}}\right) \) whose entries are the \(l \times l\) minors of A. It can be also shown that
Fiber Bunched Cocycles
We recall that \(T:\Sigma \rightarrow \Sigma \) is a topologically mixing subshift of finite type. We say that \({\mathcal {A}}:\Sigma \rightarrow GL(k, {\mathbb {R}})\) is a rHölder continuous function, if there exists \(C>0\) such that
We fix a \(\omega \in (0,1)\) and for \(r> 0\) we let \(H^{r}(\Sigma , GL(k, {\mathbb {R}}))\) be the set of rHölder continuous functions over the shift with respect to the metric \(d_{\omega }\) on \(\Sigma \).
We denote by \(h_{r}({\mathcal {A}})\) the smallest constant C in (2.2). We equip the \(H^{r}(\Sigma , GL(k, {\mathbb {R}}))\) with the distance
It is clear the locally constant functions are \(\infty \)Hölder i.e, they are rHölder for every \(r>0\), with bounded \(h_{r}({\mathcal {A}}).\)
Definition 2.2
A local stable holonomy for the linear cocycle \((T, {\mathcal {A}})\) is a family of matrices \(H_{y \leftarrow x}^{s} \in GL(k, {\mathbb {R}})\) defined for all \(x\in \Sigma \) with \(y\in W_{{{\,\mathrm{loc}\,}}}^{s}(x)\) such that

(a)
\(H_{x \leftarrow x}^{s}=Id\) and \(H_{z \leftarrow y}^{s} o H_{y \leftarrow x}^{s}=H_{z \leftarrow x}^{s}\) for any \(z,y \in W_{{{\,\mathrm{loc}\,}}}^{s}(x)\).

(b)
\({\mathcal {A}}(y)\circ H_{y \leftarrow x}^{s}=H_{T(y) \leftarrow T(x)}^{s}\circ {\mathcal {A}}(x).\)

(c)
\((x, y, v)\mapsto H_{y \leftarrow x}^{s}(v)\) is continuous.
Moreover, if \(y\in W_{{{\,\mathrm{loc}\,}}}^{u}(x)\), then similarly one defines \(H_{y \leftarrow x}^{u}\) with analogous properties.
According to (b) in the above definition, one can extend the definition to the global stable holonomy \(H_{y\leftarrow x}^{s}\) for \(y\in W^{s}(x)\) not necessarily in \(W_{{{\,\mathrm{loc}\,}}}^{s}(x)\) :
where \(n\in {\mathbb {N}}\) is large enough such that \(T^{n}(y)\in W_{{{\,\mathrm{loc}\,}}}^{s}(T^{n}(x))\). One can extend the definition the global unstable holonomy similarly.
Definition 2.3
A \(r\)Hölder continuous function \({\mathcal {A}}\) is called fiber bunched if for any \(x\in \Sigma \),
We say that the linear cocycle \((T, {\mathcal {A}})\) is fiberbunched if its generator \({\mathcal {A}}\) is fiberbunched. Recall that \(H_{b}^{r}(\Sigma , GL(k, {\mathbb {R}}))\) is the family of rHöldercontinuous and fiber bunched functions.
The geometric interpretation of the fiber bunching condition is as follows. Let \({\mathcal {A}}\in H_{b}^{r}(\Sigma , GL(k, {\mathbb {R}}))\). The projective cocycle associated to \({\mathcal {A}}\) and T is the map \({\mathbb {P}}F:\Sigma \times {\mathbb {P}}{\mathbb {R}}^{k} \rightarrow \Sigma \times {\mathbb {P}}{\mathbb {R}}^{k}\) given by
We denote by \(D{\mathcal {A}}_{v}\) the derivative of the action \({\mathbb {P}}{\mathbb {R}}^{k} \rightarrow {\mathbb {P}}{\mathbb {R}}^{k}\) on projective space at all points \(v\in {\mathbb {P}}{\mathbb {R}}^{k}\). Taking derivative
for all \(v\in {\mathbb {P}}{\mathbb {R}}^{k}.\) Therefore, the fiber bunching condition implies that rate of expansion (respectively, contraction) the projective cocycle \({\mathbb {P}}F\) at every point \(x\in \Sigma \) is bounded above by \((\frac{1}{\omega })^{r}\) (respectively, below by \(\omega ^{r}\)).
The Hölder continuity and the fiber bunched assumption on \({\mathcal {A}}\in H_{b}^{r}(\Sigma , GL(k, {\mathbb {R}}))\) implies the convergence of the canonical holonomy \(H^{s \diagup u}\) (see [13, 36]). That means, for any \(y\in W_{{{\,\mathrm{loc}\,}}}^{s \diagup u}(x)\),
In addition, when the linear cocycle is fiber bunched, the canonical holonomies vary \(r\)Hölder continuously (see [36]), i.e., there exists \(C>0\) such that for \(y\in W_{{{\,\mathrm{loc}\,}}}^{s \diagup u}(x)\),
In this paper, we will always work with the canonical holonomies for fiber bunched cocycles.
Remark 1
Even though the locally constant cocycles are not necessary fiber bunched, the canonical holonomies always exist. Indeed, for every \(y\in W^{s}(x)\) there exist m such that \(x_n=y_n\) for all \(n\ge m\). Then,
In particular \(H_{x \leftarrow y}^{s}= Id,\) for all \(y\in W_{{{\,\mathrm{loc}\,}}}^{s}(x)\). Similarly, we get the existence of the unstable holonomy.
Remark 2
If a linear cocycle is not fiber bunched, then it might admit multiple holonomies (see [37]).
Typical Cocycles
We are going to discuss typical cocycles. For details, one is referred to [2, 20] and [48].
Suppose that \(p\in \Sigma \) is a periodic point of T, we say \(p\ne z\in \Sigma \) is a homoclinic point associated to p if it is the intersection of the stable and unstable manifold of p. That is, \(z\in W^{s}(p) \cap W^{u}(p)\) (see Fig. 1). The set of homoclinic points of any periodic point is dense for hyperbolic systems such as \((\Sigma , T)\).
We define the holonomy loop
Definition 2.4
Suppose that \({\mathcal {A}}:\Sigma \rightarrow GL(k, {\mathbb {R}})\) belongs \(H_{b}^{r}(\Sigma , GL(k, {\mathbb {R}}))\). We say that \({\mathcal {A}}\) is 1typical if there exist a periodic point p and a homoclinic point z associated to p such that:

(i)
The eigenvalues of \({\mathcal {A}}^{per(p)}(p)\) have multiplicity 1 and distinct absolute values. Let \(\{v_{i}\}_{i=1}^{k}\) be the eigenvectors of \(P:={\mathcal {A}}^{per(p)}(p).\)

(ii)
For any \(1\le i, j\le k\), \(\psi _{p}^{z}(v_{i})\) does not lie in any hyperplane \(W_{j}\), where \( W_{j}\) spanned by all eigenvectors of P other than \(v_{j}\).
For \(k=2\) this second condition means that \(\psi _{p}^{z}(v_{i})\ne v_{j}\) for \(1 \le i, j \le 2\). See Fig. 2.
We refer to (i) as the (pinching) properties and to (ii) as the (twisting) properties.
The cocycles generated by \({\mathcal {A}}^{\wedge t}\), \(1\le t\le k\) also admit stable and unstable holonomies, namely \((H^{s \diagup u})^{\wedge t}.\)
Definition 2.5
Assume that \({\mathcal {A}}\) is 1typical. We say \({\mathcal {A}}\) is ttypical for \(2\le t \le k1\), if the points \(p, z \in \Sigma \) from Definition 2.4 satisfy

(I)
All the products of t distinct eigenvalues of P are distinct;

