Let π : T × X → X, written T π X, be a topological semiflow/flow on a uniform space X with T a multiplicative topological semigroup/group not necessarily discrete. We then prove:• If T π X is non-minimal topologically transitive with dense almost periodic points, then it is sensitive to initial conditions. As a result of this, Devaney chaos ⇒ Sensitivity to initial conditions, for this very general setting.Let R + π X be a C 0 -semiflow on a Polish space; then we show:• If R + π X is topologically transitive with at least one periodic point p and there is a dense orbit with no nonempty interior, then it is multi-dimensional Li-Yorke chaotic; that is, there is a uncountable set Θ ⊆ X such that for any k ≥ 2 and any distinct points x 1 , . . . , x k ∈ Θ, one can find two time sequences s n → ∞, t n → ∞ with s n (x 1 , . . . , x k ) → (x 1 , . . . , x k ) ∈ X k and t n (x 1 , . . . , x k ) → (p, . . . , p) ∈ ∆ X k .Consequently, Devaney chaos ⇒ Multi-dimensional Li-Yorke chaos.Moreover, let X be a non-singleton Polish space; then we prove:• Any weakly-mixing C 0 -semiflow R + π X is densely multi-dimensional Li-Yorke chaotic.• Any minimal weakly-mixing topological flow T π X with T abelian is densely multidimensional Li-Yorke chaotic.• Any weakly-mixing topological flow T π X is densely Li-Yorke chaotic.We in addition construct a completely Li-Yorke chaotic minimal SL(2, R)-acting flow on the compact metric space R ∪ {∞}. Our various chaotic dynamics are sensitive to the choices of the topology of the phase semigroup/group T .

The goal of this paper is to define and investigate those topological pressures, which is an extension of topological entropy presented by Feng and Huang[13], of continuous transformations. This study reveals the similarity between many known results of topological pressure. More precisely, the investigation of the variational principle is given and related propositions are also described. That is, this paper defines the measure theoretic pressure Pµ(T, f ) for any µ ∈ M(X), and shows that PB(T, f, K) = sup Pµ(T, f ) : µ ∈ M(X), µ(K) = 1 , where K ⊆ X is a non-empty compact subset and PB(T, f, K) is the Bowen topological pressure on K.Furthermore, if Z ⊆ X is an analytic subset, then PB(T, f, Z) = sup PB(T, f, K) :K ⊆ Z is compact . However, this analysis relies on more techniques of ergodic theory and topological dynamics.

By using ergodic theoretic techniques following Hillel Fürstenberg, we prove that measurable subsets of a locally compact abelian group of positive upper density contain Szemerédi-wise configurations defined by an arbitrary compact subset of the group.

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