The Specification Property for C -Semigroups 0 S. Bartoll, F. Mart´ınez-Giménez, A. Peris∗and F. Rodenas IUMPA, Universitat Politècnica de València, Departament de Matema`tica Aplicada, Edifici 7A, E-46022, València, SPAIN. 6 Abstract 1 0 We study one of the strongest versions of chaos for continuous dynamical sys- 2 tems, namely the specification property. We extend the definition of specification n property for operators on a Banach space to strongly continuous one-parameter a J semigroups of operators, that is, C -semigroups. In addition, we study the relation- 0 8 ships of the specification property for C -semigroups (SgSP) with other dynamical 0 2 properties: mixing, Devaney’s chaos, distributional chaos and frequent hypercyclic- ] ity. Concerning the applications, we provide several examples of semigroups which A exhibit the SgSP with particular interest on solution semigroups to certain linear F PDEs,whichrangefromthehyperbolicheatequationtotheBlack-Scholesequation. . h t a m 1 Introduction [ A (continuous) map on a metric space satisfies the specification property (SP) if for any 1 v choice of points, one can approximate distinct pieces of orbits by a single periodic orbit 3 with a certain uniformity. It was first introduced by Bowen [14]; since then, this property 5 8 has attracted the interest of many researchers (see, for instance, the early work [35]). In a 7 few words, the specification property requires that, for a given distance δ > 0, and for any 0 . finite family of points, there is always a periodic orbit that traces arbitrary long pieces 1 0 of the orbits of the family, up to a distance δ, allowing a minimum “jump time” N from δ 6 one piece of orbit to another one, which only depends on δ. 1 : v Definition 1. A continuous map f : X → X on a compact metric space (X,d) has the i X specificationpropertyifforanyδ > 0thereisapositiveintegerN suchthatforanyinteger δ r s ≥ 2, any set {y ,...,y } ⊂ X and any integers 0 = i ≤ j < i ≤ j < ··· < i ≤ j a 1 s 1 1 2 2 s s satisfying i −j ≥ N for r = 1,...,s−1, there is a point x ∈ X such that the following r+1 r δ conditions hold: d(fi(x),fi(y )) < δ, with i ≤ i ≤ j , for every r ≤ s, r r r fNδ+js(x) = x. This definition must be modified when one treats with bounded linear operators defined on separable Banach spaces which are never compact [3, 4]. Here, we denote the specification property for operators by OSP (see [3, 4] for definitions and properties). A ∗Corresponding author. E-mail aperis@mat.upv.es 1 continuous map on a metric space is said to be chaotic in the sense of Devaney if it is topologically transitive and the set of periodic points is dense. Although there is no com- mon agreement about what a chaotic map is, the specification property is stronger than Devaney’s definition of chaos. Recently, several properties of linear operators with the OSP and the connections of this OSP with other well-known dynamical properties, like mixing, chaosinthesenseofDevaneyandfrequenthypercyclicityhavebeenstudiedin[4], we will use these results throughout the paper. Other recent works on the specification property are [32, 33, 27]. A family (T ) of linear and continuous operators on a Banach space X is said to be t t≥0 a C -semigroup if T = Id, T T = T for all t,s ≥ 0, and lim T x = T x for all x ∈ X 0 0 t s t+s t→s t s and s ≥ 0. Let (T ) be an arbitrary C -semigroup on X. It can be shown that an operator t t≥0 0 defined by Ax := lim 1(T x−x) exists on a dense subspace of X; denoted by D(A). t→0 t t Then A, or rather (A,D(A)), is called the (infinitesimal) generator of the semigroup. It can also be shown that the infinitesimal generator determines the semigroup uniquely. If the generator A is defined on X (D(A) = X), the semigroup is expressed as {T } = t t≥0 {etA} . t≥0 Given a family of operators (T ) , we say that this family of operators is transitive t t≥0 if for every pair of non-empty open sets U,V ⊂ X there exists some t > 0 such that T (U)∩V (cid:54)= ∅. Furthermore, if there is some t such that the condition T (U)∩V (cid:54)= ∅ t 0 t holds for every t ≥ t we say that it is topologically mixing or mixing. 