We recall some of the most significative notions of linear dynamical properties, further details can be found in [2, 3].

#### Definition 3.1

*Let* 𝓐 ∈ *ML*(*X*) *and let x* ∈ *X*. *Then we say that*:

*x is a* hypercyclic vector *of* 𝓐 *if x* ∈ *D*_{∞}(𝓐) *and for each n* ∈ ℕ_{0} *and each j* ∈ {1, …, *N*} *there exists an element y*_{j,n} ∈ 𝓐^{n}*x such that the set* {*y*_{n} : *n* ∈ ℕ} *is dense in X*. *In this case*, *we say that* 𝓐 *is hypercyclic*.

*x is a periodic point of* 𝓐 *if x* ∈ *D*_{∞}(𝓐) *and there exists n* ∈ ℕ *such that x* ∈ 𝓐^{n}*x*;

𝓐 *is topologically transitive if for every pair of non*-*empty open sets U*, *V* ⊂ *X there exists n* ∈ ℕ *such that U* ∩ 𝓐^{− n}(*V*) ≠ ∅.

𝓐 *is topologically mixing if for every pair of non*-*empty open sets U*, *V* ⊂ *X there exists n*_{0} ∈ ℕ *such that U* ∩ 𝓐^{− n}(*V*) ≠ ∅ *holds for n* ≥ *n*_{0}.

𝓐 *is chaotic* (*in the sense of Devaney*) *if* 𝓐 *is topologically transitive and the set consisting on all periodic points of* 𝓐 *is dense in X*.

With a similar argument as in the case of linear operators, by Baire’s Category Theorem, the notion of hypercyclicity is equivalent to the one of transitivity, as long as the MLO’s were defined on the whole space *X* [16, Th. 3.1.6]. Let 𝓐 ∈ *ML*(*X*) and let (*O*_{n})_{n ∈ ℕ} be an open base of the topology of *X*. Then the set consisting of hypercyclic vectors of 𝓐 is denoted shortly by HC (𝓐) and it can be computed by
$$\begin{array}{}\text{HC}(\mathcal{A})=\bigcap _{n\in \mathbb{N}}\bigcup _{k\in \mathbb{N}}{\mathcal{A}}^{-k}({O}_{n}).\end{array}$$(1)

#### Definition 3.4

*Let N* ≥ 2, 1 ≤ *j* ≤ *N*, *and* 𝓐_{1}, …, 𝓐_{N} ∈ *ML*(*X*). *Let x* ∈ *X*. *Then we say that*:

*x is a* disjoint hypercyclic vector *of* 𝓐 if *x* ∈ *D*_{∞}(𝓐_{j}) *and for each n* ∈ ℕ_{0} *and each j* ∈ {1, …, *N*} *there exists an element*
$\begin{array}{}{y}_{j,n}\in {\mathcal{A}}_{j}^{n}x\end{array}$ *such that the set* {(*y*_{1,n}, &, *y*_{N,n}) : *n* ∈ ℕ} *is dense in X* ⊕&⊕ *X*. *In this case*, *we say that* 𝓐_{1}, …, 𝓐_{N} *are* disjoint hypercyclic, *or simply* d-hypercyclic.

𝓐_{1}, …, 𝓐_{N} *are* disjoint topologically transitive, *or simply* d-topologically transitive *if for any choice of non*-*empty open sets U*, *V*_{1}, …, *V*_{n} ⊂ *X there exists n* ∈ ℕ *such that*
$\begin{array}{}U\cap {\mathcal{A}}^{-n}(V)\cap \dots \cap {\mathcal{A}}_{N}^{-n}(V)\ne \mathrm{\varnothing}.\end{array}$

𝓐_{1}, …, 𝓐_{N} *are* disjoint topologically mixing, *or simply* d-topologically mixing *if for any choice of non*-*empty open sets U*, *V*_{1}, …, *V*_{n} ⊂ *X there exists n*_{0} ∈ ℕ *such that*
$\begin{array}{}U\cap {\mathcal{A}}^{-n}(V)\cap \dots \cap {\mathcal{A}}_{N}^{-n}(V)\ne \mathrm{\varnothing}\end{array}$ *for every n* ≥ *n*_{0}.