(II)
The induced map \((\psi _{p}^{z})^{\wedge t}\) on \(({\mathbb {R}}^{k})^{\wedge t}\) satisfies the analogous statement to (ii) from Definition 2.4 with respect to \(\left\{ v_{i_{1}} \wedge \ldots \wedge v_{i_{t}}\right\} _{1 \le i_{1}<\ldots <i_{t} \le k} \text { of } P^{\wedge t}\).
We say that \({\mathcal {A}}\) is typical if \({\mathcal {A}}\) is ttypical for all \(1 \le t \le k1\). We denote by \({\mathcal {W}} \subset H_{b}^{r}(\Sigma , GL(k, {\mathbb {R}}))\) the set of all typical functions.
Remark 3
Above definition for typical cocycles comes from [44] that is a slightly weaker form of typical cocycles which were first introduced by Bonatti and Viana [20]; Park [44] considered a weaker twisting assumption. We also remark that the difference between the settings of [20] and [44] does not cause any problems in translating the relevant results and statements from [20] to this paper. In spite of slight variations in the definition of typicality, such assumptions are satisfied by an open and dense subset \({\mathcal {W}}\) of maps in \(H_{b}^{r}(\Sigma , GL(k, {\mathbb {R}}))\), and its complement has infinite codimension (see [2, 20]).
The Continuity of Lyapunov Exponents
Throughout, \({\mathbb {P}}F: \Sigma ^{+} \times {\mathbb {P}} {\mathbb {R}}^{k} \rightarrow \Sigma ^{+}\times {\mathbb {P}}{\mathbb {R}}^{k}\) is the projective cocycle associated with linear cocycle \(F:\Sigma ^{+} \times {\mathbb {R}}^{k} \rightarrow \Sigma ^{+} \times {\mathbb {R}}^{k}\) that is generated by \((T, {\mathcal {A}})\).
Let \((T, \mu \)) be a Bernoulli shift. We say that a matrix cocycle is strongly irreducible when there is no finite family of proper subspaces invariant by \({\mathcal {A}}(x)\) for \(\mu \)almost every x. Furstenberg [48, Theorem 6.8] showed that the Lyapunov exponent \(\chi (\mu , {\mathcal {A}})\) of F coincides with the integral of the function \(\psi : \Sigma ^{+} \times {\mathbb {P}}{\mathbb {R}}^{k} \rightarrow {\mathbb {R}},\)
for locally constant cocycles under the strong irreducibility assumption. In other words, he showed that
for any stationary measure \(\eta \) of the associated projective cocycle \({\mathbb {P}}F\). So, one can easily show that we have the continuity of Lyapunov exponents with respect to \(({\mathcal {A}}, \mu )\) ( [48, Corollary 6.10]) under the strong irreducibility assumption.
Even though discontinuity of Lyapunov exponents is a common feature (see [7, 8]), there are some results for the continuity of Lyapunov exponents. For instance, Bocker and Viana [19] proved the continuity of Lyapunov exponents of 2dimensional locally constant cocycles under a certain assumption. Avila et al. [1] announced recently that Bocker and Viana’s result remains true in arbitrary dimensions. It was conjectured by Viana [48] that Lyapunov exponents are always continuous among \(H_{b}^{\alpha }(X ,GL(2, {\mathbb {R}}))\)cocycles, and that has been proved by Backes et al. [4].
We state the main result of Backes, Brown, and Butler as follows.
Theorem 2.6
([4, Theorem 1.1]) Lyapunov exponents vary continuously restricted to the subset of fiberbunched elements \({\mathcal {A}}:X \rightarrow GL(2,{\mathbb {R}}).\)
That improves Bocker and Viana’s result [19]. Furthermore, Butler [21] showed in the following example that the fiberbunching condition is sharp.
Example 2.7
Assume that \(T:\{0,1\}^{{\mathbb {Z}}}\rightarrow \{0,1\}^{{\mathbb {Z}}}\) is a shift map. We define a locally constant cocycle \((T, {\mathcal {A}})\) such that
where \(\sigma \) is a positive constant greater than 1. We define probability measure \(\nu _{p}\) in order to \(\nu _{p}([0])=p, \nu _{p}([1])=1p,\) and then Bernoulli measure \(\mu _{p}=\nu _{p}^{{\mathbb {Z}}}\). By definition the cocycle \((T, {\mathcal {A}})\) is fiber bunched if and only if \(\sigma ^{2}<2^{\alpha }\).^{Footnote 2}
Butler [21] showed that for above example if \(\sigma ^{4p2}\ge 2^{\alpha }\) for \(p\in (\frac{1}{2},1)\), then for each neighborhood \({\mathcal {U}}\subset H^{\alpha }(\{0, 1\}^{{\mathbb {Z}}}, SL(2, {\mathbb {R}}))\) of \({\mathcal {A}}\) and every \(\kappa \in (0, (2p1)\log \sigma ],\) there is a locally constant cocycle \({\mathcal {B}}\in {\mathcal {U}}\) such that \(\chi (x, {\mathcal {B}})=\kappa \). In particular, \({\mathcal {A}}\) is a discontinuity point for Lyapunov exponents in \(H^{\alpha }(\{0, 1\}^{{\mathbb {Z}}}, SL(2, {\mathbb {R}})).\) So, this example shows that we have discontinuity of Lyapunov exponents near fiber bunched cocycles.
By considering the exterior product cocycles \({\mathcal {A}}^{\wedge t}\), we define
if the limit exists, and set
if \(\chi _{t}({\mathcal {A}}(x))\) exists for each \(1 \le t \le k\). We define the pointwise Lyapunov spectrum of \({\mathcal {A}}\) as \(\vec {L}:=\left\{ \vec {\alpha } \in {\mathbb {R}}^{k}: \vec {\alpha }=\vec {\chi }(x)\right. \) for some \(\left. x \in \Sigma \right\} \). For simplicity, we denote \(\chi _{t}({\mathcal {A}}):=\chi _{t}({\mathcal {A}}(x))\) for all \(1\le t \le k.\)
Theorem 2.8
([44, Theorem D]) Let \({\mathcal {A}}\in {\mathcal {W}}\). Then \(\vec {L}\) is a convex and closed subset of \({\mathbb {R}}^{k}\).
We use Theorem 2.6 to show that the Lyapunov spectrum of fiber bunched cocycles is a closed and convex set.
Corollary 2.9
Let \({\mathcal {A}}\in H_{b}^{r}(X, GL(2, {\mathbb {R}})))\). Then \(\vec {L}\) is a convex and closed subset of \({\mathbb {R}}^{2}\).
Proof
Since \({\mathcal {W}}\) is open and dense in \(H_{b}^{r}(X, GL(d, {\mathbb {R}})))\) (see [20]), for every \({\mathcal {A}} \in H_{b}^{r}(X, GL(2, {\mathbb {R}})))\) there is a \({\mathcal {A}}_{k} \in {\mathcal {W}}\) such that \({\mathcal {A}}_{k}\rightarrow {\mathcal {A}}\).
By Theorem 2.6,
By Theorem 2.8, \(\vec {L}\) is a closed and convex subset of \({\mathbb {R}}^{2}.\) \(\square \)
Thermodynamic Formalism
Convex Functions
Let U be an open convex subset of \({\mathbb {R}}^{n}\) and f be a real continuous convex function on U. We say a vector \(a\in {\mathbb {R}}^{n}\) is a subgradient of f at x if for all \(z \in U\),
where the last term on the righthand side is the scalar product.
For each \(x\in {\mathbb {R}}^{n}\) set the subdifferential of f at point x
For \(x\in U\), the subdifferential \(\partial f(x)\) is always a nonempty convex compact set. Define \(\partial ^{e} f(x) := {{\,\mathrm{ext}\,}}\{\partial f(x)\}\), i.e., the set of exterior points \(\partial f(x)\). We say that f is differentiable at x if the set \(\partial ^{e} f(x)\) is singleton.
We define
Legendre Transform
Assume that \(f:{\mathbb {R}}^{k} \rightarrow {\mathbb {R}}\cup \{+\infty \}\) is a convex function that is not identically equal to \(\infty \). The Legendre transform of f is the function \(f^{*}\) of a new variable t, defined by
where tx denotes the dot product of t and x.
It is easy to show that \(f^{*}\) is a convex function and not identically equal to \(\infty \). Let \(f^{**}\) be the Legendre transform of \(f^{*}\). Assume that \(f:{\mathbb {R}}^{k}\rightarrow {\mathbb {R}}\cup \{\infty \}\) is convex and not identically equal to \(\infty \). Let \(x\in {\mathbb {R}}^{k}\). Suppose that f is lower semi continuous at x, i.e., \(\liminf _{y\rightarrow x} f(y)\ge f(x)\). Then \(f^{**}(x)=f(x).\) Feng and Huang [29, Corollary 2.5] proved the following theorem:
Theorem 3.1
Assume that S is a nonempty, convex set in \({\mathbb {R}}^{k}\) and let \(g:S\rightarrow {\mathbb {R}}\) be a concave function. Set
