This is a preprint of: “Topological entropy of continuous self-maps on closed surfaces”, Juan Luis Garcia Guirao, Jaume Llibre, Wei Gao, J. Difference Equ. Appl., vol. 26(2), 203–208, 2020. DOI: [10.1080/10236198.2019.1710141] TOPOLOGICAL ENTROPY OF CONTINUOUS SELFMAPS ON CLOSED SURFACES JUAN LUIS GARCÍA GUIRAO1 AND JAUME LLIBRE2 AND WEI GAO1,3 Abstra t. The obje tive of the present work is to present su ient on- ditions for having positive topologi al entropy for ontinuous selfmaps dened on a losed surfa e by using the a tion of this map on the homo- logi al groups of the losed surfa e. 1. Introdu tion Along this work by a losed surfa e we denote a onne ted ompa t surfa e with or without boundary, orientable or not. More pre isely, an orientable on- ne ted ompa t surfa e without boundary of genus g ≥ 0, Mg , is homeomorphi to the sphere if g = 0, to the torus if g = 1, or to the onne ted sum of g opies of the torus if g ≥ 2. An orientable onne ted ompa t surfa e with boundary of genus g ≥ 0, Mg,b , is homeomorphi to Mg minus a nite number b > 0 of open dis s having pairwise disjoint losure. In what follows Mg,0 = Mg . A nonorientable onne ted ompa t surfa e without boundary of genus g ≥ 1, Ng , is homeomorphi to the real proje tive plane if g = 1, or to the onne ted sum of g opies of the real proje tive plane if g > 1. A nonorientable onne ted ompa t surfa e with boundary of genus g ≥ 1, Ng,b , is homeomorphi to Ng minus a nite number b > 0 of open dis s having pairwise disjoint losure. In what follows Ng,0 = Ng . Let f : X → X be a ontinuous map on a losed surfa e X. A point x ∈ X is periodi of period n if f n (x) = x and f k (x) 6= x for k = 1, . . . , n − 1. The topologi al entropy of a ontinuous map f : X → X denoted by h(f ) is a nonnegative real number (possibly innite) whi h measures how mu h f mixes up the phase spa e of X. When h(f ) is positive the dynami s of the system is said to be ompli ated and the positivity of h(f ) is used as a measure of the so alled topologi al haos. Here we introdu e the topologi al entropy using the denition of Bowen [4℄. Key words and phrases. Closed surfa e, ontinuous selfmap, Lefs hetz xed point theory, periodi point, set of periods. 2010 Mathemati s Subje t Classi ation: 58F20, 37C05, 37C25, 37C30. 1 2 J.L.G. GUIRAO, J. LLIBRE, W. GAO Sin e it is possible to embedded any surfa e orientable or not in R4 by the Whitney immersion theorem see [11℄, we onsider the distan e between two points of X as the distan e of these two points in R4 . Now, we dene the distan e dn on G by dn (x, y) = max d(f i (x), f i (y)), ∀x, y ∈ G. 0≤i≤n A nite set S is alled (n, ε)separated with respe t to f if for dierent points x, y ∈ S we have dn (x, y) > ε. We denote by Sn the maximal ardinality of an (n, ε)separated set. Dene 1 h(f, ε) = lim sup log Sn . n→∞ n Then h(f ) = lim h(f, ε) ε→0 is the topologi al entropy of f . We have hosen the denition by Bowen be ause, probably it is the shorter one. The lassi al denition was due to Adler, Konheim and M Andrew [1℄. See for instan e the book of Hasselblatt and Katok [7℄ and [3℄ for other equivalent denitions and properties of the topologi al entropy. See [1, 2, 12℄ for more details on the topologi al entropy. Our main results are the following. Theorem 1. Let Mg be an orientable onne ted ompa t surfa e without bound- ary of genus g. Then the following statements hold. / {−1, 0, 1}, then the topologi al entropy of f is positive. (a) If the degree d ∈ (b) If the degree d ∈ {−1, 0, 1} and the number of roots of the hara - teristi polynomial of f∗1 equal to ±1 or 0 taking into a ount their multipli ities is not even, then the topologi al entropy of f is positive. Theorem 2. Let Mg,b , b > 0, be an orientable onne ted ompa t surfa e with boundary of genus g. Then the following statements hold. (a) If 2g + b − 1 is even and the number of roots of the hara teristi poly- nomial of f∗1 equal to ±1 or 0 taking into a ount their multipli ities is not even, then the topologi al entropy of f is positive. (b) If 2g + b − 1 is odd and the number of roots of the hara teristi poly- nomial of f∗1 equal to ±1 or 0 taking into a ount their multipli ities is not odd, then the topologi al entropy of f is positive. Theorem 3. Let Ng,b , b ≥ 0, be a nonorientable onne ted ompa t surfa e with boundary of genus g. Then the following statements hold. (a) If g + b − 1 is even and the number of roots of the hara teristi poly- nomial of f∗1 equal to ±1 or 0 taking into a ount their multipli ities is not even, then the topologi al entropy of f is positive. TOPOLOGICAL ENTROPY OF CONTINUOUS SELFMAPS ON CLOSED SURFACES 3 (b) If g+b−1 is odd and the number of roots of the hara teristi polynomial of f∗1 equal to ±1 or 0 taking into a ount their multipli ities is not odd, then the topologi al entropy of f is positive. 2. Lefs hetz zeta fun tions for surfa es Let f be a ontinuous selfmap dened on Mg,b or Ng,b , respe tively. For a losed surfa e the homologi al groups with oe ients in Q are linear ve tor spa es over Q. We re all the homologi al spa es of Mg,b with oe ients in Q, i.e. Hk (Mg,b , Q) = Q ⊕ .n.k. ⊕ Q, where n0 = 1, n1 = 2g if b = 0, n1 = 2g + b − 1 if b > 0, n2 = 1 if b = 0, and n2 = 0 if b > 0; and the indu ed linear maps f∗k : Hk (Mg,b , Q) → Hk (Mg,b , Q) by f on the homologi al group Hk (Mg,b , Q) are f∗0 = (1), f∗2 = (d) where d is the degree of the map f if b = 0, f∗2 = (0) if b > 0, and f∗1 = A where A is an n1 × n1 integral matrix (see for additional details [10, 13℄). We re all that the homologi al groups of Ng,b with oe ients in Q, i.e. Hk (Ng,b , Q) = Q ⊕ .n.k. ⊕ Q, where n0 = 1, n1 = g + b − 1 and n2 = 0; and the indu ed linear maps are f∗0 = (1) and f∗1 = A where A is an n1 × n1 integral matrix (see again for additional details [10, 13℄). Let f : X → X be a ontinuous map and let X be either Mg,b or Ng,b . Then the Lefs hetz number of f is dened by L(f ) = trace(f∗0 ) − trace(f∗1 ) + trace(f∗2 ). We shall use the Lefs hetz numbers of the iterates of f , i.e. L(f n ). In order to study the whole sequen e {L(f n )}n≥1 it is dened the formal Lefs hetz zeta fun tion of f as ∞ ! X L(f n ) n Zf (t) = exp t . n n=1 The Lefs hetz zeta fun tion is in fa t a generating fun tion for the sequen e of the Lefs hetz numbers L(f n ). From the work of Franks in [6℄ we have for a ontinuous selfmap of a losed surfa e that its Lefs hetz zeta fun tion is the rational fun tion det(I − tf∗1 ) (1) Zf (t) = , det(I − tf∗0 )det(I − tf∗2 ) 4 J.L.G. GUIRAO, J. LLIBRE, W. GAO where in I −tf∗k the I denotes the nk ×nk identity matrix, and det(I −tf∗2 ) = 1 if f∗2 = (0). Then for a ontinuous map f : Mg,b → Mg,b we have   det(I − tA)  (1 − t)(1 − dt) if b = 0,  Zf (t) =   det(I − tA) if b > 0,  1−t and for a ontinuous map f : Ng,b → Ng,b we have det(I − tA) Zf (t) = . 