𝓐_{1}, …, 𝓐_{N} *are* disjoint chaotic, *or simply* d-chaotic *if* 𝓐_{1}, …, 𝓐_{N} *are d*-*hypercyclic and the set of periodic points*, *denoted by* 𝓟(𝓐_{1}, 𝓐_{2}, …, 𝓐_{N}) := {(*x*_{1}, *x*_{2}, …, *x*_{N}) ∈ *X*^{N} : ∃*n* ∈ ℕ *with*
$\begin{array}{}{x}_{j}\in {\mathcal{A}}_{j}^{n}{x}_{j},\end{array}$ *j* ∈ {1, …, *N*}}, *is dense in X*^{N}.

It is clear that any multivalued linear extension of a hypercyclic (chaotic) single-valued linear operator is again hypercyclic (chaotic). It is also worth noting that Definition 3.1 prescribes some cases in which even zero can be a hypercyclic vector: Let 𝓐 := {0} × *W*, where *W* is a dense linear submanifold of *Y*. Then 𝓐 is hypercyclic, zero is the unique hypercyclic vector of 𝓐 and there is no single-valued linear restriction of 𝓐 that is hypercyclic (in particular, a hypercyclic MLO need not be densely defined and the inverse of a hypercyclic MLO need not be hypercyclic, in contrast to the single-valued linear case, see also [19, Prob. 1]). Furthermore, with this example, we can get that an MLO 𝓐 and some arbitraty multiples of it can be d-hypercyclic, in contrast to what is indicated in [19, p. 299]. Moreover, the non-triviality of the manifold 𝓐_{0} is not essentially connected with hypercyclicity of an MLO: Just take 𝓐:= *X* × *W*, with *W* a non-dense linear submanifold of *X* so as to 𝓐 cannot be hypercyclic.

It is an elementary fact that the point spectrum of the adjoint of a hypercyclic continuous single-valued linear operator has to be empty [4], see also [2, 3]. The same holds for hypercyclic MLO’s:

#### Theorem 3.5

*If* 𝓐 ∈ *ML*(*X*) *is hypercyclic*, *then σ*_{p}(𝓐^{*}) = ∅.

#### Proof

Let *x* be a hypercyclic vector for 𝓐. For every *n* ∈ ℕ, there exists an element *y*_{n} ∈ 𝓐^{n}*x* such that {*y*_{n} : *n* ∈ ℕ} is dense in *X*. Suppose that there exist *λ* ∈ 𝕂 and *x*^{*} ∈ *X*^{*} ∖ {0} such that *λ x*^{*} ∈ 𝓐^{*}*x*^{*}, i.e., that 〈*x*^{*}, *y*〉 = *λ*〈*x*^{*}, *x*〉, whenever *y* ∈ 𝓐*x*. It is clear that 〈*x*^{*}, *y*_{n}〉 = *λ*^{n}〈*x*^{*}, *x*〉 for all *n* ∈ ℕ, so that the assumption 〈*x*^{*}, *x*〉 = 0 implies 〈*x*^{*}, *y*_{n}〉 = 0 for all *n* ∈ ℕ, and therefore, *x*^{*} = 0. So, 〈*x*^{*}, *x*〉 ≠ O. If |*λ*| ≤ 1, then |〈*x*^{*}, *y*_{n}〉| = |*λ*^{n}〈*x*^{*}, *x*〉| ≤ |〈*x*^{*}, *x*〉| for all *n* ∈ ℕ. This would imply |〈*x*^{*}, *u*〉| ≤ |〈*x*^{*}, *x*〉| for all *u* ∈ *X*, which is a contradiction since |〈*x*^{*}, *nu*〉| diverges to ∞ for any *u* ∈ *X* such that 〈*x*^{*}, *u*〉 ≠ O. If |*λ*| > 1, then |〈*x*^{*}, *y*_{n}〉| = |*λ*^{n}〈*x*^{*}, *x*〉| ≥ |〈*x*^{*}, *x*〉| for all *n* ∈ ℕ; this would imply |〈*x*^{*}, *u*〈| ≥ |〈*x*^{*}, *x*〉| for all *u* ∈ *X*, which is a contradiction since |〈*x*^{*}, *u*/*n*〉| converges to 0 for all *u* ∈ *X*, and 〈*x*^{*}, *x*〉 ≠ 0. □

The Hypercyclicity Criterion, initially stated by Kitai [4] and by Gethner and Shapiro [5], gives some sufficient conditions in order to determine that an operator is hypercyclic. Those conditions were proved to be equivalent to be weakly mixing [21, 22]. Since then, several equivalent formulations have been given. We will state it in terms of the collapse /blow-up conditions [21, Def. 1.2] and [1, Th. 3.4] initially stated by Godefroy and Shapiro [6].

