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We study the set of eventually always hitting points for symbolic dynamics with specification. The measure and Hausdorff dimension of such fractal set are obtained. Moreover, we establish the stronger metric results by introducing a new quantity L N (ω) which describes the maximal length of string of zeros of the prefix among the first N iterations of ω in symbolic space. The Hausdorff dimensions of the level sets for this quantity are also completely determined. [ABSTRACT FROM AUTHOR]

In this paper, we calculate the Hausdorff dimension of the fractal set x ∈ d : ∏ 1 ≤ i ≤ d | T β i n (x i) − x i | < ψ (n)   for infinitely many  n ∈ ℕ , where T β i is the standard β i -transformation with β i > 1 , ψ is a positive function on ℕ and | ⋅ | is the usual metric on the torus . Moreover, we investigate a modified version of the shrinking target problem, which unifies the shrinking target problems and quantitative recurrence properties for matrix transformations of tori. Let T be a d × d non-singular matrix with real coefficients. Then, T determines a self-map of the d -dimensional torus d : = ℝ d / ℤ d . For any 1 ≤ i ≤ d , let ψ i be a positive function on ℕ and Ψ (n) : = (ψ 1 (n) , ... , ψ d (n)) with n ∈ ℕ. We obtain the Hausdorff dimension of the fractal set { x ∈ d : T n (x) ∈ L (f n (x) , Ψ (n))   for infinitely many  n ∈ ℕ } , where L (f n (x , Ψ (n))) is a hyperrectangle and { f n } n ≥ 1 is a sequence of Lipschitz vector-valued functions on d with a uniform Lipschitz constant. [ABSTRACT FROM AUTHOR]

Consider orbits (z , κ) of the fractal iterator f κ (z) : = z 2 + κ , κ ∈ ℂ , that start at initial points z ∈ K ̂ κ (m) ⊂ ℂ ̂ , where ℂ ̂ is the set of all rational complex numbers (their real and imaginary parts are rational) and K ̂ κ (m) consists of all such z whose complexity does not exceed some complexity parameter value m (the complexity of z is defined as the number of bits that suffice to describe the real and imaginary parts of z in lowest form). The set K ̂ κ (m) is a bounded-complexity approximation of the filled Julia set K κ . We present a new perspective on fractals based on an analogy with Chaitin's algorithmic information theory, where a rational complex number z is the analog of a program p , an iterator f κ is analogous to a universal Turing machine U which executes program p , and an unbounded orbit (z , κ) is analogous to an execution of a program p on U that halts. We define a real number Υ κ which resembles Chaitin's Ω number, where, instead of being based on all programs p whose execution on U halts, it is based on all rational complex numbers z whose orbits under f κ are unbounded. Hence, similar to Chaitin's Ω number, Υ κ acts as a theoretical limit or a "fractal oracle number" that provides an arbitrarily accurate complexity-based approximation of the filled Julia set K κ . We present a procedure that, when given m and κ , it uses Υ κ to generate K ̂ κ (m) . Several numerical examples of sets that estimate K ̂ κ (m) are presented. [ABSTRACT FROM AUTHOR]

This paper aims to investigate the notion of p -convex functions on fractal sets ℝ α ̂ (0 < α ̂ ≤ 1). Based on these novel ideas, we derived an auxiliary result depend on a three-step quadratic kernel by employing generalized p -convexity. Take into account the local fractal identity, we established novel Newton's type variants for the local differentiable functions. Several special cases are apprehended in the light of generalized convex functions and generalized harmonically convex functions. This novel strategy captures several existing results in the relative literature. Application is obtained in cumulative distribution function and generalized special weighted means to confirm the relevance and computational effectiveness of the considered method. Finally, we supposed that the consequences of this paper can stimulate those who are interested in fractal analysis. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to provide novel estimates of Ostrowski-type inequalities in a much simpler and shorter way of some recent significant results in the context of a fractal set ℝ α ̃ . By using our new approach, we established an auxiliary result that correlates with generalized convex () and concave functions for absolutely continuous functions with second-order local differentiable mappings. Moreover, we derived some companions of Ostrowski-type inequalities belonging to (2 α ̃) ∈ L ∞ [ s 1 , s 2 ] , (2 α ̃) ∈ L p [ s 1 , s 2 ] and (2 α ̃) ∈ L 1 [ s 1 , s 2 ] in local fractional sense. Our results generalize and offer better bounds than many known results in the existing literature associated with trapezoidal and midpoint formula. As an application perspective, we derived several estimation-type outcomes by the use of generalized α ̃ -type special means formula provided here to illustrate the usability of the obtained results. Our study contributes to a better understanding of fractal analysis and proves beneficial in exploring real-world phenomena. [ABSTRACT FROM AUTHOR]

In this paper, the definition of generalized s-preinvex function on Yang's fractal sets ℝ γ (0 < γ ≤ 1) is proposed, and the generalized Hermite–Hadamard's inequality for this class of functions is established. By using this convexity, some generalized Hermite–Hadamard type integral inequalities with parameters are established. For these inequalities, the absolute values of twice local fractional order derivative of the functions are generalized s-preinvex functions. Some special integral inequalities can be obtained by assigning special values to the obtained inequalities, and two examples are given to illustrate our results. Finally, we propose the applications of the results in numerical integration and error estimation. [ABSTRACT FROM AUTHOR]

In this paper, we study solutions to the max–min mean value problem 1 2 max q ∈ V m , p { f (q) } + 1 2 min q ∈ V m , p { f (q) } = f (p) in the Sierpinski Gasket with a prescribed Dirichlet datum at the three vertices of the first triangle. In the previous mean value, formula p is a vertex of one triangle at one stage in the construction of the Sierpinski Gasket and V m , p is the set of vertices that are adjacent to p at that stage. For this problem, it was known that there are existence and uniqueness of a continuous solution, a comparison principle holds, and, moreover, solutions are Lipschitz continuous. Here we continue the analysis of this problem and prove that the solution is piecewise linear on the segments of the Sierpinski Gasket. Moreover, we also show for which values at the three vertices of the first triangle solutions to this mean value formula coincide with infinity harmonic functions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the α -type concept of generalized h -convex function on real linear fractal sets ℝ α   (0 < α ≤ 1) , from which the known definitions of generalized convex functions and generalized s -convex functions are derived, and from this, we obtain generalized Godunova–Levin functions and generalized P -functions. Some properties of generalized h -convex functions are discussed. Lastly, some generalized Hadamard-type inequalities of these classes functions are given. [ABSTRACT FROM AUTHOR]

In this paper, a new identity with parameters involving local fractional integrals is derived. Using this identity, some general local fractional integral inequalities for generalized preinvex functions are established. A parallel development is deduced for generalized preconcave functions. Taking special values for the parameters, some generalized midpoint inequalities, trapezoidal inequalities and Simpson inequalities are obtained. Finally, as some applications, error estimates of numerical integration for local fractional integrals are given. [ABSTRACT FROM AUTHOR]

In this paper we study the solutions to nonlinear mean-value formulas on fractal sets. We focus on the mean-value problem 1 2 max q ∈ V m , p { f (q) } + 1 2 min q ∈ V m , p { f (q) } − f (p) = 0 in the Sierpiński gasket with prescribed values f (p 1) , f (p 2) and f (p 3) at the three vertices of the first triangle. For this problem we show existence and uniqueness of a continuous solution and analyze some properties like the validity of a comparison principle, Lipschitz continuity of solutions (regularity) and continuous dependence of the solution with respect to the prescribed values at the three vertices of the first triangle. [ABSTRACT FROM AUTHOR]