Abstract
The folding entropy is a quantity originally proposed by Ruelle in 1996 during the study of entropy production in the non-equilibrium statistical mechanics [53]. As derived through a limiting process to the non-equilibrium steady state, the continuity of entropy production plays a key role in its physical interpretations. In this paper, the continuity of folding entropy is studied for a general (non-invertible) differentiable dynamical system with degeneracy. By introducing a notion called degenerate rate, it is proved that on any subset of measures with uniform degenerate rate, the folding entropy, and hence the entropy production, is upper semi-continuous. This extends the upper semi-continuity result in [53] from endomorphisms to all Cr (r > 1) maps.
We further apply our result in the one-dimensional setting. In achieving this, an equality between the folding entropy and (Kolmogorov–Sinai) metric entropy, as well as a general dimension formula are established. The upper semi-continuity of metric entropy and dimension are then valid when measures with uniform degenerate rate are considered. Moreover, the sharpness of the uniform degenerate rate condition is shown by examples of Cr interval maps with positive metric (and folding) entropy.
Article PDF
Avoid common mistakes on your manuscript.
References
L. Andrey, The rate of entropy change in non-hamiltonian systems, Physics Letters. A 111 (1985), 45–46.
B. Bárány and A Käenmäki, Ledrappier–Young formula and exact dimensionality of self-affine measures, Advances in Mathematics 318 (2017), 88–129.
L Barreira, Y. Pesin and J Schmeling, Dimension and product structure of hyperbolic measures, Annals of Mathematics 149 (1999), 755–783.
M. Benedicks and L. Carleson, On iterations of 1 − ax2on (−1, 1), Annals of Mathematics 122 (1985), 1–25.
R. Bowen, Entropy expansive maps, Transactions of the American Mathematical Society 164 (1972), 323–331.
R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer, Berlin, 2008.
K. Brin and A. Katok, On local entropy, in Geometric Dynamics (Rio de Janeiro, 1981), Lecture Notes in Mathematics, Vol. 1007, Springer, Berlin, 1983, pp. 30–38.
D. Burguet, Existence of measures of maximal entropy for Crinterval maps, Proceedings of the American Mathematical Society 142 (2014), 957–968.
J. Buzzi, Intrinsic ergodicity of smooth interval maps, Israel Journal of Mathematics 100 (1997), 125–161.
W.-C. Cheng and S. Newhouse, Pre-image entropy, Ergodic Theory and Dynamical Systems 25 (2005), 1091–1113.
P. Collet and J.-P. Eckmann, On the abundance of aperiodic behaviour for maps on the interval, Communications in Mathematical Physics 73 (1980), 115–160.
W. Cowieson and L.-S. Young, SRB measures as zero-noise limits, Ergodic Theory and Dynamical Systems 25 (2005), 1115–1138.
L. Díaz, T. Fisher, M. Pacifico and J. Vieitez, Entropy-expansiveness for partially hyperbolic diffeomorphisms, Discrete and Continuous Dynamical Systems 32 (2012), 4195–4207.
J-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Reviews of Modern Physics 57 (1985), 617–656.
D. J. Evans, Response theory as a free-energy extremum, Physical Review. A 32 (1985), 2923–2925.
D. J. Evans, E. G. D. Cohen and G. P. Morriss, Viscosity of a simple fluid from its maximal Lyapunov exponents, Physical Review. A 42 (1990), 5990–5997.
D. J. Evans, E. G. D. Cohen and G. P. Morriss, Probability of second law violations in shearing steady flows, Physical Review Letters 71 (1993), 2401–2404.
K. Falconer and X. Jin, Exact dimensionality and projections of random self-similar measures and sets, Journal of the London Mathematical Society 90 (2014), 388–412.
D. J. Feng and H. Hu, Dimension theory of iterated function systems, Communications in Pure and Applied Mathematics 62 (2009), 1435–1500.
G. Gallavotti, Entropy production and thermodynamics of nonequilibrium stationary states: a point of view, Chaos 14 (2004), 680–690.
G. Gallavotti and E. G. D. Cohen, Dynamical ensembles in nonequilibrium statistical mechanics, Physical Review Letters 74 (1995), 2694–2697.
G. Gallavotti and E. G. D. Cohen, Dynamical ensembles in stationary states, Journal of Statistical Physics 80 (1995), 931–970.
G. Gallavotti and D. Ruelle, SRB states and nonequilibrium statistical mechanics close to equilibrium, Communications in Mathematical Physics 190 (1997), 279–285.
W. G. Hoover and H. A. Posch, Direct measurement of equilibrium and nonequilibrium Lyapunov spectra, Physics Letters. A 123 (1987), 227–230.
D.-Q. Jiang, M. Qian and M.-P. Qian, Mathematical Theory of Nonequilibrium Steady States, Lecture Notes in Mathematics, Vol. 1833, Springer, Berlin, 2004.
B. Kalinin and V. Sadovskaya, On pointwise dimension of non-hyperbolic measures, Ergodic Theory and Dynamical Systems 22 (2002), 1783–1801.
A. I. Khinchin, Mathematical Foundations of Statistical Mechanics, Dover, New York, 1949.
F Ledrappier, Some relations between dimension and Lyapunov exponents, Communications in Mathematical Physics 81 (1981), 229–238.
F. Ledrappier and M. Misiurewicz, Dimension of invariant measures for maps exponent zero, Ergodic Theory and Dynamical Systems 5 (1985), 595–610.
F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula; II. Relations between entropy, exponents and dimension, Annals of Mathematics 122 (1985), 509–539; 540–574.
