# Parrondo’s Paradox for Tent Maps

## Abstract

**:**

## 1. Introduction

## 2. The Results

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

#### 2.1. Maps ${g}_{{s}_{2},p}$ of Type 1

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

#### 2.2. Maps ${g}_{{s}_{2},p}$ of Type 2

**Theorem**

**1.**

**Proof.**

## 3. Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**For $p=1/2$ and ${s}_{2}\in (0,1)$, we depict the graph (

**a**) and level curves (

**b**) of the topological entropy of ${\varphi}_{{s}_{1},{s}_{2},p}$ with accuracy ${10}^{-4}$. The step size for ${s}_{1}$ and ${s}_{2}$ is ${10}^{-3}$. (

**c**) Parameter values for which $h\left({f}_{{s}_{1}}\right)-h\left({\varphi}_{{s}_{1},{s}_{2},p}\right)>2\times {10}^{-4}$. (

**d**) Parameter values for which $h\left({\varphi}_{{s}_{1},{s}_{2},p}\right)-h\left({f}_{{s}_{1}}\right)>2\times {10}^{-4}$.

**Figure 2.**For ${s}_{2}\in (0,1)$, we depict the graph of the topological entropy of ${\varphi}_{{s}_{1},{s}_{2},p}$ with accuracy ${10}^{-4}$ for: $p=1/4$ (

**a**); $p=1/3$ (

**b**); $p=2/3$ (

**c**); and $p=3/4$ (

**d**). The step size for ${s}_{1}$ and ${s}_{2}$ is ${10}^{-3}$.

**Figure 3.**For $p=1/2$ and ${s}_{2}\in (1,2]$, we depict the graph (

**a**) and level curves (

**b**) of the topological entropy of ${\varphi}_{{s}_{1},{s}_{2},p}$ with accuracy ${10}^{-4}$. The step size for ${s}_{1}$ and ${s}_{2}$ is ${10}^{-3}$. (

**c**) Parameter values for which $h\left({f}_{{s}_{1}}\right)-h\left({\varphi}_{{s}_{1},{s}_{2},p}\right)>2\times {10}^{-4}$. (

**d**) Parameter values for which $h\left({\varphi}_{{s}_{1},{s}_{2},p}\right)-h\left({f}_{{s}_{1}}\right)>2\times {10}^{-4}$.

**Figure 4.**For $1<{s}_{2}<1/p$, we depict the graph of the topological entropy of ${\varphi}_{{s}_{1},{s}_{2},p}$ with accuracy ${10}^{-4}$ for: $p=1/4$ (

**a**); $p=1/3$ (

**b**); $p=2/3$ (

**c**); and $p=3/4$ (

**d**). The step size for ${s}_{1}$ and ${s}_{2}$ is ${10}^{-3}$.

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Cánovas, J.S. Parrondo’s Paradox for Tent Maps. *Axioms* **2021**, *10*, 85.
https://doi.org/10.3390/axioms10020085

**AMA Style**

Cánovas JS. Parrondo’s Paradox for Tent Maps. *Axioms*. 2021; 10(2):85.
https://doi.org/10.3390/axioms10020085

**Chicago/Turabian Style**

Cánovas, Jose S. 2021. "Parrondo’s Paradox for Tent Maps" *Axioms* 10, no. 2: 85.
https://doi.org/10.3390/axioms10020085