It is worth noting that this criterion can be formulated, in a certain way, for MLO’s and that we do not need any type of continuity or closedness of operators under consideration for its validity; a straightforward proof of the next result is very similar to that of [23, Th. 9] (continuous version) and therefore it is omitted.

#### Criterion 3.6

(Blow-up/collapse for MLOs). *Let* 𝓐 ∈ *ML*(*X*). *Suppose that* (*m*_{n})_{n ∈ ℕ} *be a strictly increasing sequence of positive integers*.

*Let X*_{0} *be the set of those elements y* ∈ *X for which there exists a sequence* (*y*_{n})_{n ∈ ℕ} *in X such that y*_{n} ∈ 𝓐^{mn}*y*, *n* ∈ ℕ *and* lim_{n →∞}*y*_{n} = 0.

*Let X*_{∞} *consist of those elements z* ∈ *X for which there exist a null sequence* (*ω*_{n})_{n ∈ ℕ} *in X and a sequence* (*u*_{n})_{n ∈ ℕ} *in X such that u*_{n} ∈ 𝓐^{mn}*ω*_{n}, *n* ∈ ℕ *and* lim_{n → ∞}*u*_{n} = *z*.

*If X*_{0}, *X*_{∞} *are dense in X*, *then* 𝓐 *is hypercyclic*.

We will also state a version of this criterion for disjoint hypercyclicity of MLOs; see [19, Prop. 2.6 & Th. 2.7] for single-valued case of linear operators. Its proof follows the same lines.

#### Proposition 3.7

(d-Blow-up/collapse Criterion for MLO’s). *Let N* ∈ ℕ *and let* 𝓐_{j} ∈ *ML*(*X*), *j* ∈ ℕ ∩[1, *N*] *and* (*m*_{n})_{n ∈ ℕ} *a strictly increasing sequence of positive integers*.

*Let X*_{0} *be the set of those elements y* ∈ *X satisfying that for every* 1 ≤ *j* ≤ *N there exists a sequence* (*y*_{n},*j*)_{n ∈ ℕ} *in X such that y*_{n},*j* ∈
$\begin{array}{}{\mathcal{A}}_{j}^{mn}\end{array}$ *y*, *n* ∈ ℕ *and* lim_{n → ∞}*y*_{n},*j* = 0.

*For each j* ∈ ℕ ∩[1, *N*], *let us consider the set X*_{∞,j}, *consisting of those elements z* ∈ *X for which there exist elements ω*_{n,i}(*z*) *and u*_{n,i,j}(*z*) *in X*(*n* ∈ ℕ, 1 ≤ *i* ≤ *N*) *such that* (*ω*_{n,j}(*z*))_{n ∈ ℕ} *is a null sequence in X*, *u*_{n,i,j}(*z*) ∈
$\begin{array}{}{\mathcal{A}}_{j}^{{m}_{n}}\end{array}$ *ω*_{n,i}(*z*), *n* ∈ ℕ *and* lim_{n →∞}*u*_{n,i,j} = *δ*_{i,j}*z*(1 ≤ *i* ≤ *N*) *where δ*_{i,j} *denotes the Kronecker delta*.

*If X*_{0}, *X*_{∞,1}, …, *X*_{∞,N}, *then the operators* 𝓐_{1}, … 𝓐_{N} *are d*-*hypercyclic*.

Observe that the assertions of Proposition 3.6 and Proposition 3.7 can be formulated for sequences of multivalued linear operators (cf. [19, Rem. 2.8] and for finite direct sums of each operator with *k* − 1 copies of itself, *k*, *n* ∈ ℕ, namely
$\begin{array}{}(\underset{k}{\underset{\u23df}{\mathcal{A}\oplus \dots \oplus \mathcal{A}}}{)}^{n}=\underset{k}{\underset{\u23df}{{\mathcal{A}}^{n}\oplus \dots \oplus {\mathcal{A}}^{n}}},\end{array}$ *k*, *n* ∈ ℕ. In this case, we just have to pass to the subsets
$\begin{array}{}{X}_{0}^{k},{X}_{\mathrm{\infty},j}^{k},\end{array}$ 1 ≤ *j* ≤ *N*, as well to the tuples (*z*, …, *z*), (*y*_{n,j}, …, *y*_{n,j}), (*w*_{n,i}, …, *w*_{n,i}), (*u*_{n,i,j}, …, *u*_{n,i,j}), each of which having exactly *k* components, and the proof follows as indicated.