and
Then \(G(a)=g(a)\) for \(a \in \text {ri}(S)\), where \(\text {ri}(S)\) denotes the relative interior of S (cf. [46]).
Topological Entropy
We define the topological entropy of noncompact sets [17]. Assume that (X, d) is a compact metric space and \(T:X\rightarrow X\) is a continuous transformation. For any \(n\in {\mathbb {N}}\), we define a new metric \(d_{n}\) on X as follows
and for any \(\varepsilon >0\), one can define Bowen ball \(B_{n}(x, \varepsilon )\) that is an open ball of radius \(\varepsilon >0\) in the metric \(d_{n}\) around x. That is,
Let \(Y \subset X\) and \(\varepsilon >0\). We say that a countable collection of balls \({\mathcal {Y}}:=\left\{ B_{n_{i}}\left( y_{i}, \varepsilon \right) \right\} _{i}\) covers Y if \(Y \subset \bigcup _{i} B_{n_{i}}\left( y_{i}, \varepsilon \right) \). For \({\mathcal {Y}}=\left\{ B_{n_{i}}\left( y_{i}, \varepsilon \right) \right\} _{i}\), put \(n({\mathcal {Y}})=\min _{i} n_{i}\). Let \(s\ge 0\) and define
where the infimum is taken over all collections \({\mathcal {Y}}=\{B_{n_{i}}(x_{i}, \varepsilon )\}_{i}\) covering Y such that \(n({\mathcal {Y}})\ge N.\) The quantity \(S(Y, s, N, \varepsilon )\) does not decrease with N, consequently
There is a critical value of the parameter s, which we denote by \(h_{top}(T,Y, \varepsilon )\) such that
Since \(h_{top}(T,Y, \varepsilon )\) does not decrease with \(\varepsilon \), the following limit exists,
We call \(h_{top}(T, Y)\) the topological entropy of T restricted to Y or the topological entropy of Y (we denote \(h_{top}(Y)\)), as there is no confusion about T. We denote \(h_{top}(X, T)=h_{top}(T).\)
Additive Thermodynamic Formalism
A potential on X is a continuous function \(f: X \rightarrow {\mathbb {R}}\). For any \(\varepsilon >0\) a set \(E \subset X\) is said to be a \((n,\varepsilon )\)separated subset of X if \(d_{n}(x,y)> \varepsilon \) (see (3.2)) for any two different points \(x,y \in E\). Using \((n, \varepsilon )\)separated subsets, we can define the pressure \(\mathrm {P}(f)\) of f as follows:
When \(f \equiv 0\), the pressure \(\mathrm {P}(0)\) is equal to the topological entropy \(h_{top}(T)\), which measures the complexity of the system (X, T).
The pressure satisfies the variational principle:
where \(h_{\mu }(f)\) is the measuretheoretic entropy (see [45]). Any invariant measure \(\mu \in {\mathcal {M}}(X,T)\) achieving the supremum in the variational principle is called an equilbrium state of f. If the entropy map \(\mu \mapsto h_{\mu }(T)\) is upper semicontinuous, then every potential has an equilibrium state.
Subadditive Thermodynamic Formalism
Let \(\Phi =\{\log \phi _{n}\}_{n=1}^{\infty }\) be a subadditive potential over the TDS (X, T). We introduce the \(\textit{topological pressure}\) of \(\Phi \) as follows. The space X is endowed with a metric d. We define for a subadditive \(\Phi \)
Since \(P_{n}(T, \Phi , \varepsilon )\) is a decreasing function of \(\varepsilon \), We define
and
We call \(P_{\Phi }(T)\) the topological pressure of \(\Phi \). We denote by \(P_{\Phi }(q)\) the topological pressure for a subadditive potential \(q\Phi .\)
Bowen [18] showed that for any Hölder continuous \(\psi :X \rightarrow {\mathbb {R}}\) on a transitive hyperbolic set (X, T) there exists a unique equilibrium measure \(\mu \) (which is also a Gibbs state) for the additive potential \( \psi \).
Feng and Käenmäki [30] extended the Bowen’s result for subadditive potentials \(t\Phi \) on a locally constant cocycle under the assumption that the matrices in \({\mathcal {A}}\) do not preserve a common proper subspace of \({\mathbb {R}}^{d}\)(i.e. \((T, {\mathcal {A}})\) is irreducible).
Let (X, T) be a topological dynamical system, and let \(\Phi =\{\log \phi _{n}\}_{n=1}^{\infty }\) be a subadditive potential over the TDS (X, T). Define
Let \( \vec {q} = (q_{1}, ..., q_{k})\in {\mathbb {R}}_{+}^{k}\), and \(\vec {\Phi }=(\Phi _{1},...,\Phi _{k})=(\{\log \phi _{n,1}\}_{n=1}^{\infty },...,\{\log \phi _{n,k}\}_{n=1}^{\infty })\). Assume that \(\vec {q}.\vec {\Phi }=\sum _{i=1}^{k} q_{i}\Phi _{i}\) is a subadditive potential \(\{q_{i}\log \phi _{n, i}\}_{n=1}^{\infty }.\) We can write topological pressure, maximal Lyapunov exponent, and minimal Lyapunov exponent of \(\vec {\Phi }\), respectively
For \(\mu \in {\mathcal {M}}(X, T)\), we write
where \(\chi (\mu , \Phi _{i})=\lim _{n\rightarrow \infty }\frac{1}{n}\int \log \phi _{n, i}(x)d\mu (x)\) for \(i=1,...,k.\)
Cao et al. [22] proved the variational principle formula for subadditive potentials.
Theorem 3.2
([22, Theorem 1.1]) Let (X, T) be a topological dynamical system such that \(h_{top}(T)<\infty \). For \(\vec {t}\in {\mathbb {R}}_{+}^{d}\), suppose that \(\vec {\Phi }\) is a subadditive potential on the compact metric space X. Then
Let \(\vec {t}\in {\mathbb {R}}_{+}^{d}\), we denote by \({{\,\mathrm{Eq}\,}}(\vec {\Phi }, \vec {t})\) the collection of invariant measures \(\mu \) such that
If \({{\,\mathrm{Eq}\,}}(\vec {\Phi }, \vec {t})\ne \emptyset \), then each element \({{\,\mathrm{Eq}\,}}(\vec {\Phi }, \vec {t})\) is called an \(\textit{equilibrium state}\) for \(\vec {t}.\vec {\Phi }\).
In the remaining part of this section, we recall some theorems about multifractal formalism for subadditive potentials.
Theorem 3.3
([29, Theorem 1.1]) Let (X, T) be a topological dynamical system such that the topological entropy \(h_{top}(T)\) is finite. Then \(E(\beta (\Phi ))\ne \emptyset \) . Moreover,
The topological pressure is related to Lyapunov exponents in the following way.
Proposition 3.4
([29, Theorem 3.3]) Let (X, T) be a topological dynamical system such that the entropy map \(\mu \mapsto h_{\mu }(T) \) is upper semicontinuous and \(h_{top}(T)<\infty \). For \(t\in {\mathbb {R}}_{+}^{k}\), suppose that \(\vec {t}.\vec {\Phi }\) is a subadditive potential on the compact metric space X. Then,
Moreover, \({{\,\mathrm{Eq}\,}}(\vec {\Phi }, \vec {t})\) is a nonempty compact convex subset of \({\mathcal {M}}(X,T)\), for any \(t\in {\mathbb {R}}_{+}^{k}\).
Theorem 3.5
([29, Theorem 4.8]) Keep the assumption of Theorem (3.3), we also assume that the entropy map \(\mu \mapsto h_{\mu }(T)\) is upper semicontinuous on \({\mathcal {M}}(X,T)\). If \(t \in {\mathbb {R}}_{+}^{k}\) such that \(\vec {t}. \vec {\Phi }\) has a unique equilibrium state \(\mu _{\vec {t}}\in {\mathcal {M}}(X,T)\), then \(\mu _{\vec {t}}\) is ergodic, \(\nabla P_{\vec {\Phi }}(\vec {t}) = \chi (\mu _{\vec {t}}, \vec {\Phi })\), \(E(\nabla P_{\vec {\Phi }}(\vec {t}))\ne \emptyset \) and \(h_{top}(T,E(\nabla P_{\vec {\Phi }}(\vec {t}))) =h_{\mu _{\vec {t}}}(T).\)
Thermodynamic Formalism of Linear Cocycles
Our motivation for studying of subadditive (or almost additive) thermodynamic formalism is to deal with linear cocycles. In this subsection, we recall some definitions and results that were just proved for linear cocycles.
We study ergodic optimization of Lyapunov exponents. Ergodic optimization of Lyapunov exponents is concerned invariant measures in maximizing (or minimizing) the Lyapunov exponents. They were first considered by Rota and Strang [47] and by Gurvits [32], respectively. The associated growth rates are called upper joint spectral radius and lower joint spectral radius, respectively; they play an important role in Control Theory (see [9, 34]).
Let (X, T) be a topological dynamical system and let \({\mathcal {A}}:X\rightarrow GL(k,{\mathbb {R}})\) be a matrix cocycle over the topological dynamical system (X, T).
