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January  2020, 40(1): 319-329. doi: 10.3934/dcds.2020012

A shift map with a discontinuous entropy function

Department of Mathematics, The City College of New York, New York, NY, 10031, USA

* Corresponding author: Christian Wolf

Received  January 2019 Revised  July 2019 Published  October 2019

Fund Project: Christian Wolf was partially supported by a grant from the PSC-CUNY (TRADB-49-253 to Christian Wolf).

Let $ f:X\to X $ be a continuous map on a compact metric space with finite topological entropy. Further, we assume that the entropy map $ \mu\mapsto h_\mu(f) $ is upper semi-continuous. It is well-known that this implies the continuity of the localized entropy function of a given continuous potential $ \phi:X\to {\mathbb R} $. In this note we show that this result does not carry over to the case of higher-dimensional potentials $ \Phi:X\to {\mathbb R}^m $. Namely, we construct for a shift map $ f $ a $ 2 $-dimensional Lipschitz continuous potential $ \Phi $ with a discontinuous localized entropy function.

Citation: Christian Wolf. A shift map with a discontinuous entropy function. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 319-329. doi: 10.3934/dcds.2020012
References:
[1]

L. BarreiraY. Pesin and J. Schmeling, On a general concept of multifractality: Multifractal spectra for dimensions, entropies, and Lyapunov exponents. Multifractal rigidity, Chaos, 7 (1997), 27-38.  doi: 10.1063/1.166232.  Google Scholar

[2]

L. BarreiraB. Saussol and J. Schmeling, Higher-dimensional multifractal analysis, Journal de Mathématiques Pures et Appliquées, 81 (2002), 67-91.  doi: 10.1016/S0021-7824(01)01228-4.  Google Scholar

[3]

A. Blokh, Functional rotation numbers for one dimensional maps, Transactions of the American Mathematical Society, 347 (1995), 499-513.  doi: 10.1090/S0002-9947-1995-1270659-0.  Google Scholar

[4]

V. Climenhaga, Topological pressure of simultaneous level sets, Nonlinearity, 26 (2013), 241-268.  doi: 10.1088/0951-7715/26/1/241.  Google Scholar

[5]

L. DiazT. FisherM. Pacifico and J. Vieitez, Entropy-expansiveness for partially hyperbolic diffeomorphisms, Discrete Continuous Dynamical Systems, 32 (2012), 4195-4207.  doi: 10.3934/dcds.2012.32.4195.  Google Scholar

[6]

A. FanD. Feng and J. Wu, Recurrence, dimension and entropy, Journal of the London Mathematical Society, 64 (2001), 229-244.  doi: 10.1017/S0024610701002137.  Google Scholar

[7]

A. Fan, J. Schmeling and M. Wu, The multifractal spectra of V-statistics, in Further Developments in Fractals and Related Fields, Trends Math., Birkhauser/Springer, New York, 2013,135–151. doi: 10.1007/978-0-8176-8400-6_7.  Google Scholar

[8]

D. GaleV. Klee and R. T. Rockafellar, Convex functions on convex polytopes, Proceedings American Mathematical Society, 19 (1968), 867-873.  doi: 10.1090/S0002-9939-1968-0230219-6.  Google Scholar

[9]

E. Garibaldi and A. O. Lopes, Functions for relative maximization, Dynamical Systems, 22 (2007), 511-528.  doi: 10.1080/14689360701582378.  Google Scholar

[10]

W. Geller and M. Misiurewicz, Rotation and entropy, Transactions of the American Mathematical Society, 351 (1999), 2927-2948.  doi: 10.1090/S0002-9947-99-02344-2.  Google Scholar

[11]

P. GiuliettiB. KloecknerA. O. Lopes and D. Marcon, The calculus of thermodynamic formalism, Journal of the European Mathematical Society, 20 (2018), 2357-2412.  doi: 10.4171/JEMS/814.  Google Scholar

[12]

O. Jenkinson, Rotation, entropy, and equilibrium states, Transactions of the American Mathematical Society, 353 (2001), 3713-3739.  doi: 10.1090/S0002-9947-01-02706-4.  Google Scholar

[13]

T. Kucherenko and C. Wolf, Geometry and entropy of generalized rotation sets, Israel Journal of Mathematics, 199 (2014), 791-829.  doi: 10.1007/s11856-013-0053-4.  Google Scholar

[14]

T. Kucherenko and C. Wolf, Entropy and rotation sets: A toymodel approach, Communications in Contemporary Mathematics 18 (2016), 1550083, 23 pp. doi: 10.1142/S0219199715500832.  Google Scholar

[15]

T. Kucherenko and C. Wolf, Ground states and zero-temperature measures at the boundary of rotation sets, Ergodic Theory and Dynamical Systems, 39 (2019), 201-224.  doi: 10.1017/etds.2017.27.  Google Scholar

[16]

