Erro ao iniciar o TomCat

web.xml

<?xml version="1.0" encoding="UTF-8"?>
<web-app xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
    xmlns="http://java.sun.com/xml/ns/javaee"
    xsi:schemaLocation="http://java.sun.com/xml/ns/javaee http://java.sun.com/xml/ns/javaee/web-app_3_0.xsd"
    id="WebApp_ID" version="3.0">
    <display-name>rest-notas</display-name>
    <welcome-file-list>
        <welcome-file>index.html</welcome-file>
    </welcome-file-list>

    <servlet>
        <servlet-name>Jersey REST Service</servlet-name>
        <servlet-class>org.glassfish.jersey.servlet.ServletContainer</servlet-class>
        <init-param>
            <param-name>jersey.config.server.provider.packages</param-name>
            <param-value>br.com.anderson.rest_notas.rest</param-value>
        </init-param>
        <init-param>
            <param-name>com.sun.jersey.api.json.POJOMappingFeature</param-name>
            <param-value>true</param-value>
        </init-param>
        <load-on-startup>1</load-on-startup>
    </servlet>
    <servlet-mapping>
        <servlet-name>Jersey REST Service</servlet-name>
        <url-pattern>/rest/*</url-pattern>
    </servlet-mapping>
</web-app>

pom.xml

<project xmlns="http://maven.apache.org/POM/4.0.0" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://maven.apache.org/POM/4.0.0 http://maven.apache.org/xsd/maven-4.0.0.xsd">
  <modelVersion>4.0.0</modelVersion>
  <groupId>br.com.anderson</groupId>
  <artifactId>rest-notas</artifactId>
  <version>0.0.1-SNAPSHOT</version>
  <packaging>war</packaging>
  <build>
    <sourceDirectory>src</sourceDirectory>
    <plugins>
      <plugin>
        <artifactId>maven-compiler-plugin</artifactId>
        <version>3.7.0</version>
        <configuration>
          <release>10</release>
        </configuration>
      </plugin>
      <plugin>
        <artifactId>maven-war-plugin</artifactId>
        <version>3.2.1</version>
        <configuration>
          <warSourceDirectory>WebContent</warSourceDirectory>
        </configuration>
      </plugin>
    </plugins>
  </build>
  <dependencies>    
     <dependency>
         <groupId>org.glassfish.jersey.containers</groupId>
         <artifactId>jersey-container-servlet</artifactId>
         <version>2.22.1</version>
     </dependency>  
     <dependency>
         <groupId>org.glassfish.jersey.media</groupId>
         <artifactId>jersey-media-moxy</artifactId>
         <version>2.22.1</version>
     </dependency>
     <dependency>
         <groupId>org.glassfish.jersey.core</groupId>
         <artifactId>jersey-server</artifactId>
         <version>2.22.1</version>
     </dependency>
     <dependency>
         <groupId>commons-codec</groupId>
         <artifactId>commons-codec</artifactId>
         <version>1.10</version>
     </dependency>
     <dependency>
         <groupId>mysql</groupId>
         <artifactId>mysql-connector-java</artifactId>
         <version>5.1.38</version>
     </dependency>
     <dependency>
         <groupId>org.hibernate</groupId>
         <artifactId>hibernate-core</artifactId>
         <version>5.4.3.Final</version>
     </dependency>
  </dependencies>
</project>

persistence.xml

<persistence xmlns="http://java.sun.com/xml/ns/persistence"
    xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
    xsi:schemaLocation="http://java.sun.com/xml/ns/persistence
        http://java.sun.com/xml/ns/persistence/persistence_2_0.xsd"
    version="2.0">

    <!-- unidade de persistencia com o nome notes_db -->
    <persistence-unit name="notes_db">

        <!-- Implementação do JPA, no nosso caso Hibernate -->
        <provider>org.hibernate.jpa.HibernatePersistenceProvider</provider>

        <!-- Aqui são listadas todas as entidades -->
        <class>br.com.anderson.rest_notas.modelo.Nota</class>

        <properties>    
            <!-- Propriedades JDBC -->
            <property name="javax.persistence.jdbc.driver" 
                value="com.mysql.jdbc.Driver" />

            <property name="javax.persistence.jdbc.url" 
                value="jdbc:mysql://localhost/notes_db" />

            <property name="javax.persistence.jdbc.user" value="debian-sys-maint" />
            <property name="javax.persistence.jdbc.password" value="pq9C6JB9dp1aGk9A" />

            <!-- Configurações específicas do Hibernate -->
            <property name="hibernate.dialect" 
                value="org.hibernate.dialect.MySQL5InnoDBDialect" />

            <property name="hibernate.hbm2ddl.auto" value="validate" />
            <property name="hibernate.show_sql" value="true" />
            <property name="hibernate.format_sql" value="true" />
        </properties>
    </persistence-unit>
</persistence>

Como você está usando Java 11, vai precisar atualizar a versão do Jersey para

<dependency>
    <groupId>org.glassfish.jersey.core</groupId>
    <artifactId>jersey-server</artifactId>
    <version>2.29</version>
</dependency>

Boa Noite, Otávio.

Atualizei a versão do jersey, como me sugeriu, e também de todas as outras dependencias do projeto.

O problema foi resolvido, porém surgiu outra exception.

SEVERE: Servlet.init() for servlet [Jersey REST Service] threw exception java.lang.IllegalStateException: InjectionManagerFactory not found.

Pesquisando no google, descobri que isso poderia ser resolvido adicionanado a seguinte dependência.

<dependency>
        <groupId>org.glassfish.jersey.inject</groupId>
        <artifactId>jersey-hk2</artifactId>
        <version>2.29</version>
</dependency>

Agora tudo está funcionando muito bem. Muito Obrigado Otávio.

Por nada amigo, bons estudos! =)