Topological entropy for discontinuous semiflows

Abstract

We study two variations of Bowen's definitions of topological entropy based on separated and spanning sets which can be applied to the study of discontinuous semiflows on compact metric spaces. We prove that these definitions reduce to Bowen's ones in the case of continuous semiflows. As a second result, we prove that our entropies give a lower bound for the τ -entropy defined by Alves, Carvalho and Vásquez (2015). Finally, we prove that for impulsive semiflows satisfying certain regularity condition, there exists a continuous semiflow defined on another compact metric space which is related to the first one by a semiconjugation, and whose topological entropy equals our extended notion of topological entropy by using separated sets for the original semiflow.

Topological pressure for discontinuous semiflows and a variational principle for impulsive dynamical systems

“…Observe that by taking f ≡ 0 in the previous result we recover [18,Theorems 3 and 4] and, as a consequence of the proof, we recover [17,Theorem 3]. Moreover, as a subproduct of this result, we get that the four (a priori different) notions of topological pressure introduced in Section 2.1.2 actually coincide for the class of impulsive semiflows considered in the statement.…”

“…Remark 2.1. If we consider f constant and equal to zero in the above definitions we recover the definitions of topological entropy for semiflows introduced in [17] and [18].…”

“…Let ϕ : R + 0 × X → X be a semiflow (not necessarily continuous). Given δ > 0, let us consider the pseudometrics dϕ δ : X × X → R + 0 and d ϕ δ : X × X → R + 0 , introduced in [17] and [18], respectively, given by dϕ…”

“…So, as already mentioned, our intent is to provide notions of topological pressure suitable for this context and to establish an invariance principle (see Theorem 2.3) contributing to the study of thermodynamical formalism of these systems. Some previous work in this direction are [3,17,18], where a variational principle was established for the topological entropy and [2], where a notion of topological pressure (a priori different from ours) was introduced and a variational principle along with the existence of equilibrium states was obtained. As a consequence of our main result we conclude that, restricted to an appropriate class of impulsive systems, our four notions of topological pressure actually coincide with the one introduced in [2] (see Corollary 2.4) which also gives us the existence of equilibrium states.…”

Ivanov’s Theorem for Admissible Pairs Applicable to Impulsive Differential Equations and Inclusions on Tori

“…For a positive topological entropy, Theorem 1 was effectively applied to discrete chaotic dynamics in [5][6][7][8] and to chaotic impulsive differential equations on tori in [9]. Let us note that the results dealing with the topological entropy for dynamic processes and differential equations from another perspective are rather rare (see e.g., [10][11][12][13][14]), and those for multivalued dynamics are even more delicate (see e.g., [15]).…”