The following theorem extends the criterion in [24, Th. 2.1], that is a kind of reformulation of Godefroy-Shapiro [6] and Desch-Schappacher-Webb Criterion (continuous version) [25], see also [3, Th. 7.30], for MLO’s.

#### Theorem 3.8

*Let* Ω ⊆ ℂ *be an open connected set intersecting the unit circle* Ω ∩ *S*_{1}≠∅. *Let* *f* : Ω→ *X* \ {0} *be an analytic mapping such that* *λ* *f*(*λ*) ∈ 𝓐*f*(*λ*) *for all* *λ*∈ Ω. *Set*
$\begin{array}{}\stackrel{~}{X}:=\overline{span\{f(\lambda ):\lambda \in \mathrm{\Omega}\}}.\end{array}$
*Then the operator*
$\begin{array}{}{\mathcal{A}}_{|\stackrel{~}{X}}\end{array}$
*is topologically mixing in the space*
$\begin{array}{}\stackrel{~}{X}\end{array}$
*and the set of periodic points of*
$\begin{array}{}{\mathcal{A}}_{|\stackrel{~}{X}}\end{array}$
*is dense in*
$\begin{array}{}\stackrel{~}{X}\end{array}$
, *so that it is also chaotic*.

#### Proof

The proof is very similar to that of [23, Th. 5], which is based on an application of the Hahn-Banach Theorem. We will only outline the most relevant details.

Without loss of generality, we may assume that
$\begin{array}{}\stackrel{~}{X}\end{array}$
= *X*. If Ω_{0} ⊆ Ω admits a cluster point in Ω, then the (weak) analyticity of mapping *λ* ↦ *f*(*λ*), *λ* ∈ Ω shows that Ψ(Ω_{0}) := span{*f*(*λ*) : *λ* ∈ Ω_{0}} is dense in *X*.

Further on, it is clear that there exist *λ*_{0} ∈ Ω ∩ {*z* ∈ ℂ : |*z*| = 1} and *δ* > 0 such that any of the sets Ω_{0,+} := {*λ* ∈ Ω : |*λ*–*λ*_{0} | < *δ*, | *λ*| > 1} and Ω_{0,−} := {*λ* ∈ Ω : |*λ*−*λ*_{0}| < *δ*, |*λ*| < 1} admits a cluster point in Ω.

Take two open sets ∅ ≠ *U*, *V* ⊆ *X*. Then there exists *y*, *z* ∈ *X*, ϵ > 0, *p*, *q* ∈ cs (*X*) such that *B*_{p} (*y*, ϵ) ⊆ *U* and *B*_{q} (*z*, ϵ)⊆ *V*. We may assume that
$\begin{array}{}y=\sum _{i=1}^{n}{\beta}_{i}f({\lambda}_{i})\in \mathrm{\Psi}({\mathrm{\Omega}}_{0,-}),z=\sum _{j=1}^{m}{\gamma}_{j}f({\stackrel{~}{\lambda}}_{j})\in \mathrm{\Psi}({\mathrm{\Omega}}_{0,+}),\end{array}$
with *α*_{j}, *β*_{j} ∈ ℂ\ {0}, *λ*_{j} ∈ Ω_{0,−} and
$\begin{array}{}{\stackrel{~}{\lambda}}_{j}\in {\mathrm{\Omega}}_{0,+}\end{array}$
for 1 ≤ *i* ≤ *n* and 1 ≤ *j* ≤ *m*.

