# Mixing properties of nearly maximal entropy measures for ${\mathbb{Z}}^{d}$ shifts of finite type

Colloquium Mathematicae (2000)

- Volume: 84/85, Issue: 1, page 43-50
- ISSN: 0010-1354

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topRobinson, E., and Şahin, Ayşe. "Mixing properties of nearly maximal entropy measures for $ℤ^{d}$ shifts of finite type." Colloquium Mathematicae 84/85.1 (2000): 43-50. <http://eudml.org/doc/210807>.

@article{Robinson2000,

abstract = {We prove that for a certain class of $ℤ^d$ shifts of finite type with positive topological entropy there is always an invariant measure, with entropy arbitrarily close to the topological entropy, that has strong metric mixing properties. With the additional assumption that there are dense periodic orbits, one can ensure that this measure is Bernoulli.},

author = {Robinson, E., Şahin, Ayşe},

journal = {Colloquium Mathematicae},

keywords = {entropy; ergodic theory; symbolic dynamics; ; maximal measure; higher-dimensional subshift of finite type; maximal entropy; mixing; variational principles; Bernoulli measures},

language = {eng},

number = {1},

pages = {43-50},

title = {Mixing properties of nearly maximal entropy measures for $ℤ^\{d\}$ shifts of finite type},

url = {http://eudml.org/doc/210807},

volume = {84/85},

year = {2000},

}

TY - JOUR

AU - Robinson, E.

AU - Şahin, Ayşe

TI - Mixing properties of nearly maximal entropy measures for $ℤ^{d}$ shifts of finite type

JO - Colloquium Mathematicae

PY - 2000

VL - 84/85

IS - 1

SP - 43

EP - 50

AB - We prove that for a certain class of $ℤ^d$ shifts of finite type with positive topological entropy there is always an invariant measure, with entropy arbitrarily close to the topological entropy, that has strong metric mixing properties. With the additional assumption that there are dense periodic orbits, one can ensure that this measure is Bernoulli.

LA - eng

KW - entropy; ergodic theory; symbolic dynamics; ; maximal measure; higher-dimensional subshift of finite type; maximal entropy; mixing; variational principles; Bernoulli measures

UR - http://eudml.org/doc/210807

ER -

## References

top- [1] R. Burton and J. E. Steif, Non-uniqueness of measures of maximal entropy for subshifts of finite type, Ergodic Theory Dynam. Systems 14 (1994), 213-235. Zbl0807.58023
- [2] R. Burton and J. E. Steif, Some $2$-d symbolic dynamical systems: entropy and mixing, in: Ergodic Theory of ${\mathbb{Z}}^{d}$ Actions (Warwick, 1993-1994), London Math. Soc. Lecture Note Ser. 228, Cambridge Univ. Press, Cambridge, 1996, 297-305. Zbl0852.58029
- [3] A. Fieldsteel and N. A. Friedman, Restricted orbit changes of ergodic ${\mathbb{Z}}^{d}$-actions to achieve mixing and completely positive entropy, Ergodic Theory Dynam. Systems 6 (1986), 505-528. Zbl0614.28016
- [4] H R. J. Hasfura-Buenaga, The equivalence theorem for ${\mathbb{Z}}^{d}$-actions of positive entropy, ibid. 12 (1992), 725-741. Zbl0813.28007
- [5] A. del Junco and D. J. Rudolph, Kakutani equivalence of ergodic ${\mathbb{Z}}^{n}$ actions, ibid. 4 (1984), 89-104. Zbl0552.28021
- [6] F. Ledrappier, Un champ markovien peut être d'entropie nulle et mélangeant, C. R. Acad. Sci. Paris Sér. A 287 (1978), 561-563. Zbl0387.60084
- [7] M. Misiurewicz, A short proof of the variational principle for a ${\mathbb{Z}}_{+}^{N}$ action on a compact space, Astérisque 40 (1976), 147-157.
- [8] S. Mozes, A zero entropy, mixing of all orders tiling system, in: Symbolic Dynamics and its Applications (New Haven, CT, 1991), Amer. Math. Soc., Providence, RI, 1992, 319-325.
- [9] E. A. Robinson, Jr. and A. A. Şahin, On the absence of invariant measures with locally maximal entropy for a class of ${\mathbb{Z}}^{d}$ shifts of finite type, Proc. Amer. Math. Soc., to appear.
- [10] E. A. Robinson, Modeling ergodic measure preserving actions on ${\mathbb{Z}}^{d}$ shifts of finite type, preprint, 1998.
- [11] T. Ward, Automorphisms of ${\mathbb{Z}}^{d}$-subshifts of finite type, Indag. Math. (N.S.) 5 (1994), 495-504. Zbl0823.28007

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