We define the maximal Lyapunov exponent of linear cocycles as follows
Morris [42] also showed
Feng and Huang [29] gave a different proof of it.
Let us define the set of maximizing measures of \({\mathcal {A}}\) to be the set of measures on X given by
Recently, author [41] showed that the entropy of Lyapunovmaximizing measures of \(SL(2, {\mathbb {R}})\) typical onestep cocycles is zero under certain assumptions.
We also define the minimal Lyapunov exponents of linear cocycles as follows
Similarly, the set of minimizing measures is defined as follows
We remark that supremum (3.4) is attained, so \({\mathcal {M}}_{\max }\) is nonempty set. But, \({\mathcal {M}}_{\min }\) is not necessarily nonempty; see Sect. 5.
Let \(\log ^{+} \Vert {\mathcal {A}}\Vert \in L^{1}(\mu )\). We define sum of top l Lyapunov exponents of measure as follows
for \(\mu \in {\mathcal {M}}(X,T)\).
We are mainly concerned with the distribution of the Lyapunov exponents of \({\mathcal {A}}\). More precisely, for any \(\alpha \in {\mathbb {R}}\), define
which is called the \(\alpha \)level set of \(\chi _{1}({\mathcal {A}}).\)
We also define the higher dimensional of level set of all of Lyapunov exponents as follows
for \(1\le l \le k.\)
We denote \(E(\alpha )=E_{{\mathcal {A}}}(\alpha )\), when there is no confusion about \({\mathcal {A}}.\)
We denote
We say that \(\vec {\Phi }_{{\mathcal {A}}}\) is (simultaneously) quasimultiplicative if there exist \(C>0\) and \(m\in {\mathbb {N}}\) such that for every \(I, J \in {\mathcal {L}}\), there exists \(K\in {\mathcal {L}}\) with \(K\le m\) such that \(IKJ \in {\mathcal {L}}\) and
for \(1\le i \le k.\)
Park [44] proved quasimultiplicativity for typical cocycles \({\mathcal {W}}\). The approach has its roots in previous works of Feng [27, Proposition 2.8] and [28] who showed quasimultiplicativity for locally constant cocycles under the irreducibility assumption.
Theorem 3.6
([44, Theorem F]) Assume that \({\mathcal {A}}\in {\mathcal {W}}\). Then \({\mathcal {A}}\) is quasimultiplicative. Moreover, \(\vec {\Phi }_{{\mathcal {A}}}\) is (simultaneously) quasimultiplicative.
He [44] also uses the quasimultiplicative property \({\mathcal {A}}\in {\mathcal {W}}\) to show the continuity of the topological pressure which it states in the following theorem.
Theorem 3.7
([44, Theorem B]) The map \((s, {\mathcal {A}})\rightarrow P_{{\tilde{\Phi }}_{{\mathcal {A}}}}(s)\) is continuous on \([0, \infty ) \times {\mathcal {W}}\).
Theorem 3.8 shows that we have the Feng and Käenmäki’s result [30, Proposition 1.2] for typical cocycles.
Theorem 3.8
Let \({\mathcal {A}}\in {\mathcal {W}}\) be typical. Assume that \(\vec {\Phi }_{{\mathcal {A}}}\) is (simultaneously) quasimultiplicative and \(\vec {t}\in {\mathbb {R}}_{+}^{k}\). Then \(P_{\vec {\Phi }_{{\mathcal {A}}}}(\vec {t})\) has a unique equilibrium state \(\mu _{\vec {t}}\) for the subadditive potential \(\vec {t}.\vec {\Phi }_{{\mathcal {A}}} \). Furthermore, \(\mu _{\vec {t}}\) has the following Gibbs property: There exists \(C \ge 1\) such that for any \(n\in {\mathbb {N}}\), \([J]\in {\mathcal {L}}(n)\), we have
for any \(x\in [J]\). Furthermore, \(P_{\vec {\Phi }_{{\mathcal {A}}}} ( . )\) is differentiable on \({\mathbb {R}}_{+}^{k}\) and \(\nabla P_{\vec {\Phi }_{{\mathcal {A}}}} ( \vec {t} )=\chi (\mu _{\vec {t}}, \vec {\Phi }_{{\mathcal {A}}}).\)
Proof
It is easily follows from [44, Lemma 3.10] and [44, Proposition 3.9]. \(\square \)
Theorem 3.9
Assume that \(h_{top}(T)<\infty \), and \(\alpha ({\mathcal {A}})<\infty \). If \({\mathcal {A}} \in {\mathcal {W}}\), then \(P_{\Phi _{{\mathcal {A}}}}( )\) is a real continuous convex function on \({\mathbb {R}}\). Moreover, \(\alpha ({\mathcal {A}})\) exists^{Footnote 3} and it is equal to \(P_{\Phi _{{\mathcal {A}}}}^{'}(\infty ):=\lim _{t\rightarrow \infty } \frac{P_{\Phi _{{\mathcal {A}}}}(t)}{t}.\) Similarly, \(P_{\vec {\Phi }_{{\mathcal {A}}}}\) is a real continuous convex function on \({\mathbb {R}}^{k}.\) Furthermore,
Proof
See [27, Lemmas 2.2 and 2.3]. We remark that although [27, Lemmas 2.2 and 2.3] only deal with locally constant cocycles, the proof given there works for our theorem under slightly modification. Indeed, Feng uses the quasimultiplicative properties to prove the lemmas. Since \({\mathcal {A}}\in {\mathcal {W}}\), \(\vec {\Phi }_{{\mathcal {A}}}\) is (simultaneously) quasimultiplicative by Theorem 3.6. \(\square \)
Note that every almost additive potential is quasimultiplicative, but the reverse is not true; see Example (5.4). Therefore, one also can use the above theorem for almost additive potentials.
The Proofs of Theorems A, B and C
In this section, we will first prove a more general version of Theorem B, and then Theorem B helps us deduce Theorem A. Finally, we will prove Theorem C.
We discuss multifractal formalism of typical cocycles. Our motivation is to study the multifractal formalism associated to certain iterated function systems with overlaps. For instance, the Hausdorff dimension of level sets has been calculated for 2dimensionplanar affine iterated function systems satisfying strong irreducibility and the strong open set condition by Bárány et al. [14]. In additive potential setting, the Lyapunov exponents are equal the Birkhoff averages. In this case, the restricted varitional principle, topological entropy, and Hausdorff dimension level set has been studied by a lot of authors (see [24]). We remark that Feng and Huang [29] proved (1.2) for almost additive potentials under certain assumptions. In fact, the proof of Theorem 4.2 is based on some ideas from that work.
Theorem 4.1
Let \({\mathcal {A}} \in {\mathcal {W}}\). Suppose that \(P_{\vec {\Phi }_{{\mathcal {A}}}}(\vec {q})\in {\mathbb {R}}\) for each \(\vec {q}\in {\mathbb {R}}^{k}\). Then for \(\vec {\alpha } \in \vec {L}\),
Proof
By Theorem 3.6, \(\vec {\Phi _{{\mathcal {A}}}}\) is simultaneously quasimultiplicative. Then, the proof follows from [29, Theorem 4.10]. \(\square \)
Theorem 4.2
Assume that \(T:\Sigma \rightarrow \Sigma \) is a topologically mixing subshift of finite type on the compact metric space \(\Sigma \). Suppose that \({\mathcal {A}}:X\rightarrow GL(k, {\mathbb {R}})\) belongs to typical functions \({\mathcal {W}}\). Assume that \(\vec {\Omega }\) is the range of the map from \({\mathcal {M}}(\Sigma ,T)\) to \({\mathbb {R}}^{k}\)
We define
where \(\vec {\alpha }\in \Omega .\) Then,
for \(\vec {\alpha } \in \text {ri}(\vec {\Omega }).\)
Proof
It is clear that \(\vec {\Omega }\) is nonempty and convex. We define \(g:\vec {\Omega } \rightarrow {\mathbb {R}}\) by
It is easy to see that g is a realvalued concave function on \(\vec {\Omega }.\) We define
Let \(f:{\mathbb {R}}^{k} \rightarrow {\mathbb {R}}\cup \{+\infty \}\) be the function which agrees with \(g\) on \(\vec {\Omega }\) but is \(+\infty \) everywhere else. Then, f is convex and has \(\vec {\Omega }\) as its effective domain, i.e. \(\vec {\Omega }=\{\vec {x}, f(\vec {x})< \infty \}.\) By the definition of Z, Z is equal to the conjugate function of f, so Z is a convex function on \({\mathbb {R}}^{k}\) (see Subsect. 3.2).
We have
for all \(\vec {\alpha }\in \text {ri}(\vec {\Omega })\), by Theorem 3.1.
By Theorem 3.6, \(\vec {\Phi _{{\mathcal {A}}}}\) is simultaneously quasimultiplicative. Therefore, \(P_{\vec {\Phi }_{{\mathcal {A}}}}(\vec {q})\) is a convex function on \({\mathbb {R}}^{k}\) by Theorem 3.9. Then, by variational principle
