Suppose n ≥ 2 and A i ⊂ {0, 1, · · · , (n − 1)} for i = 1, · · · , l, let K i = a∈A i n −1 (K i + a) be self-similar sets contained in [0, 1]. Given m 1 , · · · , m l ∈ Z with i m i = 0, we let Sx = {(y 1 , · · · , y l ) : m 1 y 1 + · · · + m l y l = x with y i ∈ K i ∀i} .In this paper, we analyze the Hausdorff dimension and Hausdorff measure of the following set Ur = {x : Card(Sx) = r}, where Card(Sx) denotes the cardinality of Sx, and r ∈ N + . We prove under the so-called covering condition that the Hausdorff dimension of U 1 can be calculated in terms of some matrix. Moreover, if r ≥ 2, we also give some sufficient conditions such that the Hausdorff dimension of Ur takes only finite values, and these values can be calculated explicitly. Furthermore, we come up with some sufficient conditions such that the dimensional Hausdorff measure of Ur is infinity. Various examples are provided. Our results can be viewed as the exceptional results for the classical slicing problem in geometric measure theory.2000 Mathematics Subject Classification. Primary 28A80.

Abstract. The various dimensions of the intersections of the graph-directed setsorthogonal complement. We obtain the typical value of dimensions of sections for typical directions V and also provide a weaker result for exceptional directions.

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In this paper, we will di~uss the construction problems about the invariant sets and invariant measures of continuous maps which map complexes into themselves, using simplicial approximation and Markov chains. In [7], the author defined a matrix by using r-normal subdivision of the n-dimensional unit cube, considered it a Markov matrix, and constructed the invariant set and invariant measure. In fact, the matrix he defined is not Markov matrix generally. So we will give [7] and amendment in the last part of this paper. We also construct an invariant set that is the chain-recurrent set of the map by means of a non-negative matrix which only depends on the map. At last. we will prove the higher dimension Banach variation formula that can simplify the transition matrix.