December  2015, 20(10): 3363-3374. doi: 10.3934/dcdsb.2015.20.3363

Brief survey on the topological entropy

1. 

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia

Received  December 2014 Revised  February 2015 Published  September 2015

In this paper we give a brief view on the topological entropy. The results here presented are well known to the people working in the area, so this survey is mainly for non--experts in the field.
Citation: Jaume Llibre. Brief survey on the topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3363-3374. doi: 10.3934/dcdsb.2015.20.3363
References:
[1]

R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy,, Trans. Amer. Math. Soc., 114 (1965), 309. doi: 10.1090/S0002-9947-1965-0175106-9. Google Scholar

[2]

R. L. Adler and M. H. McAndrew, The entropy of Chebyshev polynomials,, Trans. Amer. Math. Soc., 121 (1966), 236. doi: 10.1090/S0002-9947-1966-0189005-0. Google Scholar

[3]

L. Alsedà, J. Llibre, F. Mañosas and M. Misiurewicz, Lower bounds of the topological entropy for continuous maps of the circle of degree one,, Nonlinearity, 1 (1988), 463. doi: 10.1088/0951-7715/1/3/004. Google Scholar

[4]

L. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One,, Second edition, 5 (2000). doi: 10.1142/4205. Google Scholar

[5]

L. Alsedà, J. Llibre, M. Misiurewicz and C. Simó, Twist periodic orbits and topological entropy for continuous maps of the circle of degree one which have a fixed point,, Ergod. Th. & Dynam. Sys., 5 (1985), 501. doi: 10.1017/S0143385700003126. Google Scholar

[6]

J. Auslander and Y. Katznelson, Continuous maps of the circle without periodic points,, Israel J. Math., 32 (1979), 375. doi: 10.1007/BF02760466. Google Scholar

[7]

L. Block and E. M. Coven, Topological conjugacy and transitivity for a class of piecewise monotone maps of the interval,, Trans. Amer. Math. Soc., 300 (1987), 297. doi: 10.1090/S0002-9947-1987-0871677-X. Google Scholar

[8]

L. Block, J. Guckenheimer, M. Misiurewicz and L.-S. Young, Periodic points and topological entropy of one dimensional maps,, in Global Theory of Dynamical Systems, (1980), 18. Google Scholar

[9]

A. M. Blokh, On sensitive mappings of the interval,, Russian Math. Surveys, 37 (1982), 189. Google Scholar

[10]

A. M. Blokh, On transitive mappings of one-dimensional branched manifolds,, Differential-Difference Equations and Problems of Mathematical Physics (in Russian), (1984), 3. Google Scholar

[11]

A. M. Blokh, On the connection between entropy and transitivity for one-dimensional mappings,, Russian Math. Surveys, 42 (1987), 209. Google Scholar

[12]

A. M. Blokh, The spectral decomposition, periods of cycles and misiurewicz conjecture for graph maps,, in Ergodic Theory and Related Topics, (1990), 24. doi: 10.1007/BFb0097525. Google Scholar

[13]

R. Bowen, Topological entropy and axiom A,, in Global Analysis (Proc. Symp. Pure Math., (1968), 23. Google Scholar

[14]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces,, Trans. Amer. Math. Soc., 153 (1971), 401. doi: 10.1090/S0002-9947-1971-0274707-X. Google Scholar

[15]

R. Bowen, Entropy for maps of the interval,, Topology, 16 (1977), 465. doi: 10.1016/0040-9383(77)90052-0. Google Scholar

[16]

R. Bowen, Entropy and the fundamental group,, in The Structure of Attractors in Dynamical Systems, (1978), 21. Google Scholar

[17]

P. Boyland, Topological methods in surface dynamics,, Topology Appl., 58 (1994), 223. doi: 10.1016/0166-8641(94)00147-2. Google Scholar

[18]

P. Boyland, Isotopy stability of dynamics on surfaces,, Geometry and Topology in Dynamics (Winston-Salem, (1998), 17. doi: 10.1090/conm/246/03772. Google Scholar

[19]

J. Franks and M. Misiurewicz, Topological methods in dynamics,, in Handbook of Dynamical Systems, (2002), 547. doi: 10.1016/S1874-575X(02)80009-1. Google Scholar

[20]

D. Fried, Entropy and twisted cohomology,, Topology, 25 (1986), 455. doi: 10.1016/0040-9383(86)90024-8. Google Scholar

[21]

