Specification on the interval

Buzzi, Jérôme

doi:10.1090/S0002-9947-97-01873-4

Specification on the interval
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by Jérôme Buzzi PDF

Trans. Amer. Math. Soc. 349 (1997), 2737-2754 Request permission

Abstract:

We study the consequences of discontinuities on the specification property for interval maps. After giving a necessary and sufficient condition for a piecewise monotonic, piecewise continuous map to have this property, we show that for a large and natural class of families of such maps (including the $\beta$-transformations), the set of parameters for which the specification property holds, though dense, has zero Lebesgue measure. Thus, regarding the specification property, the general case is at the opposite of the continuous case solved by A.M. Blokh (Russian Math. Surveys 38 (1983), 133–134) (for which we give a proof).

References

Michael Benedicks and Lennart Carleson, The dynamics of the Hénon map, Ann. of Math. (2) 133 (1991), no. 1, 73–169. MR 1087346, DOI 10.2307/2944326
A. M. Blokh, Decomposition of dynamical systems on an interval, Uspekhi Mat. Nauk 38 (1983), no. 5(233), 179–180 (Russian). MR 718829
A.M. Blokh, Decomposition of dynamical systems on an interval, Russ. Math. Surv. 38 (1983), 133–134.
Alexander M. Blokh, The “spectral” decomposition for one-dimensional maps, Dynamics reported, Dynam. Report. Expositions Dynam. Systems (N.S.), vol. 4, Springer, Berlin, 1995, pp. 1–59. MR 1346496
Rufus Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971), 401–414. MR 274707, DOI 10.1090/S0002-9947-1971-0274707-X
Manfred Denker, Christian Grillenberger, and Karl Sigmund, Ergodic theory on compact spaces, Lecture Notes in Mathematics, Vol. 527, Springer-Verlag, Berlin-New York, 1976. MR 0457675, DOI 10.1007/BFb0082364
Franz Hofbauer, On intrinsic ergodicity of piecewise monotonic transformations with positive entropy, Israel J. Math. 34 (1979), no. 3, 213–237 (1980). MR 570882, DOI 10.1007/BF02760884
Franz Hofbauer, The structure of piecewise monotonic transformations, Ergodic Theory Dynam. Systems 1 (1981), no. 2, 159–178. MR 661817, DOI 10.1017/s0143385700009202
Franz Hofbauer, Piecewise invertible dynamical systems, Probab. Theory Relat. Fields 72 (1986), no. 3, 359–386. MR 843500, DOI 10.1007/BF00334191
Franz Hofbauer, Generic properties of invariant measures for continuous piecewise monotonic transformations, Monatsh. Math. 106 (1988), no. 4, 301–312. MR 973140, DOI 10.1007/BF01295288
Olga Taussky, An algebraic property of Laplace’s differential equation, Quart. J. Math. Oxford Ser. 10 (1939), 99–103. MR 83, DOI 10.1093/qmath/os-10.1.99
Peter Walters, Equilibrium states for $\beta$-transformations and related transformations, Math. Z. 159 (1978), no. 1, 65–88. MR 466492, DOI 10.1007/BF01174569