Set
$\begin{array}{}{z}_{t}:=\sum _{j=1}^{m}\frac{{\gamma}_{j}}{{\stackrel{~}{\lambda}}_{j}^{\phantom{\rule{thinmathspace}{0ex}}t}}f({\stackrel{~}{\lambda}}_{j})\end{array}$
and *x*_{t} := *y* + *z*_{t}, *t* ≥ 0. Then {*x*_{t}, *y*, *z*_{t}} ⊆ *D*_{∞}(𝓐), *t* ≥ 0 and it can be easily seen that *z* ∈ 𝓐^{n} *z*_{n}, *n* ∈ ℕ. Consider
$\begin{array}{}{\omega}_{n}:=z+\sum _{i=1}^{n}{\beta}_{i}{\lambda}_{i}^{n}f({\lambda}_{j})\in {\mathcal{A}}^{n}{x}_{n},\end{array}$
for every *n* ∈ ℕ. There exists *n*_{0}(ϵ)∈ ℕ such that, for every *n* ≥ *n*_{0}(ϵ), *x*_{n} ∈ *B*_{p}(*y*, ϵ) and *w*_{n} ∈ *B*_{q}(*z*, ϵ). Therefore, 𝓐 is topologically mixing.

Since the set Ω ∩ exp (2*πi*ℚ) has a cluster point in Ω, the proof that the set of periodic points of 𝓐 is dense in *X* can be given as in that of [24, Th. 2.1].

Finally, the validity of implication:
$$\begin{array}{}\u3008{x}^{\ast},f(\lambda )\u3009=0,\lambda \in \mathrm{\Omega}\text{\hspace{0.17em}for some\hspace{0.17em}}{x}^{\ast}\in {X}^{\ast}\Rightarrow {x}^{\ast}=0.\end{array}$$
yields that
$\begin{array}{}\stackrel{~}{X}\end{array}$
= *X*. □

#### Example 3.10

*Suppose that A* ∈ *L*(*X*) *with closed range satisfying that there exist an open connected subset* ∅ ≠ Λ *of* ℂ *and an analytic mapping g* : Λ → *X* \ {0} *such that Ag*(*ν*) = *νg*(*ν*), *ν* ∈ Λ. *Let P*(*z*) *and Q*(*z*) *be non*-*zero complex polynomials*, *let R* := {*z* ∈ ℂ : *P*(*z*) = 0}, Λ′ := Λ\ *R*, *and let*
$\begin{array}{}\stackrel{~}{X}:=\overline{span\{g(\lambda ):\lambda \in \mathrm{\Lambda}\}}.\end{array}$
*Suppose that*
$$\begin{array}{}{\displaystyle \frac{Q}{P}({\mathrm{\Lambda}}^{\prime})\cap {S}_{1}\ne \mathrm{\varnothing}.}\end{array}$$

*Then Theorem 3.8 implies that the parts of MLO*’*s Q*(𝓐)*P*(𝓐)^{−1} *and P*(𝓐)^{−1}*Q*(𝓐) *in*
$\begin{array}{}\stackrel{~}{X}\end{array}$
*are topologically mixing in the space*
$\begin{array}{}\stackrel{~}{X}\end{array}$
. *It can be easily checked that the sets of periodic points of these operators are dense in*
$\begin{array}{}\stackrel{~}{X}\end{array}$.

Theorem 3.8 has a disjoint analogue in [18, Th. 4.3] for single-valued operators. The next result is the corresponding reformulation for MLO’s, that can be obtained with a similar proof.

#### Theorem 3.11

*Suppose that N* ∈ ℕ *and* 𝓐_{1}, …, 𝓐_{N} ∈ *ML*(*X*). *For each natural number* 0 ≤ *p* ≤ *N* *there exists a total set D*_{p} (that is, *the linear span of D*_{p} is dense in X) such that the following conditions hold for every 1 ≤ *j* ≤ *N*:

*(i) For every e* ∈ *D*_{p} there exists an eigenvalue *λ*_{j,p}(*e*) *of* 𝓐_{j} *for which* *λ*_{j,p}(*e*)*e* ∈ 𝓐_{j}*e*.

*(ii) For every e* ∈ *D*_{0} *we have* *λ*_{j,0}(*e*)∈ *int*(*S*_{1})

*(ii) For every e* ∈ *D*_{j} *we have λ*_{j,j} (*e*) ∈ *ext*(*S*_{1}).

*(iii) For every* 1 ≤ *i* ≠ *j* ≤ *N and every e* ∈ *D*_{j} *we have* |*λ*_{j,i}(*e*)| < |*λ*_{jj} (*e*)|.

*Then the operators* 𝓐_{1},…, 𝓐_{N} *are d*-*topologically mixing*.

