# On The Entropy of Continuous Flows With Uniformly Expansive Points and The Globalness of Shadowable Points With Gaps

## TL;DR

This paper investigates conditions under which continuous flows exhibit positive topological entropy, linking pointwise dynamical properties like expansiveness, shadowing, and non-wandering behavior to entropy and symbolic dynamics.

## Contribution

It establishes new criteria connecting pointwise properties of flows, including shadowing and expansiveness, to positive entropy and symbolic subshifts, even in the presence of singularities.

## Key findings

- Non-periodic nonwandering points imply positive topological entropy.
- Existence of certain shadowable, expansive points leads to symbolic subshifts.
- Pointwise shadowing properties are characterized in the context of flows.

## Abstract

In this work we study the problem of positiveness of topological entropy for flows using pointwise dynamics. We show that the existence of a non-periodic nonwandering point of an expansive and non-singular flow with shadowing is a sufficient condition to to obtain positive topological entropy. Moreover, we can deal with flows with singularities, showing that the existence of a non-wandering, non-critical, strongly-shadowable, and uniform-expansive point implies the existence of a symbolic subshift. Finally, we discuss pointwise versions of some shadowing-type properties.