D. Fried and M. Shub, Entropy linearity and chain-reccurence,, Publ. Math. de l'IHES, 50 (1979), 203. Google Scholar

[22]

S. Friedland, Entropy of holomorphic and rational maps: A survey,, in Dynamics, (2007), 113. doi: 10.1017/CBO9780511755187.005. Google Scholar

[23]

M. Gromov, Entropy, homology and semialgebraic geometry,, Séminaire Bourbaki, (1987), 1985. Google Scholar

[24]

M. Gromov, Three remarks on the geodesic dynamics and fundamental groups,, L'Enseign. Math., 46 (2000), 391. Google Scholar

[25]

J. Guaschi and J. Llibre, Periodic points of $C^1$ maps and the asymptotic Lefschetz number,, Inter. J. of Bifurcation and Chaos, 5 (1995), 1369. doi: 10.1142/S0218127495001046. Google Scholar

[26]

B. Hasselblatt and A. Katok, Handbook of Dynamical Systems,, Vol. 1A, (2002). Google Scholar

[27]

R. Ito, Rotation sets are closed,, Math. Proc. Cambridge Philos. Soc., 89 (1981), 107. doi: 10.1017/S0305004100057984. Google Scholar

[28]

B. Jiang, Nielsen theory for periodic orbits and applications to dynamical systems,, Contemp. Math., 152 (1993), 183. doi: 10.1090/conm/152/01323. Google Scholar

[29]

B. Jiang, Estimation of the number of periodic orbits,, Pacific J. of Math., 172 (1996), 151. Google Scholar

[30]

V. Y. Kaloshin, Generic diffeomorphisms with superexponential growth of number of periodic orbits,, Comm. Math. Phys., 211 (2000), 253. doi: 10.1007/s002200050811. Google Scholar

[31]

A. Katok, The entropy conjecture,, in Smooth Dynamical Systems (Russian), (1977), 181. Google Scholar

[32]

A. Katok, Fifty years of entropy in dynamics: 1958-2007,, J. Mod. Dyn., 1 (2007), 545. doi: 10.3934/jmd.2007.1.545. Google Scholar

[33]

J. Llibre and M. Misiurewicz, Horseshoes, entropy and periods for graph maps,, Topology, 32 (1993), 649. doi: 10.1016/0040-9383(93)90014-M. Google Scholar

[34]

J. Llibre and R. Saghin, Results and open questions on some of the invariants measuring the dynamical complexity of a map,, Fund. Math., 206 (2009), 307. doi: 10.4064/fm206-0-19. Google Scholar

[35]

A. Manning, Topological entropy and the first homology group,, in Dynamical systems - Warwick 1974, (1974), 185. Google Scholar

[36]

W. Marzantowicz and F. Przytycki, Entropy conjecture for continuous maps of nilmanifolds,, Israel J. of Math., 165 (2008), 349. doi: 10.1007/s11856-008-1015-0. Google Scholar

[37]

W. Marzantowicz and F Przytycki, Estimates of the topological entropy from below for continuous self-maps on some compact manifolds,, Discrete Contin. Dyn. Syst.- Series A, 21 (2008), 501. doi: 10.3934/dcds.2008.21.501. Google Scholar

[38]

J. Milnor and W. Thurston, On iterated maps of the interval,, in Dynamical Systems, (1342), 465. doi: 10.1007/BFb0082847. Google Scholar

[39]

M. Misiurewicz, Horseshoes for mappings of an interval,, Bull. Acad. Pol. Sci., 27 (1979), 167. Google Scholar

[40]

M. Misiurewicz, Horseshoes for continuous mappings of an interval,, in Dynamical Systems, (1980), 127. Google Scholar

[41]

M. Misiurewicz, Twist sets for maps of the circle,, Ergodic Theory & Dynam. Systems, 4 (1984), 391. doi: 10.1017/S0143385700002534. Google Scholar

[42]

M. Misiurewicz, Jumps of entropy in one dimension,, Fund. Math., 132 (1989), 215. Google Scholar

[43]

M. Misiurewicz and F. Przytycki, Topological entropy and degree of smooth mappings,, Bull. Ac. Pol. Sci., 25 (1977), 573. Google Scholar

[44]

M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings,, Studia Math., 67 (1980), 45. Google Scholar

[45]

S. Newhouse, J. Palis and F. Takens, Bifurcations and stability of families of diffeomorphisms,, Inst. Hautes Études Sci. Publ. Math., 57 (1983), 5. Google Scholar