We illustrate Theorem 3.11 with two interesting examples pointing out that the continuity of operators can be neglected from the formulation of [18, Theorem 4.3], as well as that there exists a great number of Banach function spaces where this extended version of the aforementioned theorem can be applied (cf. also [18, Final questions, 2.]). Other examples involving single-valued or multivalued linear operators can be similarly given, by using the analysis from Example 3.10; cf. also [26], [27, Ex. 3.8 & Ex. 3.10] and [28] for some other unbounded differential operators that we can employ here.

#### Example 3.12

*Let p* > 2 *and let X be a symmetric space of non*-*compact type of rank one*, *let P*_{p} be the parabolic domain defined in the proof of [29, *Th. 3.1*], *and let c*_{p} > 0 *be the apex of*
$$\begin{array}{}{P}_{p}={\displaystyle \left\{||\rho |{|}^{2}+{z}^{2}:\mathit{?}z\in \mathbb{C},,|\mathrm{\Im}(z)|\le ||\rho ||\cdot \left|\frac{2}{p}-1\right|\right\}\subseteq \mathbb{C}.}\end{array}$$(2)
*It is known that int*
$\begin{array}{}({P}_{p})\subseteq {\sigma}_{p}({\mathrm{\Delta}}_{X,p}^{\mathrm{\u266f}})\end{array}$
*[29*, *30]*, *where*
$\begin{array}{}{\mathrm{\Delta}}_{X,p}^{\mathrm{\u266f}}\end{array}$
*denotes the corresponding Laplace*-*Beltrami operator acting on*
$\begin{array}{}{L}_{\mathrm{\u266f}}^{p}\end{array}$
(*X*), *the space of K*-*invariant functions in L*^{p}(*X*), *see [29*, *Sec*. *2*.*3]*, *where K* *is a maximal compact subgroup of the non*-*compact semi*-*simple Lie group Isom*^{0}(*X*).

*Furthermore*, *there exists an analytic function g* : *int* (*P*_{p}) →
$\begin{array}{}{L}_{\mathrm{\u266f}}^{p}\end{array}$
(*X*) *such that*
$\begin{array}{}{\mathrm{\Delta}}_{X,p}^{\mathrm{\u266f}}\end{array}$
*g*(*λ*) = *λ* *g*(*λ*), *λ* ∈ *int*(*P*_{p}) *and that the set* Ψ(Ω) := {*g*(*λ*) : *λ* ∈ Ω} *is total in X* *for any non*-*empty open subset* Ω ⊆ *int*(*P*_{p}).

*Take N* ∈ ℕ *with N* ≥ 2 *and numbers a*_{1}, …, *a*_{N} *satisfying* − 1 − *c*_{p} < *a*_{1} < *a*_{2} < … < *a*_{N} < 1 − *c*_{p}. *For every* 1 ≤ *i* ≤ *N there exists a point* *λ*_{j} ∈ (*P*_{p}) *such that* |*λ*_{j} + *a*_{i} | > max (1, |*λ*_{j} + *a*_{j}|) *for all j* ≠ *i with* 1 ≤ *j* ≤ *N*. *Now taking* Ω_{0} *as a small ball around c*_{p}, *and D*_{j} = Ψ(Ω_{j}) *with* Ω_{j} *a small ball around* *λ*_{j}, 1 ≤ *i* ≤ *N*, *we can apply Theorem 3.11*, *and then we conclude that the operators*
$\begin{array}{}{\mathrm{\Delta}}_{X,p}^{\mathrm{\u266f}}+{a}_{1},{\mathrm{\Delta}}_{X,p}^{\mathrm{\u266f}}+{a}_{2},\dots ,{\mathrm{\Delta}}_{X,p}^{\mathrm{\u266f}}+{a}_{N}\end{array}$
*are d*-*topologically mixing*.

*Furthermore*, *by Remark 3.8*, *the set*
$\begin{array}{}\mathcal{P}({\mathrm{\Delta}}_{X,p}^{\mathrm{\u266f}}+{a}_{1},\dots ,{\mathrm{\Delta}}_{X,p}^{\mathrm{\u266f}}+{a}_{N})\end{array}$
*is dense in* (
$\begin{array}{}{L}_{\mathrm{\u266f}}^{p}\end{array}$
(*X*))^{N}. *Hence*, *the operators*
$\begin{array}{}{\mathrm{\Delta}}_{X,p}^{\mathrm{\u266f}}+{a}_{1},{\mathrm{\Delta}}_{X,p}^{\mathrm{\u266f}}+{a}_{2},\dots ,{\mathrm{\Delta}}_{X,p}^{\mathrm{\u266f}}+{a}_{N}\end{array}$
*are d*-*chaotic*.

