论文标题
关于与强烈不可还原和近端系统相关的自我措施
On self-affine measures associated to strongly irreducible and proximal systems
论文作者
论文摘要
令$μ$为$ \ mathbb {r}^{d} $上的自动措施，与Aggine ifs $φ$和正概率向量$ p $相关。假设$φ$中的地图没有共同的固定点，并且标准的不可约性和接近性假设可以通过其线性零件来满足。我们表明，每当$ d = 3 $ d = 3 $和$φ$时，$ \dimμ$等于lyapunov dim_ $ \ dim_ {l}（φ，p）$，都满足强分离条件（或较温和的强烈开放式环境）。这是根据一般标准确保$ \dimμ= \ min \ {d，\ dim_ {l}（φ，p）\} $，从中也随之而来的是平面案例。此外，我们证明$ \dimμ= d $每当$φ$是diophantine时（例如，当$φ$由代数参数定义时），并且至少由$φ$和$ p $生成的随机步行的熵是至少$（χ_{1}-χ_{d}）\ frac {（d-1）（d-2）} {2} - \ sum_ {k = 1}^{d}χ_{k {k} $，$ 0>χ_{1} {1} \ ge ge gebef gebefuneents lyapeent。我们还获得了有关$μ$的正交预测尺寸的结果。
Let $μ$ be a self-affine measure on $\mathbb{R}^{d}$ associated to an affine IFS $Φ$ and a positive probability vector $p$. Suppose that the maps in $Φ$ do not have a common fixed point, and that standard irreducibility and proximality assumptions are satisfied by their linear parts. We show that $\dimμ$ is equal to the Lyapunov dimension $\dim_{L}(Φ,p)$ whenever $d=3$ and $Φ$ satisfies the strong separation condition (or the milder strong open set condition). This follows from a general criteria ensuring $\dimμ=\min\{d,\dim_{L}(Φ,p)\}$, from which earlier results in the planar case also follow. Additionally, we prove that $\dimμ=d$ whenever $Φ$ is Diophantine (which holds e.g. when $Φ$ is defined by algebraic parameters) and the entropy of the random walk generated by $Φ$ and $p$ is at least $(χ_{1}-χ_{d})\frac{(d-1)(d-2)}{2}-\sum_{k=1}^{d}χ_{k}$, where $0>χ_{1}\ge...\geχ_{d}$ are the Lyapunov exponents. We also obtain results regarding the dimension of orthogonal projections of $μ$.