[46]

S. Newhouse, Entropy and volume,, Erg. Th. & Dyn. Syst., 8 (1988), 283. doi: 10.1017/S0143385700009469. Google Scholar

[47]

S. Newhouse, Entropy in smooth dynamical systems,, Proceedings of the International Congress of Mathematicians, (1990), 1285. Google Scholar

[48]

W. Parry, Symbolic dynamics and transformations of the unit interval,, Trans. Amer. Math. Soc., 122 (1966), 368. doi: 10.1090/S0002-9947-1966-0197683-5. Google Scholar

[49]

J. Rothschild, On the Computation of Topological Entropy,, Thesis, (1971). Google Scholar

[50]

D. Ruelle and D. Sullivan, Currents, flows and diffeomorphisms,, Topology, 14 (1975), 319. doi: 10.1016/0040-9383(75)90016-6. Google Scholar

[51]

R. Saghin and Z. Xia, The entropy conjecture for partially hyperbolic diffeomorphism with 1-D center,, Topology Appl., 157 (2010), 29. doi: 10.1016/j.topol.2009.04.053. Google Scholar

[52]

M. Shub, Dynamical Systems, filtrations and entropy,, Bull. Amer. Math. Soc., 80 (1974), 27. doi: 10.1090/S0002-9904-1974-13344-6. Google Scholar

[53]

M. Shub, All, most, some differentiable dynamical systems,, International Congress of Mathematicians, (2006), 99. Google Scholar

[54]

M. Shub and R. Williams, Entropy and stability,, Topology, 14 (1975), 329. doi: 10.1016/0040-9383(75)90017-8. Google Scholar

[55]

Y. Yomdin, Volume growth and entropy,, Israel J. Math., 57 (1987), 285. doi: 10.1007/BF02766215. Google Scholar

[56]

Y. Yomdin, $C^k$-resolution of semialgebraic mappings. Addendum to: "Volume growth and entropy",, Israel J. Math., 57 (1987), 301. doi: 10.1007/BF02766216. Google Scholar

[57]

L. S. Young, Entropy in dynamical systems,, in Entropy, (2003), 313. Google Scholar

[58]

L. S. Young, On the prevalence of horseshoes,, Trans. Amer. Math. Soc., 263 (1981), 75. doi: 10.1090/S0002-9947-1981-0590412-0. Google Scholar

show all references

References:
[1]

R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy,, Trans. Amer. Math. Soc., 114 (1965), 309. doi: 10.1090/S0002-9947-1965-0175106-9. Google Scholar

[2]

R. L. Adler and M. H. McAndrew, The entropy of Chebyshev polynomials,, Trans. Amer. Math. Soc., 121 (1966), 236. doi: 10.1090/S0002-9947-1966-0189005-0. Google Scholar

[3]

L. Alsedà, J. Llibre, F. Mañosas and M. Misiurewicz, Lower bounds of the topological entropy for continuous maps of the circle of degree one,, Nonlinearity, 1 (1988), 463. doi: 10.1088/0951-7715/1/3/004. Google Scholar

[4]

L. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One,, Second edition, 5 (2000). doi: 10.1142/4205. Google Scholar

[5]

L. Alsedà, J. Llibre, M. Misiurewicz and C. Simó, Twist periodic orbits and topological entropy for continuous maps of the circle of degree one which have a fixed point,, Ergod. Th. & Dynam. Sys., 5 (1985), 501. doi: 10.1017/S0143385700003126. Google Scholar

[6]

J. Auslander and Y. Katznelson, Continuous maps of the circle without periodic points,, Israel J. Math., 32 (1979), 375. doi: 10.1007/BF02760466. Google Scholar

[7]

L. Block and E. M. Coven, Topological conjugacy and transitivity for a class of piecewise monotone maps of the interval,, Trans. Amer. Math. Soc., 300 (1987), 297. doi: 10.1090/S0002-9947-1987-0871677-X. Google Scholar

[8]

L. Block, J. Guckenheimer, M. Misiurewicz and L.-S. Young, Periodic points and topological entropy of one dimensional maps,, in Global Theory of Dynamical Systems, (1980), 18. Google Scholar

[9]

A. M. Blokh, On sensitive mappings of the interval,, Russian Math. Surveys, 37 (1982), 189. Google Scholar

[10]