The next example is inspired by some results from [31–33].

#### Example 3.13

*Let b* > *c*/2 > 0, Ω := {*λ* ∈ ℂ : ℜ*λ* < *c* − *b*/2} *and let us consider the bounded perturbation of the one*-*dimensional Ornstein*-*Uhlenbeck operator* 𝓐_{c}u := *u*′ + 2*bxu*′ + *cu is acting on L*^{2}(ℝ), *with domain*
$\begin{array}{}D({\mathcal{A}}_{c}):=\{u\in {L}^{2}(\mathbb{R})\cap {W}_{loc}^{2,2}(\mathbb{R}):{\mathcal{A}}_{c}u\in {L}^{2}(\mathbb{R})\}.\end{array}$
*Then* 𝓐_{c} *generates a strongly continuous semigroup and* Ω ⊆ *σ*_{p}(𝓐^{c}).

*For any non*-*empty open connected subset* Ω′ ⊆ Ω, *which admits a cluster point in* Ω, *we have X* = span{*g*_{j}(*λ*) : *λ* ∈ Ω′, *i* = 1, 2}, *where g*_{1} : Ω → *X and g*_{2} : Ω → *X* *are defined by g*_{1}(*λ*) :=
$\begin{array}{}{\mathcal{F}}^{-1}({e}^{-\frac{{\xi}^{2}}{2b}}\xi |\xi {|}^{-(2+\frac{\lambda -c}{b})})(\cdot ),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{g}_{2}(\lambda ):={\mathcal{F}}^{-1}({e}^{-\frac{{\xi}^{2}}{2b}}|\xi {|}^{-(1+\frac{\lambda -c}{b})})(\cdot ),\lambda \in \mathrm{\Omega}\end{array}$
*denoting by* 𝓕^{−1} *the inverse Fourier transform on the real line*.

*Furthermore*, 𝓐^{c}*g*_{j}(*λ*) = *λ* *g*_{j}(*λ*), *λ*∈ Ω, *i* = 1, 2. *Suppose that P*_{j}(*z*) *and Q*_{j}(*z*) *are non*-*zero complex polynomials* (1 ≤ *i* ≤ *N*), *and* Ω′ *denotes the set obtained by removing all zeroes of polynomials P*_{j}(*z*) *from* Ω. *Let* Λ ⊆ ℂ *be a non*-*empty open connected subset intersecting* *S*_{1} = {*z* ∈ ℂ : |*z*| = 1} *such that*
$$\begin{array}{}{\displaystyle \mathrm{\Lambda}\subseteq \bigcap _{1\le j\le N}\frac{{Q}_{j}}{{P}_{j}}({\mathrm{\Omega}}^{\prime}).}\end{array}$$(3)

*In addition*, *let us suppose that there exist non*-*empty open connected subsets* Ω_{p} ⊆ Ω′, 0 ≤ *p* ≤ *N*, *such that*
$$\begin{array}{}{\displaystyle \frac{{Q}_{j}}{{P}_{j}}({\mathrm{\Omega}}_{0})\subseteq \text{int}({S}_{1}),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\frac{{Q}_{j}}{{P}_{j}}({\mathrm{\Omega}}_{j})\subseteq \text{ext}({S}_{1}),\phantom{\rule{thinmathspace}{0ex}}1\le j\le N.}\end{array}$$(4)

*Finally*, *applying Theorem 3.11 and Remark 3.9*, *as in Example 3.10*, *we get that the multivalued linear operators Q*_{1}(𝓐_{c})*P*_{1}(𝓐_{c})^{−1}, …, *Q*_{N}(𝓐_{c})*P*_{N}(𝓐_{c})^{−1} *are d*-*topologically mixing and that the set of periodic points of these operators are dense in X*^{N}.

*Furthermore*, *the same conclusion holds if we replace some of the operators Q*_{i}(𝓐_{c})^{−1}*P*_{i}(𝓐_{c}) *with P*_{i}(𝓐_{c})*Q*_{i}(𝓐_{c})^{−1}, 1 ≤ *i* ≤ *N*.

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