
-
Previous Article
Pullback attractors for 2D MHD equations on time-varying domains
- DCDS Home
- This Issue
-
Next Article
Propagating fronts for a viscous Hamer-type system
Forward triplets and topological entropy on trees
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 $.
References:
[1] |
R. Adler, A. Konheim and M. McAndrew,
Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319.
doi: 10.1090/S0002-9947-1965-0175106-9. |
[2] |
L. Alsedà, J. Guaschi, J. Los, F. Mañosas and P. Mumbrú,
Canonical representatives for patterns of tree maps, Topology, 36 (1997), 1123-1153.
doi: 10.1016/S0040-9383(96)00039-0. |
[3] |
L. Alsedà, D. Juher and F. Mañosas,
Topological and algebraic reducibility for patterns on trees, Ergodic Theory Dynam. Systems, 35 (2015), 34-63.
doi: 10.1017/etds.2013.52. |
[4] |
L. Alsedà, D. Juher and F. Mañosas,
On the minimum positive entropy for cycles on trees, Tran. Amer. Math. Soc., 369 (2017), 187-221.
doi: 10.1090/tran6677. |
[5] |
L. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, 2$^{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. |
show all references
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. |











[1] |
Boris Hasselblatt, Zbigniew Nitecki, James Propp. Topological entropy for nonuniformly continuous maps. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 201-213. doi: 10.3934/dcds.2008.22.201 |
[2] |
Dongkui Ma, Min Wu. Topological pressure and topological entropy of a semigroup of maps. Discrete and Continuous Dynamical Systems, 2011, 31 (2) : 545-557 . doi: 10.3934/dcds.2011.31.545 |
[3] |
Dante Carrasco-Olivera, Roger Metzger Alvan, Carlos Arnoldo Morales Rojas. Topological entropy for set-valued maps. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3461-3474. doi: 10.3934/dcdsb.2015.20.3461 |
[4] |
Lluís Alsedà, David Juher, Pere Mumbrú. Minimal dynamics for tree maps. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 511-541. doi: 10.3934/dcds.2008.20.511 |
[5] |
José S. Cánovas. Topological sequence entropy of $\omega$–limit sets of interval maps. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 781-786. doi: 10.3934/dcds.2001.7.781 |
[6] |
José M. Amigó, Ángel Giménez. Formulas for the topological entropy of multimodal maps based on min-max symbols. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3415-3434. doi: 10.3934/dcdsb.2015.20.3415 |
[7] |
João Ferreira Alves, Michal Málek. Zeta functions and topological entropy of periodic nonautonomous dynamical systems. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 465-482. doi: 10.3934/dcds.2013.33.465 |
[8] |
Daniel Wilczak, Piotr Zgliczyński. Topological method for symmetric periodic orbits for maps with a reversing symmetry. Discrete and Continuous Dynamical Systems, 2007, 17 (3) : 629-652. doi: 10.3934/dcds.2007.17.629 |
[9] |
Xueting Tian, Paulo Varandas. Topological entropy of level sets of empirical measures for non-uniformly expanding maps. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5407-5431. doi: 10.3934/dcds.2017235 |
[10] |
Ghassen Askri. Li-Yorke chaos for dendrite maps with zero topological entropy and ω-limit sets. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 2957-2976. doi: 10.3934/dcds.2017127 |
[11] |
Steven M. Pederson. Non-turning Poincaré map and homoclinic tangencies in interval maps with non-constant topological entropy. Conference Publications, 2001, 2001 (Special) : 295-302. doi: 10.3934/proc.2001.2001.295 |
[12] |
Wacław Marzantowicz, Feliks Przytycki. Estimates of the topological entropy from below for continuous self-maps on some compact manifolds. Discrete and Continuous Dynamical Systems, 2008, 21 (2) : 501-512. doi: 10.3934/dcds.2008.21.501 |
[13] |
Katrin Gelfert. Lower bounds for the topological entropy. Discrete and Continuous Dynamical Systems, 2005, 12 (3) : 555-565. doi: 10.3934/dcds.2005.12.555 |
[14] |
Jaume Llibre. Brief survey on the topological entropy. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3363-3374. doi: 10.3934/dcdsb.2015.20.3363 |
[15] |
Vincent Delecroix. Divergent trajectories in the periodic wind-tree model. Journal of Modern Dynamics, 2013, 7 (1) : 1-29. doi: 10.3934/jmd.2013.7.1 |
[16] |
Piotr Oprocha, Paweł Potorski. Topological mixing, knot points and bounds of topological entropy. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3547-3564. doi: 10.3934/dcdsb.2015.20.3547 |
[17] |
Michał Misiurewicz. On Bowen's definition of topological entropy. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 827-833. doi: 10.3934/dcds.2004.10.827 |
[18] |
Jan Philipp Schröder. Ergodicity and topological entropy of geodesic flows on surfaces. Journal of Modern Dynamics, 2015, 9: 147-167. doi: 10.3934/jmd.2015.9.147 |
[19] |
Xiaomin Zhou. Relative entropy dimension of topological dynamical systems. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6631-6642. doi: 10.3934/dcds.2019288 |
[20] |
Yun Zhao, Wen-Chiao Cheng, Chih-Chang Ho. Q-entropy for general topological dynamical systems. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 2059-2075. doi: 10.3934/dcds.2019086 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]