A. M. Blokh, On transitive mappings of one-dimensional branched manifolds,, Differential-Difference Equations and Problems of Mathematical Physics (in Russian), (1984), 3. Google Scholar

[11]

A. M. Blokh, On the connection between entropy and transitivity for one-dimensional mappings,, Russian Math. Surveys, 42 (1987), 209. Google Scholar

[12]

A. M. Blokh, The spectral decomposition, periods of cycles and misiurewicz conjecture for graph maps,, in Ergodic Theory and Related Topics, (1990), 24. doi: 10.1007/BFb0097525. Google Scholar

[13]

R. Bowen, Topological entropy and axiom A,, in Global Analysis (Proc. Symp. Pure Math., (1968), 23. Google Scholar

[14]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces,, Trans. Amer. Math. Soc., 153 (1971), 401. doi: 10.1090/S0002-9947-1971-0274707-X. Google Scholar

[15]

R. Bowen, Entropy for maps of the interval,, Topology, 16 (1977), 465. doi: 10.1016/0040-9383(77)90052-0. Google Scholar

[16]

R. Bowen, Entropy and the fundamental group,, in The Structure of Attractors in Dynamical Systems, (1978), 21. Google Scholar

[17]

P. Boyland, Topological methods in surface dynamics,, Topology Appl., 58 (1994), 223. doi: 10.1016/0166-8641(94)00147-2. Google Scholar

[18]

P. Boyland, Isotopy stability of dynamics on surfaces,, Geometry and Topology in Dynamics (Winston-Salem, (1998), 17. doi: 10.1090/conm/246/03772. Google Scholar

[19]

J. Franks and M. Misiurewicz, Topological methods in dynamics,, in Handbook of Dynamical Systems, (2002), 547. doi: 10.1016/S1874-575X(02)80009-1. Google Scholar

[20]

D. Fried, Entropy and twisted cohomology,, Topology, 25 (1986), 455. doi: 10.1016/0040-9383(86)90024-8. Google Scholar

[21]

D. Fried and M. Shub, Entropy linearity and chain-reccurence,, Publ. Math. de l'IHES, 50 (1979), 203. Google Scholar

[22]

S. Friedland, Entropy of holomorphic and rational maps: A survey,, in Dynamics, (2007), 113. doi: 10.1017/CBO9780511755187.005. Google Scholar

[23]

M. Gromov, Entropy, homology and semialgebraic geometry,, Séminaire Bourbaki, (1987), 1985. Google Scholar

[24]

M. Gromov, Three remarks on the geodesic dynamics and fundamental groups,, L'Enseign. Math., 46 (2000), 391. Google Scholar

[25]

J. Guaschi and J. Llibre, Periodic points of $C^1$ maps and the asymptotic Lefschetz number,, Inter. J. of Bifurcation and Chaos, 5 (1995), 1369. doi: 10.1142/S0218127495001046. Google Scholar

[26]

B. Hasselblatt and A. Katok, Handbook of Dynamical Systems,, Vol. 1A, (2002). Google Scholar

[27]

R. Ito, Rotation sets are closed,, Math. Proc. Cambridge Philos. Soc., 89 (1981), 107. doi: 10.1017/S0305004100057984. Google Scholar

[28]

B. Jiang, Nielsen theory for periodic orbits and applications to dynamical systems,, Contemp. Math., 152 (1993), 183. doi: 10.1090/conm/152/01323. Google Scholar

[29]

B. Jiang, Estimation of the number of periodic orbits,, Pacific J. of Math., 172 (1996), 151. Google Scholar

[30]

V. Y. Kaloshin, Generic diffeomorphisms with superexponential growth of number of periodic orbits,, Comm. Math. Phys., 211 (2000), 253. doi: 10.1007/s002200050811. Google Scholar

[31]

A. Katok, The entropy conjecture,, in Smooth Dynamical Systems (Russian), (1977), 181. Google Scholar

[32]

A. Katok, Fifty years of entropy in dynamics: 1958-2007,, J. Mod. Dyn., 1 (2007), 545. doi: 10.3934/jmd.2007.1.545. Google Scholar

[33]

J. Llibre and M. Misiurewicz, Horseshoes, entropy and periods for graph maps,, Topology, 32 (1993), 649. doi: 10.1016/0040-9383(93)90014-M. Google Scholar

[34]