0 A family of operators (T ) is said to be universal if there exists some x ∈ X such t t≥0 that {T x : t ≥ 0} is dense in X. When (T ) is a C -semigroup we particularly refer to t t t≥0 0 it as hypercyclic. In this setting, transitivity coincides with universality, but it is strictly weaker than mixing [9]. In addition, two notions of chaos are recalled: Devaney chaos and distributional chaos. An element x ∈ X is said to be a periodic point of (T ) if there exists some t > 0 such t t≥0 0 that T x = x. A family of operators (T ) is said to be chaotic in the sense of Devaney t0 t t≥0 if it is hypercyclic and there exists a dense set of periodic points in X. On the other hand, it is distributionally chaotic if there are an uncountable set S ⊂ X and δ > 0, so that for each ε > 0 and each pair x,y ∈ S of distinct points we have Dens{s ≥ 0 : ||T x−T y|| ≥ δ} = 1 and s s Dens{s ≥ 0 : ||T x−T y|| < ε} = 1, s s where Dens(B) is the upper density of a Lebesgue measurable subset B ⊂ R+ defined as 0 µ(B ∩[0,t]) limsup , t t→∞ with µ standing for the Lebesgue measure on R+. A vector x ∈ X is said to be distribu- 0 tionally irregular for the C -semigroup (T ) if for every ε > 0 we have 0 t t≥0 Dens{s ≥ 0 : ||T x|| ≥ ε−1} = 1 and s Dens{s ≥ 0 : ||T x|| < ε} = 1. s Suchvectorswereconsideredin[8]soastogetafurtherinsightintothephenomenonof distributional chaos, showing the equivalence between a distributionally chaotic operator and an operator having a distributionally irregular vector. This equivalence was later generalized for C -semigroups in [1]. 0 2 Devaney chaos, hypercyclicity and mixing properties have been widely studied for linear operators on Banach and more general spaces [9, 12, 18, 21, 23, 24, 34]. The recent books [7] and [25] contain the basic theory, examples, and many results on chaotic linear dynamics. A stronger concept than hypercyclic operators is the notion of frequently hypercyclic operators introduced by Bayart and Grivaux [6] (see [25] and the references therein) trying to quantify the frequency with which an orbit meets an open set. This concept was extended to C -semigroups in [2]. 0 A C -semigroup (T ) is said to be frequently hypercyclic if there exists x ∈ X (called 0 t t≥0 frequently hypercyclic vector) such that Dens({t ≥ 0 : T x ∈ U}) > 0 for every non-empty t open subset U ⊂ X, where Dens(B) is the lower density of a Lebesgue measurable subset B ⊂ R+ defined as 0 µ(B ∩[0,t]) liminf . t→∞ t In [16] it was proved that if x ∈ X is a frequently hypercyclic vector for (T ) , then x is t t≥0 a frequently hypercyclic vector for every the operator T , t > 0. t In[13]BonillaandGrosse-Erdmann,basedonaresultofBayartandGrivaux,provided a Frequent Hypercyclicity Criterion for operators. Later, Mangino and Peris [28] obtained a continuous version of the criterion based on Pettis integrals, which is called the Frequent Hypercyclicity Criterion for semigroups. The aim of this work is to study the specification property for strongly continuous semigroups of operators on Banach spaces, that is, for C -semigroups and its relationship 0 with other dynamical properties, like hypercyclicity, mixing, chaos and frequent hypercyclicity; and to provide useful tools ensuring