本文研究帶重疊之迭代函數系統所生成之自相似集合的維度問題。在無重疊的情形下，其 Hausdorff 維度由相似維度給出；然而當存在重疊時，維度可能下降，且其精確計算變得相當困難。 本文首先整理與分析維度下降與超指數凝聚的相關結果，包括 Hochman 的逆定理，其將維度下降與柱集合之間的超指數接近建立聯繫，以及 Rapaport 對於代數參數情形下精確重疊的結果。此外，我們亦介紹存在超指數凝聚但不具有精確重疊的例子，說明維度下降並非僅由精確重疊所決定。 本文的主要貢獻在於提出一種有限狀態矩陣方法，用於計算特定迭代函數系統吸引子的精確維度。在一維且具有共同收縮率 1/a 且平移量為有理數的情形下，我們利用有理網格對吸引子進行分割，並將其局部結構表示為有限個狀態。覆蓋數的成長可由轉移矩陣描述，進而透過其譜半徑得到維度。 此方法進一步推廣至高維及非均勻但可公度之收縮率 a^(−n_i) 的情形。在此設定下，局部狀態可能同時涉及不同層級的方格，使得狀態空間較為複雜，但仍保持有限狀態結構。 最後，本文透過具體例子說明此方法，包括具有精確重疊的情形，並討論此方法的限制，例如狀態空間的快速增長以及對有理平移的限制。;This thesis studies dimension phenomena for self-similar sets generated by iterated function systems with overlaps. In the absence of overlaps， the Hausdorff dimension is given by the similarity dimension. However， in the presence of overlaps， the dimension may drop， and its precise determination becomes significantly more complicated. We first review several fundamental results concerning dimension drop and super-exponential concentration. In particular， we discuss Hochman′s inverse theorem， which relates dimension drop to super-exponential closeness of cylinder sets， and Rapaport′s result on exact overlaps for algebraic parameters. We also present examples of super-exponential concentration without exact overlaps， illustrating that dimension drop is not solely determined by exact overlaps. The main contribution of this thesis is a finite-state matrix method for computing the exact dimension of certain iterated function systems. For one-dimensional systems with a common contraction ratio 1/a and rational translations， the attractor is decomposed using a rational grid， and its local structure is encoded by finitely many states. The growth of covering numbers is then described by a transition matrix， and the dimension is obtained from the spectral radius of this matrix. This construction is further extended to higher-dimensional systems and to non-uniform but commensurable contraction ratios of the form a^ (−n_i). In this setting， local states may involve cubes from multiple grid levels， leading to a more complicated but still finite-state representation. Several examples are provided to illustrate the method， including cases with exact overlaps where redundant maps can be removed. Finally， we discuss the limitations of the approach， such as the growth of the state space and the restriction to rational translations.