J. Llibre and R. Saghin, Results and open questions on some of the invariants measuring the dynamical complexity of a map,, Fund. Math., 206 (2009), 307. doi: 10.4064/fm206-0-19. Google Scholar

[35]

A. Manning, Topological entropy and the first homology group,, in Dynamical systems - Warwick 1974, (1974), 185. Google Scholar

[36]

W. Marzantowicz and F. Przytycki, Entropy conjecture for continuous maps of nilmanifolds,, Israel J. of Math., 165 (2008), 349. doi: 10.1007/s11856-008-1015-0. Google Scholar

[37]

W. Marzantowicz and F Przytycki, Estimates of the topological entropy from below for continuous self-maps on some compact manifolds,, Discrete Contin. Dyn. Syst.- Series A, 21 (2008), 501. doi: 10.3934/dcds.2008.21.501. Google Scholar

[38]

J. Milnor and W. Thurston, On iterated maps of the interval,, in Dynamical Systems, (1342), 465. doi: 10.1007/BFb0082847. Google Scholar

[39]

M. Misiurewicz, Horseshoes for mappings of an interval,, Bull. Acad. Pol. Sci., 27 (1979), 167. Google Scholar

[40]

M. Misiurewicz, Horseshoes for continuous mappings of an interval,, in Dynamical Systems, (1980), 127. Google Scholar

[41]

M. Misiurewicz, Twist sets for maps of the circle,, Ergodic Theory & Dynam. Systems, 4 (1984), 391. doi: 10.1017/S0143385700002534. Google Scholar

[42]

M. Misiurewicz, Jumps of entropy in one dimension,, Fund. Math., 132 (1989), 215. Google Scholar

[43]

M. Misiurewicz and F. Przytycki, Topological entropy and degree of smooth mappings,, Bull. Ac. Pol. Sci., 25 (1977), 573. Google Scholar

[44]

M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings,, Studia Math., 67 (1980), 45. Google Scholar

[45]

S. Newhouse, J. Palis and F. Takens, Bifurcations and stability of families of diffeomorphisms,, Inst. Hautes Études Sci. Publ. Math., 57 (1983), 5. Google Scholar

[46]

S. Newhouse, Entropy and volume,, Erg. Th. & Dyn. Syst., 8 (1988), 283. doi: 10.1017/S0143385700009469. Google Scholar

[47]

S. Newhouse, Entropy in smooth dynamical systems,, Proceedings of the International Congress of Mathematicians, (1990), 1285. Google Scholar

[48]

W. Parry, Symbolic dynamics and transformations of the unit interval,, Trans. Amer. Math. Soc., 122 (1966), 368. doi: 10.1090/S0002-9947-1966-0197683-5. Google Scholar

[49]

J. Rothschild, On the Computation of Topological Entropy,, Thesis, (1971). Google Scholar

[50]

D. Ruelle and D. Sullivan, Currents, flows and diffeomorphisms,, Topology, 14 (1975), 319. doi: 10.1016/0040-9383(75)90016-6. Google Scholar

[51]

R. Saghin and Z. Xia, The entropy conjecture for partially hyperbolic diffeomorphism with 1-D center,, Topology Appl., 157 (2010), 29. doi: 10.1016/j.topol.2009.04.053. Google Scholar

[52]

M. Shub, Dynamical Systems, filtrations and entropy,, Bull. Amer. Math. Soc., 80 (1974), 27. doi: 10.1090/S0002-9904-1974-13344-6. Google Scholar

[53]

M. Shub, All, most, some differentiable dynamical systems,, International Congress of Mathematicians, (2006), 99. Google Scholar

[54]

M. Shub and R. Williams, Entropy and stability,, Topology, 14 (1975), 329. doi: 10.1016/0040-9383(75)90017-8. Google Scholar

[55]

Y. Yomdin, Volume growth and entropy,, Israel J. Math., 57 (1987), 285. doi: 10.1007/BF02766215. Google Scholar

[56]

Y. Yomdin, $C^k$-resolution of semialgebraic mappings. Addendum to: "Volume growth and entropy",, Israel J. Math., 57 (1987), 301. doi: 10.1007/BF02766216. Google Scholar

[57]

L. S. Young, Entropy in dynamical systems,, in Entropy, (2003), 313. Google Scholar

[58]

L. S. Young, On the prevalence of horseshoes,, Trans. Amer. Math. Soc., 263 (1981), 75. doi: 10.1090/S0002-9947-1981-0590412-0. Google Scholar

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