A dynamical system is a pair made out of a space of points and a function that determines how the points move in the space. Usually some extra properties are required from the space and the function to make them susceptible for studying. Topological dynamical systems require the space to be a compact metric space, meaning we can measure distances between the points and all sequences of points contain subsequences that converge to a single point. A function is required to be continuous meaning close enough points remain close to each other after one iteration of the function.

Cellular automata are examples of such systems. The space, called the configuration space, is made out of a regular lattice of symbols. Usually the lattice gets its structure from a group and the set of symbols is finite. The elements of the group are called cells and the symbols are called states. The continuous function, called the global rule, is made out of a neighbourhood vector and a local rule. The global rule applies the local rule for each cell independently and simultaneously and its value depends on the value of the cells in the neighbourhood of each cell.

Topological entropy is a measure of complexity of a given topological dynamical system. Simple systems tend to often have low or even zero entropy and by contrast complex systems often have high entropy. Two systems are called conjugate if they are equivalent in some sense. For example they have the same orbits and they share many dynamical properties. Conjugate systems have the same topological entropy, which makes it an useful value if we want to show that two given systems are not conjugate. The topological entropy has its measure-theoretic counterpart which we will also study. The entropies are related to each other by a variational principle.

Another measure of complexity are Lyapunov exponents. They can be thought of as the speed of the propagation of differences in a given system. The connections between Lyapunov exponents and various dynamical properties have been widely studied. In this dissertation we give an answer to the conjecture which states that a sensitive cellular automaton must have a configuration with a non-zero Lyapunov exponent. We construct a cellular automaton, which has no such configurations. The Lyapunov exponents are also related to measure-theoretic entropy. One can for instance calculate an upper bound for the measure-theoretic entropy of a given cellular automaton by first calculating the Lyapunov exponents. The Lyapunov exponents are only defined for one-dimensional cellular automata. In this dissertation we generalize the Lyapunov exponents for cellular automata over finitely generated groups and the measure-theoretic entropy for cellular automata over amenable groups and show that an analogous connection exists between them in the more general case.

A decision problem is a question with two possible answers: Yes or no. For example: ”Is a given natural number a prime number?” and ”Is a given route between two cities the shortest possible route?” are examples of decision problems. A decision problem is called decidable if there exists an algorithm (one can think of it as a computer program) that always gives the right answer to the problem for every possible input. The problem is called undecidable if no such algorithm exists. In this dissertation we study several decision problems related to reversible Turing machines, reversible cellular automata and group cellular automata. Namely these problems ask for example: ”Is the topological entropy of a given system zero?” and ”Are the Lyapunov exponents of a given system zero?”. Some related decision problems are also considered.