\(\square \)
Proof of Theorem B
Theorem B is just the onedimensional version of Theorems 4.1 and 4.2. Also, note that \(\mathring{\Omega }=\text {ri}(\Omega )\) (see Fig. 3). \(\square \)
Remark 4
In the locally constant cocycles case with Bernoulli measures, Theorem 4.2 is true for \(\vec {\alpha }\in \vec {\Omega }\) under strongly irreducible assumption, which means we do not need pinching assumption in this case.
Now, we are going to show that the closure of the set where the entropy spectrum is positive is equal the Lyapunov spectrum for typical cocycles. This result is first attempt to characterize Lyapunov spectrum as a set of positive entropy spectrum. The main input of our argument will be the fact that the topological pressure is convex for typical cocycles, and Theorems 4.1 and 4.2. Then, we can show the concavity of the entropy spectrum of Lyapunov exponents by Theorem B.
We recall that \(T:\Sigma \rightarrow \Sigma \) is a topologically mixing subshift of finite type and \({\mathcal {A}}:\Sigma \rightarrow GL(k, {\mathbb {R}})\) is a Hölder continuous function. We always assume \(h_\mathrm{top}(T)>0\).
Lemma 4.3
Let \({\mathcal {A}} \in {\mathcal {W}}\). Then, \(h_\mathrm{top}(E(\alpha ))\) is concave on the convex set \(\mathring{L}\).
Proof
The topological pressure \(P_{\Phi _{{\mathcal {A}}}}( .)\) is convex by Theorem 3.9 and \(\mathring{L}\) is convex by Theorem 2.8. Moreover, by Theorem 4.1, we have
Since the Legendre transform of the convex function is concave (cf. [33, Theorem 1.1.2]), \(h_\mathrm{top}(E(\alpha ))\) is concave. \(\square \)
Lemma 4.4
Assume that a nonnegative function f defined on a convex domain D is concave and achieves a positive value at some point \(x\in D\). Then \(f(y)>0\) for all \(y\in \mathring{D}\).
Proof
Let x be in D such that \(f(x)>0\). For any point y in the interior of D, we can always choose a point \(z\in \mathring{D}\) such that:
for some \(\lambda \in (0,1)\). Hence,
Therefore, \(f(y)>0\) for all \(y\in \mathring{D}\). \(\square \)
Theorem 4.5
For \(\alpha \in \mathring{L}\), \(h_\mathrm{top}(E(\alpha ))>0.\)
Proof
By Lemma 4.3, \(h_\mathrm{top}(E(\alpha ))\) is concave. Moreover, by Theorems 3.3, 4.1 and 4.2,
Since the measuretheoretic entropy is nonnegative, \(h_\mathrm{top}(E(\alpha ))\ge 0\).
We claim that there is \(\alpha \) such that \(h_\mathrm{top}(E(\alpha ))>0.\) Let us assume \(h_\mathrm{top}(E(\alpha ))=0\) for all \(\alpha \in \mathring{L}\). Then, as by Oseledets Theorem for every ergodic measure \(\mu \) supported on \((\Sigma , T)\) there exist a common value of Lyapunov exponent shared \(\mu \)almost everywhere, we must have \(h(\mu )=0\) for every ergodic measure \(\mu \). Thus, by variational principle, \(h_\mathrm{top}(T) = \sup _{\mu } h(\mu ) = 0\), which is a contradiction. Consequently, by Lemma 4.4, \(h_\mathrm{top}(E(\alpha ))>0\) for all \(\alpha \in \mathring{L}.\) \(\square \)
Remark 5
Entropy spectrum at boundary of Lyapunov spectrum is not necessarily positive. In fact, there is a conjecture, which is known as Meta conjecture, that says that under generic assumptions the entropy spectrum at boundary of Lyapunov spectrum is zero (which would mean that \(h_{top}(E(\beta ({\mathcal {A}})))=0\)); this phenomenon is often referred to as ergodic optimization of Lyapunov exponents, see for example [9, 10]. In the additive potential case, instead, this phenomenon is often referred to as ergodic optimization of Birkhoff averages, see for example [34].
Proof of Theorem A
That is a direct consequence of Theorem 4.5. \(\square \)
Park [44] proved Theorem 2.8 for higher dimensional case. That means, \(\vec {L}\) is closed and convex. So, we can obtain the following generalization of Theorem A to the Lyapunov spectrum of of all Lyapunov exponents.
Theorem 4.6
\(\overline{\{ \vec {\alpha } \in {\mathbb {R}}^{k}, \quad h_{top}(E(\vec {\alpha }))>0 \}}= \vec {L}.\)
We remark that the concavity of a function defined on a convex set implies the continuity of the function in the interior, and that the continuity of the entropy under the change of the Lyapunov exponents implies the continuity of the Lyapunov spectrum.
We will discuss the continuity of the entropy spectrum of Lyapunov exponents, that is, the topological entropy of level sets of points with a common given Lyapunov exponent. In the locally constant cocycles case, Lemma 4.7 follows from [31].
Lemma 4.7
Assume \({\mathcal {A}}_{k}, {\mathcal {A}}\in {\mathcal {W}}\) with \({\mathcal {A}}_{k} \rightarrow {\mathcal {A}}\). For \(t_{k}, t>0\), let \(t_{k}\rightarrow t\). Suppose that \(\alpha _{t_{k}}\) and \(\alpha _{t}\) are the derivatives of \(P_{\Phi _{{\mathcal {A}}_{k}}}( )\) and \(P_{\Phi _{{\mathcal {A}}}}( )\) at \(t_{k}\) and t, respectively. Then,
Proof
According to Theorem 3.8, \(P_{\Phi _{{\mathcal {A}}}}( )\) is differentiable for any \(t>0\) and there is a unique equilibrium measure \(\mu _{t}\) for the subadditive potential \(t\Phi _{{\mathcal {A}}}\) . Therefore, we have
where \(P_{\Phi _{{\mathcal {A}}}}^{'}(t)=\alpha _{t}\), by Theorem 3.5.
Taking into account the observation above, to prove the theorem it is enough to show that \(h_{\mu _{t_{k}}}(T) \rightarrow h_{\mu _{t}}(T)\).
By the definition of \({{\,\mathrm{Eq}\,}}(\Phi _{{\mathcal {A}}_{k}}, t_{k})\),
Notice that the Lyapunov exponents are upper semicontinuous. Moreover, the topologically mixing subshift of finite type \(T:\Sigma \rightarrow \Sigma \) implies upper semicontinuity of the entropy map \(\mu \mapsto h_{\mu }(T)\). Now, we conclude from above observations and Theorem 3.7,
This shows \(\mu _{t} \in {{\,\mathrm{Eq}\,}}(\Phi _{{\mathcal {A}}}, t)\). By weak* compactness \(\mu _{t_{k}}\) has a accumulation point, let us call \(\mu _{t}\). According the above observation, \(\mu _{t}\) is an equilibrium measure for \(t\Phi _{{\mathcal {A}}}\). Then uniqueness of equilibrium measure implies \(\mu _{t_{k}} \rightarrow \mu _{t}\). Moreover, we have equality in the above, which gives the claim. Furthermore, it shows the continuity of the Lyapunov exponent of equilibrium measures. \(\square \)
Assume that \((\mu _{t})\) is a sequence of equilibrium measures for a subadditive potential \(t\log \Phi \), where \(t>0\). The author [40] investigated the behavior of the equilibrium measure \((\mu _{t})\) as \(t\rightarrow \infty \). In the thermodynamic interpretation of the parameter t, it is the inverse temperature. The limits \(t\rightarrow \infty \) is called zero temperature limits, and the accumulation points of the measure \((\mu _{t})\) as \(t\rightarrow \infty \) are called ground states.
Zero temperature limit is also related to ergodic optimization, as for \(t\rightarrow \infty \) any accumulation points of equilibrium measure \(\mu _{t}\) is a maximizing measure \(\Phi .\)
Theorem 4.8
([40, Theorem 1.1]) Let (X, T) be a topological dynamical system for which the entropy map \(\mu \mapsto h_{\mu }(T)\) is upper semicontinuous and topological entropy \(h_{top}(T)<\infty .\) Assume that \(\Phi =\{\log \phi _{n}\}_{n=1}^{\infty }\) is a subadditive potential on the compact metric X. Then a family of equilibrium measures \((\mu _{t})\) for potentials \(t\Phi \), where \(t>0\), has a weak* accumulation point as \(t \rightarrow \infty .\) Any such accumulation point is a Lyapunov maximizing measure for \(\Phi \). Furthermore,

