Topological transitivity for a class of monotonic mod one transformations

Abstract

Suppose that f : [0, 1] → [0, 2] is a continuous strictly increasing piecewise differentiable function, and define T f Mathematics Subject Classification (2000). Primary 37E05; Secondary 37B05 · 37D20 · 54H20.

Density of periodic measures and large deviation principle for generalised mod one transformations

“…This family has recently attracted attention in the setting beyond specification, since T β does not satisfy the specification property for Lebesgue almost parameter β > 1 [2,26]. After that, various generalisations of T β has been considered by many authors: (α, β)-transformations [4,5,9,11,24], (−β)-transformations [18][19][20]27], generalised β-transformations [9,10,29,32] (see also figure 1).…”

Density of periodic measures and large deviation principle for generalized $(α, β)$-transformations

“…This family has recently attracted attention in the setting beyond specification, since T β does not satisfy the specification property for Lebesgue almost parameter β > 1 ( [2,22]). Inspired by β-transformations, many authors have considered various generalizations of T β : (α, β)-transformations ( [4,5,7,9,20]), (−β)-transformations ( [15,16,17,23]), generalized βtransformations ( [7,8,24,26]).…”

TOPOLOGICAL PRESSURE OF THE SET OF GENERIC POINTS FOR Zd-ACTIONS

“…While the definition of (α-β)-shift is similar to that of β-shift, some of dynamical properties are different. For example, it is known that all β-shifts are topologically mixing, but (α-β)-shift is not transitive for some 1 < β < 2 (see [17]). Moreover, it is not difficult to see that the saturatedness is failed for certain non-transitive (α-β)-shifts.…”