G. Liao, M. Viana and J. Yang, The entropy conjecture for diffeomorphisms away from tangencies, Journal of the European Mathematical Society 15 (2013), 2043–2060.
G. Liao and S. Wang, Ruelle inequality of folding type for C1+αmaps, Mathematische Zeitshrift 290 (2018), 509–519.
P.-D. Liu, Ruelle inequality relating entropy, folding entropy and negative Lyapunov exponents, Communications in Mathematical Physics 240 (2003), 531–538.
R. Mañé, A proof of Pesin’s formula, Ergodic Theory and Dynamical Systems 1 (1981), 95–102.
R. Mñné, The Hausdorff dimension of invariant probabilities of rational maps, in Dynamical Systems, Valparaiso 1986, Lecture Notes in Mathematics, Vol. 1331, Springer, Berlin, 1986, pp. 86–117.
A. Manning, The dimension of the maximal measure for a polynomial map, Annals of Mathematics 119 (1984), 425–430.
E. Mihailescu, Physical measures for multivalued inverse iterates near hyperbolic repellors, Journal of Statistical Physics 139 (2010), 800–819.
E. Mihailescu, Thermodynamic formalism for invariant measures in iterated function systems with overlaps, Communications in Contemporary Mathematics 24 (2022), 931–970.
E. Mihailescu and M. Urbański, Entropy production for a class of inverse SRB measures, Journal of Statistical Physics 150 (2013), 881–888.
E. Mihailescu and M. Urbański, Measure-theoretic degrees and topological pressure for non-expanding transformations, Journal of Functional Analysis 267 (2014), 2823–2845.
E. Mihailescu and M. Urbański, Random countable iterated function systems with overlaps and applications, Advances in Mathematics 298 (2016), 726–758.
M. Misiurewicz, On non-continuity of topological entropy, Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 19 (1971), 319–320.
M. Misiurewicz, Diffeomorphism without any measure with maximal entropy, Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 21 (1973), 903–910.
S. Newhouse, Continuity properties of entropy, Annals of Mathematics 129 (1989), 215–235.
V. I. Oseledec, A multiplicative ergodic theorem, Trudy Moskovskogo Matematičeskogo Obščestva 19 (1968), 179–210.
Y. Pesin and H. Weiss, On the dimension of deterministic and random cantor-like sets, symbolic dynamics, and the Eckmann–Ruelle conjecture, Communications in Mathematical Physics 182 (1996), 105–153.
Y. B. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory, Russian Mathematical Surveys 32 (1977), 55–114.
Y. B. Pesin, Dimension Theory in Dynamical Systems, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997.
F. Przytycki and M. Urbański, Conformal Fractals: Ergodic Theory Methods, London Mathematical Society Lecture Note Series, Vol. 371, Cambridge University Press, Cambridge, 2010.
M.-P. Qian, M. Qian and G.-L. Gong, The reversibility and the entropy production of Markov processes, in Probability Theory and its Applications in China, Contemporary Mathematics, Vol. 118, American Mathematical Society, Providence, RI, 1991, pp. 255–261.
V. A. Rokhlin, Lectures on the entropy theory of measure-preserving transformations, Russian Mathematical Surveys 22 (1967), 1–52.
D. Ruelle, An inequality for the entropy of differentiable maps, Boletim da Sociedade Brasileira de Matemática 9 (1978), 83–87.
D. Ruelle, Positivity of entropy production in nonequilibrium statistical mechanics, Journal of Statistical Physics 85 (1996), 1–23.
D. Ruelle, Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics, Journal of Statistical Physics 95 (1999), 393–468.
S. Ruette, Mixing Crmaps of the interval without maximal measure, Israel Journal of Mathematical 127 (2002), 253–277.
U. Seifert, Stochastic thermodynamics, ñuctuation theorems and molecular machines, Reports on Progress in Physics 75 (2012), Article no. 126001.
P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, Vol. 79, Springer, New York–Berlin, 1982.
W. Wu and Y. Zhu, On preimage entropy, folding entropy and stable entropy, Ergodic Theory and Dynamical Systems 41 (2021), 1217–1249.
W. Wu and Y. Zhu, Entropy via preimage structure, Advances in Mathematics 406 (2022), Article no. 108483.
Y. Yomdin, Volume growth and entropy, Israel Journal of Mathematics 57 (1987), 285–300.
L.-S. Young, Dimension, entropy and Lyapunov exponents, Ergodic Theory and Dynamical Systems 2 (1982), 109–124.
X.-J. Zhang, H. Qian and M. Qian, Stochastic theory of nonequilibrium steady states and its applications. Part I, Physics Reports 510 (2012), 1–86.
Acknowledgement
The authors are grateful to the anonymous referees whose thoughtful comments helped improve the exposition.
Author information
Authors and Affiliations
Corresponding author
Additional information
G. Liao was partially supported by the National Key R&D Program of China (2022YFA1005802), NSFC(11701402, 11790274, 12071328, 12122113), BK20170327, BK20211595, Double Innovation Plan of Jiangsu Province of China and Tang Scholar.
S. Wang was supported by NSFC(12201244, 12271204), and a faculty development grant from Jilin University.
Rights and permissions
About this article
Cite this article
Liao, G., Wang, S. Continuity properties of folding entropy. Isr. J. Math. (2024). https://doi.org/10.1007/s11856-024-2653-6
Received:
Revised:
Published:
DOI: https://doi.org/10.1007/s11856-024-2653-6