# Livschitz Theorem in Suspension Flows and Markov Systems: Approach in Cohomology of Systems

## Abstract

**:**

## 1. Introduction

## 2. From Cohomological to Periodic Information

#### 2.1. Cocycles and Cohomology Defined on a General Group

#### 2.2. Cocycles in Continuous Time and Relation to Periodic Orbits

## 3. Formulating and Demonstrating Livschitz Theorem to Hyperbolic Flows

**Property**

**1.**

**Property**

**2.**

**Property**

**3.**

**Property**

**4.**

**Lemma**

**1**(Anosov Closing Lemma)

**.**

**Theorem**

**1**(Livschitz Theorem for flows)

**.**

**Hypothesis**

**1.**

**Hypothesis**

**2.**

**Proof.**

## 4. Livschitz Theorem for Suspension Flows

**Theorem**

**2.**

**Property**

**5.**

**Property**

**6.**

**Property**

**7.**

**Theorem**

**3**

**.**Let $f:\phantom{\rule{3.33333pt}{0ex}}M\to M$ be a diffeomorphism with a locally maximal compact hyperbolic set ${\mathsf{\Lambda}}_{f}\subset M$ such that $f\left|{}_{{\mathsf{\Lambda}}_{f}}\right.$ is topologically transitive and $\tau :M\to (0,\infty )$ is a Lipschitz function. Let $\mathsf{\Psi}={\left\{{\psi}^{t}\right\}}_{t\in \mathbb{R}}$ be a suspension flow in Y over f with length function τ and for the set

**Proof.**

## 5. Markov Systems

- $\mathsf{\Lambda}={\bigcup}_{t\in [0,\epsilon ]}{\phi}^{t}\left({\bigcup}_{i=1}^{p}{R}_{i}\right)$;
- for each $i\ne j$ we have ${\phi}^{t}{R}_{i}\cap {R}_{j}=\phantom{\rule{3.33333pt}{0ex}}\varnothing $ for all $t\in [0,\epsilon ]$ or ${\phi}^{t}{R}_{j}\cap {R}_{i}=\varnothing $ for all $t\in [0,\epsilon ]$.

**Theorem**

**4.**

**Property**

**8.**

**Property**

**9.**

## 6. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Laureano, R.D.
Livschitz Theorem in Suspension Flows and Markov Systems: Approach in Cohomology of Systems. *Symmetry* **2020**, *12*, 338.
https://doi.org/10.3390/sym12030338

**AMA Style**

Laureano RD.
Livschitz Theorem in Suspension Flows and Markov Systems: Approach in Cohomology of Systems. *Symmetry*. 2020; 12(3):338.
https://doi.org/10.3390/sym12030338

**Chicago/Turabian Style**

Laureano, Rosário D.
2020. "Livschitz Theorem in Suspension Flows and Markov Systems: Approach in Cohomology of Systems" *Symmetry* 12, no. 3: 338.
https://doi.org/10.3390/sym12030338