We study some notions of cohomology for asymptotically additive sequences and prove a Livšic-type result for almost additive sequences of potentials. As a consequence, we are able to characterize almost additive sequences based on their equilibrium measures and also show how to obtain almost (and asymptotically) additive sequences of Hölder continuous functions satisfying the bounded variation condition (with a unique equilibrium measure) and which are not physically equivalent to any additive sequence generated by a Hölder continuous function. Moreover, we also use our main result to suggest a classification of almost additive sequences based on physical equivalence relations with respect to the classical additive setup.
REFERENCES
1.
D.-J.
Feng
and W.
Huang
, “Lyapunov spectrum of asymptotically sub-additive potentials
,” Commun. Math. Phys.
297
, 1
–43
(2010
).2.
N.
Cuneo
, “Additive, almost additive and asymptotically additive potential sequences are equivalent
,” Commun. Math. Phys.
377
, 2579
–2595
(2020
).3.
D.
Ruelle
, Thermodynamic Formalism
, Encyclopedia of Mathematics and its Applications
(Addison-Wesley
, 1978
), Vol. 5
.4.
A. C. D.
van Enter
, R.
Fernández
, and A. D.
Sokal
, “Regularity properties and pathologies of position-space renormalization-group transformations: Scope and limitations of Gibbsian theory
,” J. Stat. Phys.
72
, 879
–1167
(1993
).5.
A.
Lopes
, S.
Lopes
, and P.
Varandas
, “Bayes posterior convergence for loss functions via almost additive thermodynamic formalism
,” J. Stat. Phys.
186
, 35
(2022
).6.
L.
Barreira
, “Nonadditive thermodynamic formalism: Equilibrium and Gibbs measures
,” Discrete Contin. Dyn. Syst. A
16
, 279
–305
(2006
).7.
R.
Bowen
, “Some systems with unique equilibrium states
,” Math. Syst. Theory
8
, 193
–202
(1974/75
).8.
A.
Mummert
, “The thermodynamic formalism for almost-additive sequences
,” Discrete Contin. Dyn. Syst. A
16
, 435
–454
(2006
).9.
J.
Barral
and M.
Mensi
, “Gibbs measures on self-affine Sierpinski carpets and their singularity spectrum
,” Ergod. Theory Dyn. Syst.
27
, 1419
–1443
(2007
).10.
J.
Bochi
, “Ergodic optimization of Birkhoff averages and Lyapunov exponents
,” in International Congress of Mathematicians (ICM 2018)
(World Scientific
, 2019
), pp. 1825
–1846
.11.
J.
Bochi
and M.
Rams
, “The entropy of Lyapunov-optimizing measures of some matrix cocycles
,” J. Mod. Dyn.
10
, 255
–286
(2016
).12.
T.
Bomfim
, R.
Huo
, P.
Varandas
, and Y.
Zhao
, “Typical properties of ergodic optimization for asymptotically additive potentials
,” Stoch. Dyn.
23
, 2250024
(2023
).13.
J.-R.
Chazottes
and M.
Hochman
, “On the zero-temperature limit of Gibbs states
,” Commun. Math. Phys.
297
, 265
–281
(2010
).14.
G.
Contreras
, “Ground states are generically a periodic orbit
,” Invent. Math.
205
, 383
–412
(2016
).15.
G.
Contreras
, A. O.
Lopes
, and P.
Thieullen
, “Lyapunov minimizing measures for expanding maps of the circle
,” Ergod. Theory Dyn. Syst.
21
, 1379
–1409
(2001
).16.
O.
Jenkinson
, “Ergodic optimization
,” Discrete Contin. Dyn. Syst. A
15
, 197
–224
(2006
).17.
O.
Jenkinson
, “Ergodic optimization in dynamical systems
,” Ergod. Theory Dyn. Syst.
39
, 2593
–2618
(2019
).18.
O.
Jenkinson
and M.
Pollicott
, “Joint spectral radius, Sturmian measures and the finiteness conjecture
,” Ergod. Theory Dyn. Syst.
38
, 3062
–3100
(2018
).19.
I. D.
Morris
, “Maximizing measures of generic Hölder functions have zero entropy
,” Nonlinearity
21
, 993
–1000
(2008
).20.
Y.
Zhao
, “Constrained ergodic optimization for asymptotically additive potentials
,” J. Math. Anal. Appl.
474
, 612
–639
(2019
).21.
E.
Garibaldi
and J. T. A.
Gomes
, “Aubry set for asymptotically sub-additive potentials
,” Stoch. Dyn.
16
, 1660009
(2016
).22.
L.
Barreira
and P.
Doutor
, “Almost additive multifractal analysis
,” J. Math. Pures Appl.
92
, 1
–17
(2009
).23.
L.
Barreira
, Y.
Cao
, and J.
Wang
, “Multifractal analysis of asymptotically additive sequences
,” J. Stat. Phys.
153
, 888
–910
(2013
).24.
L.
Barreira
and B.
Saussol
, “Variational principles and mixed multifractal spectra
,” Trans. Am. Math. Soc.
353
, 3919
–3944
(2001
).25.
L.
Barreira
, B.
Saussol
, and J.
Schmeling
, “Higher-dimensional multifractal analysis
,” J. Math. Pures Appl.
81
, 67
–91
(2002
).26.
A. N.
Livsic
, “Cohomology of dynamical systems
,” Math. U.S.S.R. Izv.
6
, 1278
–1301
(1972
).27.
B.
Bárány
, A.
Käenmäki
, and I.