(i)
\(\chi (\Phi , \mu )=\lim _{i\rightarrow \infty } \chi (\Phi , \mu _{t_{i}}),\)

(ii)
\(h_{\mu }(T)=\lim _{i \rightarrow \infty } h_{\mu _{t_{i}}}(T).\)
Moreover, above result works for higher dimensional case.
We use the above theorem to prove the following theorem.
Theorem 4.9
Suppose that \(\mathcal {{\mathcal {A}}} \in {\mathcal {W}}\). If \(\alpha _{t}= P_{\Phi _{{\mathcal {A}}}}^{'}(t)\) for \(t>0.\) Then
Proof
Since \(\mathcal {{\mathcal {A}}}\in {\mathcal {W}}\), Theorem 3.8 implies that for \(t>0\), there is a unique equilibrium state \(\mu _{t}\) for the subadditive potential \(t \Phi _{{\mathcal {A}}}\) such that
By Theorem 3.5,
We know that
by Theorem 3.3. So, we only need to show that
That follows from Theorem 4.8. \(\square \)
Proof of Theorem C
It follows from Theorems 4.9 and 4.7. \(\square \)
Theorem 4.10
Suppose \({\mathcal {A}}_{l}, {\mathcal {A}}\in {\mathcal {W}}\) with \({\mathcal {A}}_{l} \rightarrow A\), and \(\vec {t_{l}},\vec {t}\in {\mathbb {R}}_{+}^{k}\) such that \(t_{l}\rightarrow t.\) Assume that \(\vec {\alpha _{t_{l}}}= \nabla P_{\vec {\Phi }_{{\mathcal {A}}_{l}}}(\vec {t_{l}})\) and \(\vec {\alpha _{t}}=\nabla P_{\vec {\Phi }_{{\mathcal {A}}}}(\vec {t})\). Then,
Moreover,
Proof
The proof is similar to Theorems 4.9 and 4.8 and is omitted. \(\square \)
The Proof of Theorem D
In this section, we are going to prove the continuity of the lower joint spectral radius for general cocycles under certain assumptions. This kind of result is known by Bochi and Morris [12] under 1domination assumption for locally constant cocycles. Breuillard and Sert [15] extended their result to the joint spectrum of locally constant cocycles. Moreover, they gave a counterexample [15, Example 4.13] that shows that we have discontinuity the lower joint spectral for typical cocycles. Even though, we have a lot of results for the upper spectral radius, we have few result about the lower spectral radius, which shows that working on the latter case is much harder than the former case.
Assume that \(T:X\rightarrow X\) is a diffeomorphism on a compact invariant set X. Let \(V\oplus W\) be a splitting of the tangent bundle over X that is invariant by the tangent map DT. In this case, if vectors in V are uniformly contracted by DT and vectors in W are uniformly expanded, then the splitting is called hyperbolic. The more general notion is the dominated splitting, if at each point all vectors in V are more contracted than all vectors in W. Domination could also be called uniform projective hyperbolicity. Indeed, domination is equivalent to V being hyperbolic repeller and W being hyperbolic attractor in the projective bundle.
In the linear cocycles, we are interested in bundles of the form \(X\times {\mathbb {R}}^{k}, \) where the linear cocycles are generated by \((T, {\mathcal {A}})\). Bochi and Gourmelon [11] showed that a cocycle admits a dominated splitting \(V\oplus W\) with \(\dim V=k\) if and only if when \(n\rightarrow \infty \), the ratio between the \(kth\) and \((k+ 1)th\) singular values of the matrices of the \(nth\) iterate increase uniformly exponentially. In fact, they extended the Yoccoz’s result [50] that was proved for 2dimensional vector bundles.
Definition 5.1
We say that \({\mathcal {A}}\) is idominated if there exist constants \(C >1\), \(0<\tau <1\) such that
According to the multilinear algebra properties, \({\mathcal {A}}\) is idominated if and only if \({\mathcal {A}}^{\wedge i}\) is 1dominated.
Let (X, T) be a TDS. We say that \({\mathcal {A}}:X\rightarrow GL(k, {\mathbb {R}})\) is almost multiplicative if there is a constant \(C>0\) such that
We note that since clearly \({\mathcal {A}}^{m+n}(x) \le {\mathcal {A}}^m(x) {\mathcal {A}}^n(T^m(x)) \quad \) for all \(x\in X, m,n\in {\mathbb {N}}\), the condition of almost multiplicativity of \({\mathcal {A}}\) is equivalent to the statement that \(\Phi _{{\mathcal {A}}}\) is almost additive.
Almost Additive Thermodynamic Formalism
In this subsection, we state a theorem that shows we have the Bowen’s result for almost additive sequences.
Let \(T:\Sigma \rightarrow \Sigma \) be a topologically mixing subshift of finite type. We say that a subadditive sequences \(\Phi :=\{\log \phi _{n}\}_{n=1}^{\infty }\) over \((\Sigma , T)\) has bounded distortion: there exists \(C\ge 1\) such that for any \(n\in {\mathbb {N}}\) and \(I \in {\mathcal {L}}(n)\), we have
for any \(x, y \in [I].\)
Theorem 5.2
([5, Theorem 10.1.9]) Let \(\Phi =\{\log \phi _{n}\}_{n=1}^{\infty }\) be an almost additive sequence over a topologically mixing subshift of finite type \((\Sigma , T)\). Assume that \(\Phi \) has bounded distortion. Then :