that many natural solution semigroups associated to linear PDEs satisfy the specification property for C -semigroups. The paper 0 is structured as follows: In Section 2 we introduce the notion of the specification property for C -semigroups, from now on denoted by SgSP. Section 3 is devoted to study the SgSP 0 in connection with other dynamical properties. Finally, in Section 4 we provide several applications of the results in previous sections to solution semigroups of certain linear PDEs, and a characterization of translation semigroups which exhibit the SgSP. 2 Specification property for C -semigroups 0 A first notion of the specification property for a one-parameter family of continuous maps actingonacompactmetricspacewasgivenin[15]. Whentryingtostudythespecification property in the context of semigroups of linear operators defined on separable Banach spaces, the first crucial problem is that these spaces are never compact, therefore, our first task should be to adjust the SP in this context, in the vain of the discrete case, and the following definition can be considered the natural extension in this setting. Definition 2 (Specification property for semigroups, SgSP). A C -semigroup (T ) on 0 t t≥0 a separable Banach space X has the SgSP if there exists an increasing sequence (K ) of n n T-invariant sets with 0 ∈ K and ∪ K = X and there exists a t > 0, such that for 1 n∈N n 0 each n ∈ N and for any δ > 0 there is a positive real number M ∈ R such that for any δ,n + integer s ≥ 2, any set {y ,...,y } ⊂ K and any real numbers: 0 = a ≤ b < a ≤ b < 1 s n 1 1 2 2 ··· < a ≤ b satisfying b +M ∈ N·t and a −b ≥ M for r = 1,...,s−1, there s s s δ,n 0 r+1 r δ,n 3 is a point x ∈ K such that, for each t ∈ [a ,b ], r = 1,2,...,s, the following conditions n r r r hold: (cid:107)T (x)−T (y )(cid:107) < δ, tr tr r T (x) = x, where t = M +b . t δ,n s Analogously to the discrete case, the meaning of this property is that if the semigroup has the SgSP then it is possible to approximate simultaneously several finite pieces of orbits by one periodic orbit. Obviously, parameter intervals for the approximations must be disjoint. The following result is an immediate consequence of the corresponding definitions. Proposition 3. Let (T ) be a C -semigroup on a separable Banach spaceX. Then the t t≥0 0 following assertions are equivalent: 1. (T ) has the SgSP. t t≥0 2. Some operator T has the OSP. t0 3 SgSP and other dynamical properties for C -semigroups 0 In this section, we study the relation between the specification property and topological mixing, Devaney chaos, distributional chaos and frequent hypercyclicity. The following observations are useful to characterize mixing semigroups (see [25]). Remark 4. From the definition, the semigroup (T ) is mixing if and only if for every t t≥0 pair of non-empty open sets U,V ⊂ X, such that the complementary of the return set R(U,V) := {t ≥ 0 : T (U)∩V (cid:54)= ∅} is bounded. t Remark 5. Let (T ) be a C -semigroup on a separable Banach space X. The semigroup t t≥0 0 (T ) is mixing if and only if for every non-empty open set U ⊂ X and every open t t≥0 0-neighbourhood W, the complementary of the return sets R(U,W) and R(W,U) are bounded. Proposition 6. Let (T ) be a C -semigroup on a separable Banach space X. If (T ) t t≥0 0 t t≥0 has the SgSP, then (T ) is mixing. t t≥0 Proof. Let us consider a non-empty open set U and a 0-neighbourhood W. We claim that there exists some t > 0 such that t ∈ R(U,W)∩R(W,U), ∀t > t , and this implies 1 1 (T ) is mixing. t t≥0 Fixu ∈ U andδ > 0suchthatB(u,2δ) ⊂ U andB(0,2δ) ⊂ W. Byhypothesis, (T ) t t≥0 has the SgSP, then there are t > 0 and a T -invariant set K such