1−t 3. Basi results In this se tion we present the main result stated in Theorem 7 for proving Theorems 1, 2 and 3. Sin e its proof is short and important for this work we provide it here. For a polynomial H(t) we dene H ∗ (t) by H(t) = (1 − t)α (1 + t)β tγ H ∗ (t), where α, β and γ are non-negative integers su h that 1 − t, 1 + t and t do not divide H ∗ (t). The spe tral radii of the maps f∗k are denoted sp(f∗k ), and they are equal to the largest modulus of all the eigenvalues of the linear map f∗k . The spe tral radius of f∗ is sp(f∗ ) = max sp(f∗k ). k=0,...,m The next result is due to Manning [9℄. Theorem 4. Let f : X → X be a ontinuous map on a losed surfa e X. Then log max{1, sp(f∗1 )} ≤ h(f ). Lemma 5. Let f : X → X be a ontinuous map and let X be a losed surfa e. If the topologi al entropy of f is zero, then all the eigenvalues of the indu ed homomorphism f∗1 are zero or root of unity. Proof. Sin e the topologi al entropy is zero, by Theorem 4 we have sp(f∗1 ) = 1. So, all the eigenvalues of f∗1 have modulus in the interval [0, 1] and at least one of them is 1. Then the hara teristi polynomial of f∗1 is of the form tm p(t), where m is a nonnegative integer, positive if the zero is an eigenvalue. And p(t) is a polynomial with integer oe ients and whose independent term a0 is nonzero. Sin e the produ t of all non-zeros eigenvalues of f∗1 is the integer a0 and, these eigenvalues have modulus in (0, 1], we have that any of these eigenvalues an have modulus smaller than one, otherwise we are in ontradi tion with the fa t a0 is an integer. In short, all the nonzero eigenvalues have modulus one, and onsequently a0 = 1. TOPOLOGICAL ENTROPY OF CONTINUOUS SELFMAPS ON CLOSED SURFACES 5 Sin e if a polynomial has integer oe ients, onstant term 1 and all of whose roots have modulus 1, then all of its roots are roots of unity, see [14℄, the lemma follows.  The nth y lotomi polynomial is dened by Y cn (t) = (wk − t), k being wk = e2πik/n a primitive nth root of unity and where k runs over all the relative primes ≤ n. See [8℄ for the properties of these polynomials. Y   1 For a positive integer n the Euler fun tion is ϕ(n) = n 1− . p p|n,p prime It is known that the degree of the polynomial cn (t) is ϕ(n). Note that ϕ(n) is even for n > 2. A proof of the next result an be found in [8℄. Proposition 6. Let ξ be a primitive nth root of the unity and P (t) a poly- nomial with rational oe ients. If P (ξ) = 0 then cn (t)|P (t). The proofs of our results are strongly based in the next theorem. Theorem 7 (Theorem 3.2 of [5℄). Let X be a losed surfa e, f : X → X be a ontinuous selfmap, and let Zf (t) = P (t)/Q(t) be its Lefs hetz zeta fun tion. If P ∗ (t) or Q∗ (t) has odd degree, then the topologi al entropy of f is positive. Proof. From the denitions of a polynomial H ∗ and of the Lefs hetz zeta fun - tion we have P (t) P ∗ (t) Zf (t) = = (1 − t)a (1 + t)b tc ∗ , Q(t) Q (t) where a, b and c are integers. Assume now that the topologi al entropy h(f ) = 0. Then by Lemma 5 all the eigenvalues of the indu ed homomorphisms f∗1 's are zero or roots of unity. Therefore, by (1) all the roots of the polynomials P ∗ (t) and Q∗ (t) are roots of the unity dierent from ±1 and zero. Hen e, by Proposition 6 the polynomials P ∗ (t) and Q∗ (t) are produ t of y lotomi polynomials dierent from c1 (t) = 1 − t and c2 (t) = 1 + t. Consequently P ∗ (t) and Q∗ (t) have even degree be ause all the y lotomi polynomials whi h appear in them have even degree due to the fa t that the Euler fun tion ϕ(n) for n > 2 only takes even values. But this is a ontradi tion with the assumption that P ∗ (t) or Q∗ (t) has odd degree.  