1.
There is a unique equilibrium measure for \(\Phi \),

2.
There is a unique invariant Gibbs measure for \(\Phi \),

3.
The two measures coincide and are ergodic.
Theorem 5.3
Let \(t\in {\mathbb {R}}^{k}\) and let \(\vec {\Phi }\) be an almost additive potential. Then, Proposition 3.4 holds for \(t\in {\mathbb {R}}^{k}\).
Proof
It follows from [29, Theorem 3.3]. \(\square \)
Given an almost additive potential \(\Phi =\{\log \phi _{n}\}_{n=1}^{\infty }\). Feng and Huang [29, Lemma A.4.] proved the following lemma:
Lemma 5.4
Let \(\mu \in {\mathcal {M}}(X, T)\). Then the map \(\mu \mapsto \chi (\mu , \Phi )\) is continuous on \({\mathcal {M}}(X, T)\).
A Cocycle is Almost Multiplicativite Under a Cone Condition
Let \({\mathcal {A}}:X\rightarrow GL(k, {\mathbb {R}})\) be a matrix cocycle over a TDS (X, T). We say that a family of convex cones \({\mathcal {C}}:=(C_x)_{x\in X}\) is a \(\ell \)dimensional invariant cone field on X if
where \(\ell \) is the dimension \({\mathcal {C}}.\)
Let V be a vector space over the reals.
Definition 5.5
Fix a convex cone \(C \subset V\). Given \(v, w \in C\), let
with \(\alpha = 0\) and/or \(\beta = \infty \) if the corresponding set is empty. The cone distance between v and w is
The distance \(d_{c}\) is called Hilbert projective (pseudo) metric.
Theorem 5.6
Let \(C_{1} \subset V_{1}\) and \(C_{2} \subset V_{2}\) be convex cones, and \(L:V_{1}\rightarrow V_{2}\) be a linear map such that \(L(C_{1}) \subset C_{2}^{o}\). Assume that \(\triangle :=\sup _{{\hat{v}}, {\hat{\psi }}\in L(C_{1})} d_{C_{2}}({\hat{v}}, {\hat{w}}).\) Then for all \(v,w\in C_{1}\), we have
where we use the convention that \(\tanh \infty = 1.\)
Proof
See [39]. \(\square \)
We are also going to use the following lemma in Proposition 5.8.
Lemma 5.7
Let V be a finite dimensional vector space. Suppose that \(C_{1}\) and \(C_{2}\) are two convex cones in V such that \(C_1\subset C_2^o\) and \(d_{C_{2}}\) is the Hilbert metric on \(C_2\). Then \(C_1\) is bounded in metric \(d_{C_2}\).
Proof
Let us denote d as the usual distance on the projective space. Since \(C_{1} \subset C_{2}^{o}\), \(d(C_{1}, \partial C_{2})>0\). Hence, for every \(v,w \in C_{1}\) the distances d(A, v), d(B, w) are uniformly bounded from below by \(c_1=d(C_1, \partial C_2)\), where A, B are the intersections of the line \({\overline{vw}}\) with \(\partial C_1\). On the other hand, d(v, w) is uniformly bounded from above by \(c_2={{\,\mathrm{diam}\,}}_d(C_1)\). Thus, \(d_{C_2}(v,w) \le \log ( (c_1+c_2)/c_1)\).
\(\square \)
Proposition 5.8
Let X be a compact metric space, and let \({\mathcal {A}}: X\rightarrow GL(k, {\mathbb {R}})\) be a matrix cocycle over a TDS (X, T). Let \((C_r)_{r\in X}\) be a 1dimensional invariant cone field on X. Then, there exists \(\kappa >0\) such that for every \(m,n>0\) and for every \(x\in X\) we have
Proof
Let us start from the notation. Denote by \(\pi \) the natural projection from \({\mathbb {R}}^{k}\) to the projective space \({\mathbb {P}}{\mathbb {R}}^{k}\) and by d the natural metric on \({\mathbb {P}}{\mathbb {R}}^{k}\). For a family of convex cones \((C_r)_{r\in X}\), all contained in the interior \(C^o\) of another convex cone C, we define their convex hull as
The Hausdorff distance in metric d between C and \({{\,\mathrm{conv}\,}}(C_r)\) equals the supremum of Hausdorff distances between C and \(C_r\) (to be absolutely precise, the Hausdorff distance is defined for compact sets and the metric d is defined on the projective space, so we mean here the Hausdorff distance between \(\overline{\pi (C)}\) and \(\overline{\pi ({{\,\mathrm{conv}\,}}(C_r))}\)).This supremum is positive, hence \({{\,\mathrm{conv}\,}}(C_r)\subset C^o\).
For every \(x\in X\) the set \(T^{1}(x)\) is compact. Thus, we can define
for \(x\in T(X)\) and, by compactness, we have \(D_x\subset C_x^o\). We choose \(D_x\) as any convex cone contained in \(C_x^o\) for \(x\in X\setminus T(X)\), we only demand that \(x\rightarrow D_x\) is a continuous map (this can be done because X is compact, hence \(X\setminus T(X)\) is open in X). One can check that, as \(D_x\subset C_x^o\), we have
Hence, \((D_x)\) is another invariant cone field, strictly contained in \((C_x)\).
Let for each \(x\in X\) \(d_x\) be the Hilbert metric in \(C_x\). Let d be the usual metric on \({\mathbb {P}}{\mathbb {R}}^{k}\). We have the following properties:

Each \(D_x\) is bounded in \(d_x\) by Lemma 5.7. By compactness of X, there exists \(K_1>0\) such that \({{\,\mathrm{diam}\,}}_{d_x}(D_x)<K_1\) for all \(x\in X\).

In each \(D_x\) the metric \(d_x\) is equivalent to d. By compactness of X, there exists \(K_2>1\) such that for every \(x\in X\) for every \(v,w\in D_x\) we have \(K_2^{1} d_x(v,w) \le d(v,w) \le K_2 d_x(v,w)\).

Each \({\mathcal {A}}(x): D_x \rightarrow D_{T(x)}\) is a contraction by Theorem 5.6. By compactness of X, there exists \(\lambda <1\) such that for every \(x\in X\) for every \(v,w\in D_x\) we have \(d_{T(x)}({\mathcal {A}}(x)v, {\mathcal {A}}(x)w) \le \lambda d_x(v,w)\).

For \(v\in C_x\) denote
$$\begin{aligned} \gamma _x(v):= \log (\Vert {\mathcal {A}}(x)v\Vert /\Vert v\Vert ) \in {\mathbb {R}}. \end{aligned}$$(5.3)The map \(v\rightarrow \gamma _x(v)\) is Lipschitz (in metric d) on \(D_x\). By compactness of X, there exists \(K_3>0\) such that for every \(x\in X\) the map \(\gamma _x\) is \(K_3\)Lipschitz (in metric d) on \(D_x\).

For every \(x\in X\) the convex cone \(D_x\) contains (for some \(v_x\in D_x\cap {\mathbb {P}}{\mathbb {R}}^{k}\) and \(r_x>0\)) a ball \(B(v_x,r_x)=\{w\in {\mathbb {P}}{\mathbb {R}}^{k}; d(v_x,w)<r_x\}\). By compactness of X, there exists \(r>0\) such that for every \(x\in X\) we have \(D_x \supset B(v_x,r)\) for some \(v_x\in D_x\cap {\mathbb {P}}{\mathbb {R}}^{k} \).
Choose some \(x\in X\) and \(v,w\in D_x\). Fix \(m>0\). Denote
Note three obvious properties of this function:

\(\gamma _x\) is a projective function, that is \(\gamma _x (v)=\gamma _x (\alpha v)\) for \(\alpha >0\). Thus, we can define \(\gamma _x\) on the projective space \({\mathbb {P}}{\mathbb {R}}^{k}\). The same holds for \(\gamma _x^m\).