that the restriction of 0 t0 T to K has the SP and K ∩B(u,δ) (cid:54)= ∅. From Definition 2, there exists M (depending t0 on K and δ, which we suppose M ∈ N · t ) such that if we choose y ∈ K ∩ B(u,δ), 0 1 y = 0, s > 0 with s ∈ N·t , and 0 = a = b < a = M < b = M +s then there exists 2 0 1 1 2 2 a periodic point x ∈ K with period 2M +s such that (cid:107)T (x)−T (y )(cid:107) < δ, a ≤ t ≤ b , t t 1 1 1 (cid:107)T (x)−T (y )(cid:107) < δ, a ≤ t ≤ b . t t 2 2 2 4 This implies (cid:107)x − y (cid:107) < δ, so (cid:107)x − u(cid:107) < 2δ and hence x ∈ U. From the second line of 1 the previous equation, we have T (x) ∈ B(0,δ) ⊂ W for M ≤ t ≤ M + s. Therefore t t ∈ R(U,W). Moreover, for t(cid:48) := 2M + s − t ∈ [M,M + s], we have (cid:107)T (x)(cid:107) < δ t(cid:48) too, hence w := T (x) ∈ B(0,δ) ⊂ W. Since x is periodic with period 2M + s, then t(cid:48) T (w) = T (T (x)) = x ∈ U. Therefore t ∈ R(W,U)∩R(U,W). Since s ∈ N·t can be t t t(cid:48) 0 selected arbitrary big, we conclude [M,+∞[⊂ R(W,U)∩R(U,W). Proposition 7. Let (T ) be a C -semigroup on a separable Banach space X. If (T ) t t≥0 0 t t≥0 has the SgSP then (T ) is Devaney chaotic. t t≥0 Proof. By Proposition 6, (T ) is topologically transitive and, by the definition of SgSP, t t≥0 it is clear that any vector in the space may be approximated by a periodic point. Proposition 8. Let (T ) be a C -semigroup on a separable Banach space X. If (T ) t t≥0 0 t t≥0 has the SgSP with respect to an increasing sequence (K ) of invariant compact sets, then n n (T ) is distributionally chaotic. t t≥0 Proof. We first recall that for single maps on compact metric spaces, Oprocha [32] showed that the SP implies distributional chaos in our sense. Since there is t > 0 such that T 0 t0 has the OSP, and by hypothesis the associated increasing sequence (K ) of invariant sets n n consists of compact sets, then T | is distributionally chaotic for every n ∈ N, thus the t0 Kn operator T is distributionally chaotic. Applying Theorem 3.1 in [1] we obtain that the t0 semigroup is distributionally chaotic. It is well-known [25, 16, 28] that a C -semigroup is hypercyclic (respectively mixing, 0 frequently hypercyclic) if and only if it admits a hypercyclic (resp. mixing, frequently hypercyclic) discretization (Tn) for certain t > 0. In particular, it is useful for our t0 n 0 purposes the following characterization of frequent hypercyclicity for semigroups in terms of the frequent hypercyclicity of some of its operators [16, 28]. Proposition 9 ([16, 28]). Let (T ) be a C -semigroup on a separable Banach space X. t t≥0 0 Then the following conditions are equivalent: (i) (T ) is frequently hypercyclic. t t≥0 (ii) For every t > 0 the operator T is frequently hypercyclic. t (iii) There exists t > 0 such that T is frequently hypercyclic. 0 t0 We point out the connection between the frequent hypercyclicity for semigroups and the specification property SgSP. Proposition 10. Let(T ) beaC -semigrouponaseparableBanachspaceX. If(T ) t t≥0 0 t t≥0 has the SgSP, then (T ) is frequently hypercyclic. t t≥0 Proof. By Proposition 3, if (T ) has the SgSP, then there exists t > 0 such that the t t≥0 0 operator T has the OSP, so the operator T is frequently hypercyclic by Theorem 13 t0 t0 in [4] and, therefore, the C -semigroup (T ) is frequently hypercyclic (see Proposition 0 t t≥0 9). We want to provide a useful sufficient condition for a C -semigroup to have the SgSP. 0 It goes through a criterion that we will not recall here to avoid unnecessary additional notation, but we will pick up some computable conditions ensuring that the criterion is fulfilled. 