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3. Constructed by Bassam Fayad, Topologically mixing and minimal but not ergodic, analytic transformation on $\mathbb{T}^5$, 2000.

4. Constructed by Furstenberg, Strict ergodicity and transformation of the torus, 1961.

6. As Ian Morris points out in the comments, the identity map on a singleton set works here. A less trivial example was given by Karl Petersen, A topologically strongly mixing symbolic minimal set, 1970.

3. Constructed by Bassam Fayad, Topologically mixing and minimal but not ergodic, analytic transformation on $\mathbb{T}^5$, 2000.

4. Constructed by Furstenberg, Strict ergodicity and transformation of the torus, 1961.

6. As Ian Morris points out in the comments, the identity map on a singleton set works here. A less trivial example was given by Karl Petersen, A topologically strongly mixing symbolic minimal set, 1970.

It is very often the case that one wishes to study a topological dynamical system as a measure-preserving system by equipping it with an invariant measure, and in this case it is quite reasonable to ask about the relationships between the two different classes of properties. But this depends on which invariant measure you choose, because in general there may be very many of them. One may ask what properties of $(X,T)$ let you pick invariant measures $\mu$ with certain nice properties, and this is a whole different story which would expand this answer far beyond the bounds of propriety.

It is very often the case that one wishes to study a topological dynamical system as a measure-preserving system by equipping it with an invariant measure, and in this case it is quite reasonable to ask about the relationships between the two different classes of properties. But this depends on which invariant measure you choose, because in general there may be very many of them. One may ask what properties of $(X,T)$ let you pick invariant measures $\mu$ with certain nice properties, and this is a whole different story which would expand this answer far beyond the bounds of propriety.

Relationships between topological properties http://www.math.uh.edu/%7Eclimenha/pics/topological-properties.png

Ergodic hierarchy http://www.math.uh.edu/%7Eclimenha/pics/ergodic-properties.png