6 J.L.G. GUIRAO, J. LLIBRE, W. GAO 4. Proof of Theorems 1, 2 and 3 Proof of Theorem 1. Sin e Mg is an orientable onne ted ompa t surfa e with- out boundary of genus g , then the Lefs hetz zeta fun tion of f is equal to det(I − tA) Zf (t) = , (1 − t)(1 − dt) where d is the degree of f and 2g is the dimension of the hara teristi polyno- mial det(I−tA) of f∗1 = A. Note here that if d ∈ / {−1, 0, 1}, then Q∗ (t) = 1−dt and therefore by Theorem 7 statement (a) of Theorem 1 is proved. Assume now that d ∈ {−1, 0, 1}. Note that in this ase Q(t) = (1−t)(1−dt) and Q∗ (t) = 1. So, by Theorem 7 the main role will be play by the 2g degree polynomial P (t) = det(I −tA) where f∗1 = A. Sin e 2g is even and the number of roots of the hara teristi polynomial of f∗1 equal to ±1 or 0 taking into a ount their multipli ities is not even, then P ∗ (t) has odd degree. Therefore, statement (b) of Theorem 1 follows by the appli ation of Theorem 7.  Proof of Theorem 2. Note now, sin e Mg,b is an orientable onne ted ompa t surfa e with boundary (b > 0) of genus g , then the Lefs hetz zeta fun tion of f is equal to det(I − tA) Zf (t) = 1−t being 2g + b − 1 the degree of the hara teristi polynomial det(I − tA) of f∗1 = A. Now the proof is similar to the statements (b) and ( ) of Theorem 1.  Proof of Theorem 3. For a nonorientable onne ted ompa t surfa e with or without boundary (b ≥ 0) of genus g ≥ 1, the Lefs hetz zeta fun tion of f is equal to det(I − tA) Zf (t) = 1−t being g + b − 1 the degree of the hara teristi polynomial det(I − tA) of f∗1 = A. Then the proof if this theorems follows in a similar way to the proof of statements (b) and ( ) of Theorem 1.  A knowledgements The se ond author is partially supported by the Ministerio de E onomía, Industria y Competitividad, Agen ia Estatal de Investiga ión grants MTM- 2016-77278-P (FEDER) and MDM-2014-0445, the Agèn ia de Gestió d'Ajuts Universitaris i de Re er a grant 2017SGR1617, and the H2020 European Re- sear h Coun il grant MSCA-RISE-2017-777911. TOPOLOGICAL ENTROPY OF CONTINUOUS SELFMAPS ON CLOSED SURFACES 7 Referen es [1℄ R.L. Adler, A.G. Konheim and M.H. M Andrew, Topologi al entropy, Trans. Amer. Math. So . 114 (1965), 309319. [2℄ L. Alseda, J. Llibre and M. Misiurewi z, Combinatorial dynami s and entropy in dimension one, Se ond edition, Advan ed Series in Nonlinear Dynami s Vol. 5, World S ienti Publishing Co., In ., River Edge, NJ, 2000. [3℄ F. Balibrea, On problems of Topologi al Dynami s in non-autonomous dis rete sys- tems, Applied Mathemati s and Nonlinear S ien es 1(2) (2016), 391404. [4℄ R. Bowen, Entropy for group endomorphisms and homogeneous spa es, Trans. Amer. Math. 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Walters, An Introdu tion to Ergodi Theory. Springer-Verlag, 1992. [13℄ J.W. Vi ks, Homology theory. An introdu tion to algebrai topology, SpringerVerlag, New York, 1994. A ademi Press, New York, 1973. [14℄ L.C. Washington, Introdu tion to y lotomi elds, Springer, Berlin, 1982. 1 Departamento de Matemáti a Apli ada y Estadísti a. Universidad Polité - ni a de Cartagena, Hospital de Marina, 30203-Cartagena, Región de Mur ia, Spain. E-mail address : juan.gar iaup t.es 2 Departament de Matemàtiques. Universitat Autònoma de Bar elona, Bel- laterra, 08193-Bar elona, Catalonia, Spain E-mail address : jllibremat.uab. at 3 S hool of Information S ien e and Te hnology, Yunnan Normal Univer- sity, Kunming 650500, China E-mail address : gaoweiynnu.edu. n