| 1. | Departament de Matemàtiques and Centre de Recerca Matemàtica, Edifici Ciències, Universitat Autònoma de Barcelona, Cerdanyola del Vallès, Barcelona 08913, Spain |
| 2. | Departament Informàtica, Matemàtica Aplicada i Estadística, Universitat de Girona, c/ Universitat de Girona, 6, Girona 17003, Spain |
| 3. | Departament de Matemàtiques, Edifici Ciències, Universitat Autònoma de Barcelona, Cerdanyola del Vallès, Barcelona 08913, Spain |
We provide a new and very simple criterion of positive topological entropy for tree maps. We prove that a tree map $ f $ has positive entropy if and only if some iterate $ f^k $ has a periodic orbit with three aligned points consecutive in time, that is, a triplet $ (a,b,c) $ such that $ f^k(a) = b $, $ f^k(b) = c $ and $ b $ belongs to the interior of the unique interval connecting $ a $ and $ c $ (a forward triplet of $ f^k $). We also prove a new criterion of entropy zero for simplicial $ n $-periodic patterns $ P $ based on the non existence of forward triplets of $ f^k $ for any $ 1\le k<n $ inside $ P $. Finally, we study the set $ \mathcal{X}_n $ of all $ n $-periodic patterns $ P $ that have a forward triplet inside $ P $. For any $ n $, we define a pattern that attains the minimum entropy in $ \mathcal{X}_n $ and prove that this entropy is the unique real root in $ (1,\infty) $ of the polynomial $ x^n-2x-1 $.
| [1] |
R. Adler, A. Konheim and M. McAndrew,
Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319.
doi: 10.1090/S0002-9947-1965-0175106-9. |
| [2] |
L. Alsedà, J. Guaschi, J. Los, F. Mañosas and P. Mumbrú,
Canonical representatives for patterns of tree maps, Topology, 36 (1997), 1123-1153.
doi: 10.1016/S0040-9383(96)00039-0. |
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L. Alsedà, D. Juher and F. Mañosas,
Topological and algebraic reducibility for patterns on trees, Ergodic Theory Dynam. Systems, 35 (2015), 34-63.
doi: 10.1017/etds.2013.52. |
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L. Alsedà, D. Juher and F. Mañosas,
On the minimum positive entropy for cycles on trees, Tran. Amer. Math. Soc., 369 (2017), 187-221.
doi: 10.1090/tran6677. |
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L. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, 2$^{nd}$ edition, Advanced Series in Nonlinear Dynamics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000.
doi: 10.1142/4205. |
| [6] |
L. Alsedà and X. Ye,
No division and the set of periods for tree maps, Ergodic Theory Dynam. Systems, 15 (1995), 221-237.
doi: 10.1017/S0143385700008348. |
| [7] |
S. Baldwin,
Generalizations of a theorem of Sarkovskii on orbits of continuous real-valued functions, Discrete Math., 67 (1987), 111-127.
doi: 10.1016/0012-365X(87)90021-5. |
| [8] |
F. Blanchard, E. Glasner, S. Kolyada and A. Maas,
On Li-Yorke pairs, J. Reine Angew. Math., 547 (2002), 51-68.
doi: 10.1515/crll.2002.053. |
| [9] |
L. Block, J. Guckenheimer, M. Misiurewicz and L. S. Young, Periodic points and topological entropy of one-dimensional maps, Global Theory of Dynamical Systems, Lecture Notes in Math., Springer, Berlin, 819 (1980), 18–34. |
| [10] |
A. Blokh,
Trees with snowflakes and zero entropy maps, Topology, 33 (1994), 379-396.
doi: 10.1016/0040-9383(94)90019-1. |
| [11] |
M. A. Blokh, R. Fokkink, J. Mayer, L. Oversteegen and E. Tymchatyn, Fixed point theorems for plane continua with applications, Mem. Amer. Math. Soc., 224 (2013), 97 pp.
doi: 10.1090/S0065-9266-2012-00671-X. |
| [12] |
T.-Y. Li and J. Yorke,
Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992.
doi: 10.1080/00029890.1975.11994008. |
| [13] |
J. Llibre and M. Misiurewicz,
Horseshoes, entropy and periods for graph maps, Topology, 32 (1993), 649-664.