Morris
, “Domination, almost additivity, and thermodynamical formalism for planar matrix cocycles
,” Isr. J. Math.
239
, 173
–214
(2020
).28.
L.
Barreira
and J.
Schmeling
, “Sets of ‘non-typical’ points have full topological entropy and full Hausdorff dimension
,” Isr. J. Math.
116
, 29
–70
(2000
).29.
A.
Katok
and B.
Hasselblatt
, Introduction to the Modern Theory of Dynamical Systems
, Encyclopedia of Mathematics and its Applications
(Cambridge University Press
, 1995
), Vol. 54
.30.
M.
Viana
and K.
Oliveira
, Foundations of Ergodic Theory
, Cambridge Studies in Advanced Mathematics
(Cambridge University Press
, Cambridge
, 2016
), Vol. 151
.31.
T.
Bomfim
and P.
Varandas
, “Multifractal analysis of the irregular set for almost-additive sequences via large deviations
,” Nonlinearity
28
(10
), 3563
–3585
(2015
).32.
T.
Bousch
, “La condition de Walters
,” Ann. Sci. Éc. Norm. Supér.
34
, 287
–311
(2001
).33.
P.
Walters
, “Invariant measures and equilibrium states for some mappings which expand distances
,” Trans. Am. Math. Soc.
236
, 121
–153
(1978
).34.
S.
Silverman
, “On maps with dense orbits and the definition of chaos
,” Rocky Mt. J. Math.
22
, 353
–375
(1992
).35.
B.
Kalinin
, “Livšic theorem for matrix cocycles
,” Ann. Math.
173
, 1025
–1042
(2011
).36.
D.-J.
Feng
and K.
Lau
, “The pressure function for products of non-negative matrices
,” Math. Res. Lett.
9
, 363
–378
(2002
).37.
L.
Barreira
and K.
Gelfert
, “Multifractal analysis for Lyapunov exponents on nonconformal repellers
,” Commun. Math. Phys.
267
, 393
–418
(2006
).38.
R.
Mohammadpour
, “Lyapunov spectrum properties and continuity of the lower joint spectral radius
,” J. Stat. Phys.
187
, 23
(2022
).39.
S.
Crovisier
and R.
Potrie
, Introduction to Partially Hyperbolic Dynamics
, Notes
(International Centre for Theoretical Physics
, Trieste, Italy
, 2015
).40.
J.
Bochi
and N.
Gourmelon
, “Some characterizations of domination
,” Math. Z.
263
, 221
–231
(2009
).41.
L.
Barreira
, “A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems
,” Ergod. Theory Dyn. Syst.
16
, 871
–927
(1996
).42.
R.
Bowen
, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms
, Springer Lectures Notes in Mathematics
(Springer Verlag
, 1975
), Vol. 470
.43.
R. B.
Israel
and R. R.
Phelps
, “Some convexity questions arising in statistical mechanics
,” Math. Scand.
54
, 133
–156
(1984
).44.
E.
de Faria
and P.
Guarino
, “Dynamics of circle mappings
,” in Coloquio Brasileiro de Matematica
(IMPA
, 2021
), Vol. 33
.45.
G.
Iommi
and Y.
Yayama
, “Weak Gibbs measures as Gibbs measures for asymptotically additive sequences
,” Proc. Am. Math. Soc.
145
(4
), 1599
–1614
(2017
).46.
T.
Bousch
, “Une représentation des cobords faibles d’un système dynamique
,” Ann. Fac. Sci. Toulose: Math.
32
, 817
–821
(2023
).47.
T.
Bousch
and O.
Jenkinson
, “Cohomology classes of dynamically non-negative Ck functions
,” Invent. Math.
148
, 207
–217
(2002
).48.
A.
Katok
and E. A.
Robinson
, Jr., “Cocycles, cohomology and combinatorial constructions in ergodic theory
,” in Smooth Ergodic Theory and its Applications (Seattle, WA, 1999)
, Proceedings of Symposium in Pure Mathematics
(American Mathematical Society
, Providence, RI
2001
), Vol. 69
, pp. 107
–173
.49.
W.
Krieger
, “On quasi-invariant measures in uniquely ergodic systems
,” Invent. Math.
14
, 184
–196
(1971
).50.
A.
Kocsard
, “On cohomological C0-(in)stability
,” Bull. Braz. Math. Soc.
44
, 489
–495
(2013
).51.
F.
Hofbauer
, “Examples for the nonuniqueness of the equilibrium state
,” Trans. Am. Math. Soc.
228
, 223
–241
(1977
).52.
V.
Climenhaga
and D.
Thompson
, “Equilibrium states beyond specification and the Bowen property
,” J. London Math. Soc.
87
(2
), 401
–427
(2013
).53.
H.
Hu
, “Equilibriums of some non-Hölder potentials
,” Trans. Am. Math. Soc.
360
(4
), 2153
–2190
(2008
).54.
G.
Iommi
and M.
Todd
, “Transience in dynamical systems
,” Ergod. Theory Dyn. Syst.
33
, 1450
–1476
(2012
).55.
Y.
Pesin
and K.
Zhang
, “Phase transitions for uniformly expanding maps
,” J. Stat. Phys.
122
(6
), 1095
–1110
(2006
).56.
P.
Walters
, “A natural space of functions for the Ruelle operator theorem
,” Ergod. Theory Dyn. Syst.
27
, 1323
–1348
(2007
).© 2024 Author(s). Published under an exclusive license by AIP Publishing.
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