5 Proposition 11. Let(T ) beaC -semigrouponaseparableBanachspaceX. If(T ) t t≥0 0 t t≥0 satisfies the Frequently Hypercyclic Criterion for semigroups of [28], then every operator T , t > 0, has the OSP and, therefore, the semigroup (T ) has the SgSP. t t t≥0 Proof. If (T ) satisfies the Frequently Hypercyclic Criterion for semigroups of [28] then t t≥0 every operator T (t > 0) satisfies the Frequently Hypercyclic Criterion for operators of t [13]. Using a result from [4] about the OSP on operators which says that if an operator T on a Banach space satisfies the Frequently Hypercyclic Criterion then it has the OSP, we obtain that the operator T has the OSP for every t > 0, and finally, by using Proposition t 3, we conclude that the semigroup (T ) has the SgSP. t t≥0 Corollary 2.3 in [28] ensured that, under some conditions expressed in terms of eigen- vector fields for the infinitesimal generator A of the C -semigroup (T ) , the semigroup 0 t t≥0 is frequent hypercyclic. In fact, it was proved in [28] that it satisfies the Frequently Hypercyclic Criterion. As a consequence, we obtain the SgSP under the same conditions. Proposition 12. Let (T ) be a C -semigroup on a separable complex Banach space X t t≥0 0 andletAbeitsinfinitesimalgenerator. Assumethatthereexistsafamily(f ) oflocally j j∈Γ bounded measurable maps f : I → X such that I is an interval in R, f (I ) ⊂ D(A), j j j j j where D(A) denotes the domain of the generator, Af (t) = itf (t) for every t ∈ I , j ∈ Γ j j j and span{f (t) : j ∈ Γ, t ∈ I } is dense in X. If either j j a) f ∈ C2(I ,X), j ∈ Γ, j j or b) X does not contain c and (cid:104)ϕ,f (cid:105) ∈ C1(I ), ϕ ∈ X(cid:48), j ∈ Γ, 0 j j then the semigroup (T ) has the SgSP. t t≥0 Proof. The result directly follows from the Corollary 2.3 in [28] and Proposition 11. Remark 13. It was pointed out in Remark 2.4 in [28] that the spectral criterion for chaos in [19] of C - semigroups implies frequent hypercyclicity. As a consequence, if a semigroup 0 satisfies the spectral criterion, then it has the SgSP. 4 Applications and examples In this section we will present several examples of C -semigroups exhibiting the spec- 0 ification property, with particular interest in solution semigroups to certain PDEs. A characterization of translation semigroups with the SgSP is also provided. In the following examples, in order to ensure whether the solution semigroup has the SgSP, we will check the conditions of Proposition 12 (i.e., the conditions of Corollary 2.3 in [28]) or the spectral criterion in [19] for chaos. Example 14 (The solution semigroup of the hyperbolic heat transfer equation). Let us consider the hyperbolic heat transfer equation (HHTE):  τ∂2u + ∂u = α∂2u,  ∂t2 ∂t ∂x2    u(0,x) = ϕ (x),x ∈ R, 1      ∂u(0,x) = ϕ (x),x ∈ R ∂t 2 6 whereϕ andϕ representtheinitialtemperatureandtheinitialvariationoftemperature, 1 2 respectively, α > 0 is the thermal diffusivity, and τ > 0 is the thermal relaxation time. The dynamical behaviour presented by the solutions of the classical heat equation was studied by Herzog [26] on certain spaces of analytic functions with certain growth control. Later, the dynamical properties of the solution semigroup for the hyperbolic heat transfer equation were also established in [17, 25]. The HHTE can be expressed as a first-order equation on the product of a certain function space with itself X ⊕ X. We set u = u and u = ∂u. Then the associated 1 2 ∂t first-order equation is:  (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) u 0 I u  ∂ 1 = 1 .  ∂t u2 ατ ∂∂x22 −τ1I u2  (cid:18) u (0,x) (cid:19) (cid:18) ϕ (x) (cid:19)  1 = 1 ,x ∈ R  u (0,x) ϕ (x) 2 2 We fix ρ > 0 and consider the space [26] (cid:88)∞ a ρn X = {f : R → C;f(x) = n xn,(a ) ∈ c } ρ n n≥0 0 n! n=0 endowed with the norm ||f|| = sup |a |, where c is the Banach space of complex n≥0 n 0 sequences tending to 0. Since (cid:18) (cid:19) 0 I A := . α ∂2 −1I τ ∂x2 τ is an operator on X := X ⊕ X , we have that (T ) = (etA) is the C -semigroup ρ ρ t t≥0 t≥0 0 solution of the HHTE. We know from [17] and [25] that, given α, τ and ρ such that ατρ > 2, the solution semigroup (etA) defined on X ⊕ X is mixing and chaotic t≥0 ρ ρ since it satisfies the hypothesis of the spectral criterion [19]. Therefore, it satisfies the hypothesis of Corollary 2.3 in [28] which implies that the solution semigroup fulfilles the Frequent Hypercyclicity Criterion and, by the Proposition 11, it follows that the solution semigroup of the HHTE has the SgSP. Remark 15. With minor changes, we can apply the previous argument to the wave equation   u = αu  tt xx  u(0,x) = ϕ (x), x ∈ R 1  u (0,x) = ϕ (x), x ∈ R  t 2 which can be expressed as a first order equation in X ⊕X (see [25]), in order to state ρ ρ that its semigroup solution has the SgSP. Remark 16. This result can be extended to the solution semigroup of an abstract Cauchy problem of the form: (cid:26) (cid:27) u = Au t , u(0,x) = ϕ(x) where A is a linear operator on a Banach space X and the generator of the solution semigroup. If A satisfy the conditions of Corollary 2.3 in [28], then the semigroup (T ) t t≥0 with generator A has the SgSP. 7 Example 17 (C -semigroup solution of the Black-Scholes equation). Black and Scholes 0 proposed in [10] a mathematical model which gives a theoretical estimate of the price of stock options. The model is based on a partial differential equation, called the Black- Scholes equation, which estimates the price of the option over time. They proved that undersomeassumptionsaboutthemarket,thevalueofastockoptionu(x,t),asafunction of the current value of the underlying asset x ∈ R+ = [0,∞) and time, satisfies the final value problem:  ∂u = −1σ2x2∂2u −rx∂u +ru in R+ ×[0,T]  ∂t 2 ∂x2 ∂x u(0,t) = 0 for t ∈ [0,T]  u(x,T) = (x−p)+ for x ∈ R+ where p > 0 represents a given strike price, σ > 0 is the volatility, r > 0 is the interest rate and (cid:26) x−p if x > p (x−p)+ = 0 if x ≤ p. Let v(x,t) = u(x,T −t), then it satisfies the forward Black-Scholes equation, defined for all time t ∈ R+ by  ∂v = 1σ2x2∂2v +rx∂v −rv in R+ ×R+  ∂t 2 ∂x2 ∂x v(0,t) = 0 for t ∈ R+  v(x,0) = h(x) for x ∈ R+ where h(x) = (x − p)+, although we consider the general case where h(x) is arbitrary. This problem can be expressed in an abstract form:  ∂v = Bv,  ∂t v(0,t) = 0,  v(x,0) = h(x) for x ∈ R+. where B = (D )2 +γ(D )−rI, being D = νx ∂ with ν = √σ and γ = r −ν. ν ν ν ∂x 2 ν It was shown that the Black-Scholes equation admits a C -semigroup solution which 0 can be represented by T := f(tD ), where t ν f(z) = eg(z) with g(z) = z2 +γz −r. In [22] (see also [11] for alternative analytical solutions), a new explicit formula for the solution of the Black-Scholes equation was given in certain spaces of functions Ys,τ defined by (cid:26) (cid:27) u(x) u(x) Ys,τ = u ∈ C((0,∞)) ; lim = 0, lim = 0 x→∞ 1+xs x→0 1+x−τ endowed with the norm (cid:12) (cid:12) (cid:12) u(x) (cid:12) ||u||Ys,τ = sup(cid:12)(cid:12)(1+xs)(1+x−τ)(cid:12)(cid:12). x>0 Later, it was proved in [20] that the Black-Scholes semigroup is strongly continuous and chaotic for s > 1,τ ≥ 0 with sν > 1 and it was showed in [31] that it satisfies the spectral criterion in [19] under the same restrictions on the parameters and, therefore, the hypothesis of Corollary 2.3 of [28] and, consequently, the Black-Scholes semigroup has the SgSP. 8 There exist other C -semigroups related with PDEs which present the SgSP. In fact, 0 the