S. Newhouse, Continuity properties of entropy, Annals of Mathematics, 129 (1989), 215-235.  doi: 10.2307/1971492.  Google Scholar

[17]

K. R. Parthasarathy, On the category of ergodic measures, Illinois Journal of Mathematics, 5 (1961), 648-656.  doi: 10.1215/ijm/1255631586.  Google Scholar

[18]

R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970.  Google Scholar

[19]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[20]

K. Ziemian, Rotation sets for subshifts of finite type, Fundamenta Mathematicae, 146 (1995), 189-201.   Google Scholar

show all references

References:
[1]

L. BarreiraY. Pesin and J. Schmeling, On a general concept of multifractality: Multifractal spectra for dimensions, entropies, and Lyapunov exponents. Multifractal rigidity, Chaos, 7 (1997), 27-38.  doi: 10.1063/1.166232.  Google Scholar

[2]

L. BarreiraB. Saussol and J. Schmeling, Higher-dimensional multifractal analysis, Journal de Mathématiques Pures et Appliquées, 81 (2002), 67-91.  doi: 10.1016/S0021-7824(01)01228-4.  Google Scholar

[3]

A. Blokh, Functional rotation numbers for one dimensional maps, Transactions of the American Mathematical Society, 347 (1995), 499-513.  doi: 10.1090/S0002-9947-1995-1270659-0.  Google Scholar

[4]

V. Climenhaga, Topological pressure of simultaneous level sets, Nonlinearity, 26 (2013), 241-268.  doi: 10.1088/0951-7715/26/1/241.  Google Scholar

[5]

L. DiazT. FisherM. Pacifico and J. Vieitez, Entropy-expansiveness for partially hyperbolic diffeomorphisms, Discrete Continuous Dynamical Systems, 32 (2012), 4195-4207.  doi: 10.3934/dcds.2012.32.4195.  Google Scholar

[6]

A. FanD. Feng and J. Wu, Recurrence, dimension and entropy, Journal of the London Mathematical Society, 64 (2001), 229-244.  doi: 10.1017/S0024610701002137.  Google Scholar

[7]

A. Fan, J. Schmeling and M. Wu, The multifractal spectra of V-statistics, in Further Developments in Fractals and Related Fields, Trends Math., Birkhauser/Springer, New York, 2013,135–151. doi: 10.1007/978-0-8176-8400-6_7.  Google Scholar

[8]

D. GaleV. Klee and R. T. Rockafellar, Convex functions on convex polytopes, Proceedings American Mathematical Society, 19 (1968), 867-873.  doi: 10.1090/S0002-9939-1968-0230219-6.  Google Scholar

[9]

E. Garibaldi and A. O. Lopes, Functions for relative maximization, Dynamical Systems, 22 (2007), 511-528.  doi: 10.1080/14689360701582378.  Google Scholar

[10]

W. Geller and M. Misiurewicz, Rotation and entropy, Transactions of the American Mathematical Society, 351 (1999), 2927-2948.  doi: 10.1090/S0002-9947-99-02344-2.  Google Scholar

[11]

P. GiuliettiB. KloecknerA. O. Lopes and D. Marcon, The calculus of thermodynamic formalism, Journal of the European Mathematical Society, 20 (2018), 2357-2412.  doi: 10.4171/JEMS/814.  Google Scholar

[12]

O. Jenkinson, Rotation, entropy, and equilibrium states, Transactions of the American Mathematical Society, 353 (2001), 3713-3739.  doi: 10.1090/S0002-9947-01-02706-4.  Google Scholar

[13]

T. Kucherenko and C. Wolf, Geometry and entropy of generalized rotation sets, Israel Journal of Mathematics, 199 (2014), 791-829.  doi: 10.1007/s11856-013-0053-4.  Google Scholar

[14]

T. Kucherenko and C. Wolf, Entropy and rotation sets: A toymodel approach, Communications in Contemporary Mathematics 18 (2016), 1550083, 23 pp. doi: 10.1142/S0219199715500832.  Google Scholar

[15]

T. Kucherenko and C. Wolf, Ground states and zero-temperature measures at the boundary of rotation sets, Ergodic Theory and Dynamical Systems, 39 (2019), 201-224.  doi: 10.1017/etds.2017.27.  Google Scholar

[16]

S. Newhouse, Continuity properties of entropy, Annals of Mathematics, 129 (1989), 215-235.  doi: 10.2307/1971492.  Google Scholar

[17]

K. R. Parthasarathy, On the category of ergodic measures, Illinois Journal of Mathematics, 5 (1961), 648-656.  doi: 10.1215/ijm/1255631586.  Google Scholar

[18]

R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970.  Google Scholar

[19]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[20]

K. Ziemian, Rotation sets for subshifts of finite type, Fundamenta Mathematicae, 146 (1995), 189-201.   Google Scholar

Figure 1.  The set $ {\mathcal R} = {\mathcal R}(\Phi) $ in Example 1
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