doi: 10.1016/0040-9383(93)90014-M. |
| [14] |
M. Misiurewicz,
Minor cycles for interval maps, Fund. Math., 145 (1994), 281-304.
doi: 10.4064/fm-145-3-281-304. |
| [15] |
M. Misiurewicz and Z. Nitecki, Combinatorial patterns for maps of the interval, Mem. Amer. Math. Soc., 94 (1991), 112 pp.
doi: 10.1090/memo/0456. |
| [16] |
A. N. Sharkovsky,
Co-existence of cycles of a continuous mapping of the line into itself, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 5 (1995), 1263-1273.
doi: 10.1142/S0218127495000934. |
| [17] |
P. Štefan,
A theorem of Šarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line, Comm. Math. Phys., 54 (1977), 237-248.
doi: 10.1007/BF01614086. |
show all references
| [1] |
R. Adler, A. Konheim and M. McAndrew,
Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319.
doi: 10.1090/S0002-9947-1965-0175106-9. |
| [2] |
L. Alsedà, J. Guaschi, J. Los, F. Mañosas and P. Mumbrú,
Canonical representatives for patterns of tree maps, Topology, 36 (1997), 1123-1153.
doi: 10.1016/S0040-9383(96)00039-0. |
| [3] |
L. Alsedà, D. Juher and F. Mañosas,
Topological and algebraic reducibility for patterns on trees, Ergodic Theory Dynam. Systems, 35 (2015), 34-63.
doi: 10.1017/etds.2013.52. |
| [4] |
L. Alsedà, D. Juher and F. Mañosas,
On the minimum positive entropy for cycles on trees, Tran. Amer. Math. Soc., 369 (2017), 187-221.
doi: 10.1090/tran6677. |
| [5] |
L. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, 2$^{nd}$ edition, Advanced Series in Nonlinear Dynamics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000.
doi: 10.1142/4205. |
| [6] |
L. Alsedà and X. Ye,
No division and the set of periods for tree maps, Ergodic Theory Dynam. Systems, 15 (1995), 221-237.
doi: 10.1017/S0143385700008348. |
| [7] |
S. Baldwin,
Generalizations of a theorem of Sarkovskii on orbits of continuous real-valued functions, Discrete Math., 67 (1987), 111-127.
doi: 10.1016/0012-365X(87)90021-5. |
| [8] |
F. Blanchard, E. Glasner, S. Kolyada and A. Maas,
On Li-Yorke pairs, J. Reine Angew. Math., 547 (2002), 51-68.
doi: 10.1515/crll.2002.053. |
| [9] |
L. Block, J. Guckenheimer, M. Misiurewicz and L. S. Young, Periodic points and topological entropy of one-dimensional maps, Global Theory of Dynamical Systems, Lecture Notes in Math., Springer, Berlin, 819 (1980), 18–34. |
| [10] |
A. Blokh,
Trees with snowflakes and zero entropy maps, Topology, 33 (1994), 379-396.
doi: 10.1016/0040-9383(94)90019-1. |
| [11] |
M. A. Blokh, R. Fokkink, J. Mayer, L. Oversteegen and E. Tymchatyn, Fixed point theorems for plane continua with applications, Mem. Amer. Math. Soc., 224 (2013), 97 pp.
doi: 10.1090/S0065-9266-2012-00671-X. |
| [12] |
T.-Y. Li and J. Yorke,
Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992.
doi: 10.1080/00029890.1975.11994008. |
| [13] |
J. Llibre and M. Misiurewicz,
Horseshoes, entropy and periods for graph maps, Topology, 32 (1993), 649-664.
doi: 10.1016/0040-9383(93)90014-M. |
| [14] |
M. Misiurewicz,
Minor cycles for interval maps, Fund. Math., 145 (1994), 281-304.
doi: 10.4064/fm-145-3-281-304. |
| [15] |
M. Misiurewicz and Z. Nitecki, Combinatorial patterns for maps of the interval, Mem. Amer. Math. Soc., 94 (1991), 112 pp.
doi: 10.1090/memo/0456. |
| [16] |
A. N. Sharkovsky,
Co-existence of cycles of a continuous mapping of the line into itself, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 5 (1995), 1263-1273.
doi: 10.1142/S0218127495000934. |
| [17] |
P. Štefan,
A theorem of Šarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line, Comm. Math. Phys., 54 (1977), 237-248.
doi: 10.1007/BF01614086. |
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