examples given in [31] in the context of strong mixing measures, satisfy either the conditions of Corollary 2.3 of [28] or the spectral criterion in [19] and, therefore, they have the SgSP. The examples provided in [31] include the semigroup generated by a linear perturbation of the one-dimensional Ornstein-Uhlenbeck operator, the solution C - 0 semigroup of a partial differential equation of population dynamics, the solution C - 0 semigroup associated to Banasiak and Moszyn´ski models of birth-and-death processes. Example 18 (Translation semigroups). Let 1 ≤ p < ∞ and let v : R → R be a strictly + (cid:82)b positive locally integrable function, that is, v is measurable with v(x)dx < ∞ for all 0 b > 0. We consider the space of weighted p-integrable functions defined as X = Lp(R ) = {f : R → K ; f is measurable and (cid:107)f(cid:107) < ∞}, v + + where (cid:16)(cid:90) ∞ (cid:17)1/p (cid:107)f(cid:107) = |f(x)|pv(x)dx . 0 The translation semigroup is then given by T f(x) = f(x+t), t,x ≥ 0. t This defines a C -semigroup on Lp(R ) if and only if there exist M ≥ 1 and w ∈ R such 0 v + that, for all t ≥ 0, the following condition v(x) ≤ Mewtv(x+t) for almost all x ≥ 0. is satisfied. In that case, v is called an admissible weight function and we will assume in the sequel that v belongs to this class of weight functions. For the translation semigroup defined on Lp(R ), there was proved in [28] that (T ) v + t t≥0 ischaoticifandonlyifitsatisfiestheFrequentHypercyclicityCriterionforsemigroupsand that (T ) is chaotic if and only if every operator T satisfies the Frequent Hypercyclicity t t≥0 t Criterion for operators. A more complete characterization of the frequently hypercyclic criterion for the translation semigroup on Lp(R ) was given in [29]: v + Theorem 19 (Theorem 3.10, [29]). Let v be an admissible weight function on R. The following assertions are equivalent: (1) The translation semigroup (T ) is frequently hypercyclic on Lp(R ). t t≥0 v + (cid:80) (2) v(k) < ∞. k∈Z (cid:82)∞ (3) v(t)dt < ∞. −∞ (4) (T ) is chaotic on Lp(R ). t t≥0 v + (4) (T ) satisfies the Frequently Hypercyclicity Criterion. t t≥0 This result allows us to give a characterization of the SgSP for the translation semigroup on the space X = Lp(R ). v + Theorem 20. LetusconsiderthetranslationsemigrouponthespaceX = Lp(R ), where v + 1 ≤ p < ∞ and v : R → R is an admissible weight function. We claim that the following + assertions are equivalent: 9 (cid:82)∞ (i) v(x)dx < ∞. 0 (ii) (T ) has SgSP. t t≥0 (iii) (T ) is Devaney chaotic. t t≥0 (iv) (T ) satisfies the Frequently Hypercyclicity Criterion. t t≥0 (v) The translation semigroup (T ) is frequently hypercyclic. t t≥0 Proof. By Theorem 19 [29] and Propositions 10 and 11, it is obvious that for the translation semigroup the SgSP is equivalent to satisfy the Frequently Hypercyclicity Criterion and the SgSP is equivalent to frequently hypercyclic. Acknowledgements The authors are supported by MEC Project MTM2013-47093-P. The second and third authors are also supported by GVA, Project PROMETEOII/2013/013. References [1] A.A. Albanese, X.Barrachina, E.M. Mangino and A.Peris, Distributional chaos for strongly continuous semigroups of operators, Commun. Pure Appl. Anal.12 (2013), pp. 2069–2082. [2] C. Badea and S. Grivaux, Unimodular eigenvalues, uniformly distributed sequences and linear dynamics, Adv. 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Peris, Hypercyclic, topologically mixing and chaotic semigroups on Banach spaces, Studia Math. 170 (2005), pp. 57– 75. [10] F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ. 81 (1973), pp. 637–654. 10

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