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.

[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.

[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.

[4]

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

[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.

[6]

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

[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.

[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.

[9]

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

[10]

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

[11]

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

[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.

[13]

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

[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.

[15]

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

[16]

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

[17]

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

[18]

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

[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.

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[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.

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[32]

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

[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.

[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.

[35]

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

[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.

[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.

[38]

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

[39]

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

[40]

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

[41]

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

[42]

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

[43]

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

[44]

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

[45]

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

[46]

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

[47]

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

[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.

[49]

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

[50]

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

[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.

[52]

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

[53]

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

[54]

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

[55]

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

[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.

[57]

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

[58]

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

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.

[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.

[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.

[4]

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

[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.

[6]

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

[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.

[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.

[9]

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

[10]

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

[11]

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

[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.

[13]

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

[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.

[15]

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

[16]

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

[17]

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

[18]

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

[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.

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[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.

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[32]

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

[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.

[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.

[35]

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

[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.

[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.

[38]

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

[39]

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

[40]

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

[41]

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

[42]

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

[43]

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

[44]

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

[45]

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

[46]

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

[47]

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

[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.

[49]

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

[50]

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

[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.

[52]

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

[53]

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

[54]

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

[55]

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

[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.

[57]

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

[58]

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

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