Projections of ( × m , × n )-invariant Gibbs measures preserve dimension
Projections of ( × m , × n )-invariant Gibbs measures preserve dimension
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( × m , × n ) 不变吉布斯测度的投影保留维度
DOI：
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发表时间：
2014
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通讯作者：
J. Almarza
中科院分区：
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作者：
J. Almarza
通过Marstrand、Mattila、Hunt和Kaloshin的工作，几乎所有投影的维度守恒都得到了很好的建立。最近，Hochman和Shmerkin利用Furstenberg最先提出的工具CP-Chains证明了[0，1]2上的所有投影保持度量的维度，这些度量是时不变测度和×n-不变测度(对于m，n乘法独立)的乘积。利用这些工具，Ferguson，Fraser和Sahlsten将守恒结果推广到(×m，×n)-不变度量，这些不变度量是(m，n)-adi符号编码下Bernoulli方案的推进。他们的证明依赖于条件测度的参数化，而条件测度不能扩展到伯努利情形之外。在这项工作中，我们将他们的结果从Bernoulli测度推广到任何传递SFT上的Gibbs测度。不是尝试类似的参数化，而是通过将问题简化为双遍历平均的逐点收敛来实现证明，当系统是精确的时，已知双遍历平均的逐点收敛。
Dimension conservation for almost every projection has been wellestablished by the work of Marstrand, Mattila and Hunt and Kaloshin. 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 timesm-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 pushforward 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.