Almarza, J.I."CP-chains and dimension preservation for projections of (×m,×n)-invariant Gibbs measures" (2017) Advances in Mathematics. 304:227-265
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Dimension conservation for almost every projection has been well-established by the work of Marstrand, Mattila and Hunt and Kaloshin. More recently, Hochman and Shmerkin used CP-chains, a tool first introduced by Furstenberg, to prove all projections preserve dimension of measures on [0,1]2 that are the product of a ×m-invariant and a ×n-invariant measure (for m, n multiplicatively independent). Using these tools, Ferguson, Fraser and Sahlsten extended that conservation result to (×m,×n)-invariant measures that are the push-forward of a Bernoulli scheme under the (m,n)-adic symbolic encoding. Their proof relied on a parametrization of conditional measures which could not be extended beyond the Bernoulli case. In this work, we extend their result from Bernoulli measures to Gibbs measures on any transitive SFT. Rather than attempt a similar parametrization, the proof is achieved by reducing the problem to that of the pointwise convergence of a double ergodic average which is known to hold when the system is exact. © 2016


Documento: Artículo
Título:CP-chains and dimension preservation for projections of (×m,×n)-invariant Gibbs measures
Autor:Almarza, J.I.
Filiación:Departamento de Computación, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Int. Güiraldes 2160, Buenos Aires, Argentina
Palabras clave:Dynamical systems; Ergodicity; Fractal; Gibbs measures
Página de inicio:227
Página de fin:265
Título revista:Advances in Mathematics
Título revista abreviado:Adv. Math.


  • Alonso, A., A counterexample on the continuity of conditional expectations (1988) J. Math. Anal. Appl., 129, pp. 1-5
  • Baladi, V., Positive Transfer Operators and Decay of Correlations (2000), World Scientific Singapore; Billingsley, P., Convergence of Probability Measures (1968), John Wiley & Sons NY; Blackwell, D., Dubins, L., Merging of opinions with increasing information (1962) Ann. Statist., 33 (2), pp. 882-886
  • Bowen, R., Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (1975), Springer-Verlag NY; Derrien, J.-M., Lesigne, E., Un théorème ergodique polynômial ponctuel pour les endomorphismes exacts et les k-systèmes (1996) Ann. Inst. Henri Poincaré, 32, pp. 765-778
  • Falconer, K., Techniques in Fractal Geometry (1997), John Wiley & Sons Ltd. Chichester; Ferguson, A., Fraser, J., Sahlsten, T., Scaling scenery of (×m,×n) invariant measures (2015) Adv. Math., 268, pp. 564-602
  • Furstenberg, H., Intersection of Cantor sets and transversality of semigroups (1970) Problems in Analysis, , Princeton Univ. Press Princeton, NJ
  • Furstenberg, H., Recurrence in Ergodic Theory and Combinatorial Number Theory (1981), Princeton University Press Princeton, NJ; Furstenberg, H., Ergodic fractal measures and dimension conservation (2008) Ergodic Theory Dynam. Systems, 28, pp. 405-422
  • Hochman, M., Geometric rigidity of ×m invariant measures (2012) J. Eur. Math. Soc., 14, pp. 1539-1563
  • Hochman, M., Shmerkin, P., Local entropy averages and projections of fractal measures (2012) Ann. of Math., 175 (3), pp. 1001-1059
  • Kenyon, R., Peres, Y., Hausdorff dimensions of sofic affine-invariant sets (1996) Israel J. Math., 94, pp. 157-178
  • Lalley, S., Regenerative representation for one-dimensional Gibbs states (1986) Ann. Probab., 14 (4), pp. 1262-1271
  • Lyons, R., Strong laws of large numbers for weakly correlated random variables (1988) Michigan Math. J., 35 (3), pp. 353-359
  • Przytycki, F., Urbanski, M., Conformal Fractals: Ergodic Theory Methods (2010)


---------- APA ----------
(2017) . CP-chains and dimension preservation for projections of (×m,×n)-invariant Gibbs measures. Advances in Mathematics, 304, 227-265.
---------- CHICAGO ----------
Almarza, J.I. "CP-chains and dimension preservation for projections of (×m,×n)-invariant Gibbs measures" . Advances in Mathematics 304 (2017) : 227-265.
---------- MLA ----------
Almarza, J.I. "CP-chains and dimension preservation for projections of (×m,×n)-invariant Gibbs measures" . Advances in Mathematics, vol. 304, 2017, pp. 227-265.
---------- VANCOUVER ----------
Almarza, J.I. CP-chains and dimension preservation for projections of (×m,×n)-invariant Gibbs measures. Adv. Math. 2017;304:227-265.