\(\gamma _x^m(v) \le \log {\mathcal {A}}^m(x)\),

\(\gamma _x^{m+n}(v) = \gamma _x^m(v) + \gamma _{T^m(x)}^n({\mathcal {A}}^m(x) v)\).
We have
Hence (see (5.3)),
for every \(v,w\in D_x\).
To finish the proof we need the following lemma.
Lemma 5.9
Let \(A\in GL(k, {\mathbb {R}})\) and let \(K,r>0\). Assume \(\gamma (v)\gamma (w)<K\) for some \(v\in {\mathbb {P}}{\mathbb {R}}^{k}\) and all \(w\in B(v,r)\), where \(\gamma (v)=\log \Vert Av\Vert \). Then there exists a constant \(\rho =\rho (K,r)\), depending on K and r but not on A, such that \(\gamma (v) \ge \log A  \rho (K,r)\).
Before proving Lemma 5.9 let us observe that it indeed implies the assertion of Proposition 5.8. As \(D_x\) contains some ball B(v, r) with \(v\in D_x\cap {\mathbb {P}}{\mathbb {R}}^{k}\), we can apply the lemma to the matrix \({\mathcal {A}}^m(x)\), obtaining \(\log {\mathcal {A}}^m(x) \le \rho (K_4,r) + \gamma _x^m(v)\). Hence, for every \(w\in D_x\) we have
Similarly, \(D_{T^m(x)}\) contains a ball of size r, hence for every \(u\in D_{T^m(x)}\) we have
Thus, choosing \(u={\mathcal {A}}^m(x) w\) we get
which is our assertion.
Now, let us come back and prove Lemma 5.9.
Proof
We start by a decomposition \(A=O_1 D O_2\), where \(O_1, O_2\) are orthogonal matrices and D is a diagonal matrix with elements \(\pm (\sigma _i(A))\) (the singular values of A). It is enough to prove the assertion for the matrix D.
So, let D be a diagonal matrix. Let e be the eigenvector corresponding to the maximal eigenvalue: \(De=D\). Even when \(v.e=0\), the ball B(v, r) still must contain a vector w such that \(w.e\ge 1/2 \cdot \sin r\). We have \(w=w.ee + (1(w.e)^2)^{1/2}e'\), where \(e.e'=0\). Hence,
Thus, for every \(u\in B(v,r)\) we have
We are done. \(\square \)
\(\square \)
Given ldimensional subspace \(E(x)\subset T_{x} {\mathbb {R}}^{k}\), we define the convex cone
We say that a differentiable map \(f: {\mathbb {R}}^{k} \rightarrow {\mathbb {R}}^{k}\) satisfies a \(l\)cone condition on a set \(\Lambda \subset {\mathbb {R}}^{k}\) if for each \(x \in \Lambda \) a ldimensional subspace \(E(x) \subset T_{x} {\mathbb {R}}^{k}\) varying continuously with x such that
Corollary 5.10
Let M be a compact manifold in dimension d. Let \(T\in {{\,\mathrm{Diff}\,}}^1(M)\). Let \({\mathcal {A}}(x)=DT(x)\in GL(k,{\mathbb {R}})\) satisfying a \(1\)cone condition. Then, there exists \(\kappa >0\) such that for every \(m,n>0\) and for every \(x\in M\) we have
Proof
It follows from the proof of Proposition 5.8. \(\square \)
One needs to be careful that quasimultiplicativity is not equivalent of almost additivity. For instance, let \(T:\{0,1\}^{{\mathbb {Z}}}\rightarrow \{0,1\}^{{\mathbb {Z}}}\) be a shift map. We define a linear cocycle \((T, {\mathcal {A}})\) such that
where \(\theta \) is an irrational angle. It is easy to see that the locally constant cocycle \((T, {\mathcal {A}})\) is strongly irreducible. Feng [27, Proposition 2.8] showed that the irreducible matrix cocycles are quasimultiplicative.
The Proof of Theorem D
We say that \({\hat{\rho }}({\mathcal {A}})\) is the upper joint spectral radius of \({\mathcal {A}}: X \rightarrow GL(k, {\mathbb {R}})\) if
Similarly, we say that \({\check{\rho }}({\mathcal {A}})\) is the lower joint spectral radius of \({\mathcal {A}}: X \rightarrow GL(k, {\mathbb {R}})\) if
Assume that \(f:X\rightarrow X\) is a convex continuous function on the compact metric space X. We have \(\overline{\partial f({\mathbb {R}})}=\partial f({\mathbb {R}}) \cup \{f^{'}(\infty )\}\cup \{f^{'}(\infty )\}\), where \(\partial f({\mathbb {R}}) \) is defined as in (3.1).
Theorem 5.11
Let (X, T) be a TDS such that the entropy map \(\mu \mapsto h_{\mu }(T)\) is upper semicontinuous and \(h_{top}(T)<\infty \). Suppose that \({\mathcal {A}}:X\rightarrow GL(k, {\mathbb {R}})\) is a matrix cocycle over the TDS (X, T). Assume that \((C_x)_{x\in X}\) is a 1dimensional invariant cone field on X. Then \(\alpha ({\mathcal {A}})\) can be approximated by the Lyapunov exponents of the equilibrium measures for the almost additive potential \(t\Phi _{{\mathcal {A}}}\), where \(t\in {\mathbb {R}}\). Moreover, a Lyapunov minimizing measure for \({\mathcal {A}}\) exists.
Proof
We know that \({\mathcal {A}}\) is almost multiplicative by Proposition 5.8. Let \(\alpha :=\alpha ({\mathcal {A}})=P_{\Phi _{{\mathcal {A}}}}^{'}(\infty )\) (see Theorem 3.9).
By the convexity of \(P_{\Phi _{{\mathcal {A}}}}()\), there exists a sequence \((t_{j} )\) such that \( P_{\Phi _{{\mathcal {A}}}}^{'}(t_j )=: \alpha _{j}\) exists for every \(j\in {\mathbb {N}}\) and \(\alpha _{j}\rightarrow \alpha \) as \(j \rightarrow \infty \). There exists \(\mu _{j} \in {{\,\mathrm{Eq}\,}}(\Phi _{{\mathcal {A}}}, t_{j} )\) such that \(\chi (\mu _{j}, \Phi )=\alpha _{j}\) for all j, by Theorem 5.3. Extract a subsequence if necessary so that \(\mu _{j} \rightarrow \mu \) (weak* topology)^{Footnote 4} as \(j\rightarrow \infty \). By Lemma 5.4, we have
Furthermore, our proof shows that a Lyapunov minimizing measure exists.
\(\square \)
Now, we can show the continuity of the minimal Lyapunov exponent.
Proof of Theorem D
According to Theorem 5.11, \(\alpha ({\mathcal {A}})\) can be approximated by Lyapunov exponents of equilibrium measures for the almost additive potential \(t\Phi _{{\mathcal {A}}}\), where \(t\in {\mathbb {R}}\). Thus, it is enough to show that
where \(\mu , \mu _{n}\) are the equilibrium measures.
By Proposition 5.8, \({\mathcal {A}}\) is almost multiplicative and \(\Phi _{{\mathcal {A}}}\) has bounded distortion. Hence, there exist a unique equilibrium measure for the almost additive potential \(t\Phi _{{\mathcal {A}}}\) by Theorem 5.2. Therefore, (5.5) follows from Lemma 5.4. \(\square \)
Domination can be characterized in terms of the existence of invariant cone fields for derivative cocycles ( [23, Theorem 2.6]). This fact shows that 1domination implies that \({\mathcal {A}}\) is almost multiplicative. Hence, one can prove Theorem D for fiber bunched cocycles^{Footnote 5} under 1domination assumption.
It is possible to obtain the generalization of Theorem 5.11 to the joint spectrum of all Lyapunov exponents. One can also obtain the continuity of the lower joint spectral radius for all Lyapunov exponents.
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Acknowledgements
The author thanks Michał Rams for his careful reading of an earlier version of this paper and many helpful discussions. In particular, the author thanks Michał for helping with Proposition 5.8. He would also like to thank Aaron Brown and Cagri Sert for introducing papers [44] and [15]. Finally, he thanks the anonymous referee for corrections and suggestions that helped improve the paper. Much of this work was completed at IMPAN in 20162020, and was partially supported by the National Science Center Grant 2019/33/B/ST1/00275 (Poland).
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Mohammadpour, R. Lyapunov Spectrum Properties and Continuity of the Lower Joint Spectral Radius. J Stat Phys 187, 23 (2022). https://doi.org/10.1007/s1095502202910w
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DOI: https://doi.org/10.1007/s1095502202910w
Keywords
 Lyapunov spectrum
 Typical cocycles
 Thermodynamic formalism
 Lower joint spectral radius
 Entropy spectrum
Mathematics Subject Classification
 37A60
 37D30
 37D35
 37H15
 37N40