Part1 -- Beginner level confusion regarding terminologies -- symbolic dynamics, trajectory, phase space I came across the topic of symbolic dynamics when studying about time series analysis. Since I have not formally taken any course on chaotic dynamics, I have some difficulties in understanding some terms and need help.
The document which I am following is available here
Link1 : http://london.ucdavis.edu/~reu/REU10/smith.pdf
Link2: Incompatible implementations of physical symbol systems
Confusion 1: In the document, the definition of symbolic dynamics is that the symbolic dynamics is obtained from the "state-space" partitioning. My understanding of state-space is the plot of variables. In case of Tent Map, this will be plot of $x_{n+1}$ vs. $x_n$. A symbol is assigned if a state which is now a 2 d coordinate expressed as, $\mathbf{x} = [x_{n+1}, x_{n}]$ lies within an interval or not. How is this symbolizing procedure different from the symbolizing using the time series which I wrote above? In Link2, figure 2 shows the symbolic sequence from the time series, but figure 1 describes theoretically and pictorially using state space concept. What is the correct way? Is my example /approach correct?
Confusion 2: Is trajectory another name for time series or is it the curve obtained in phase space? In the document, sometimes it is indicated that the trajectory is a time series whereas sometime they say that is a curve / orbit in phase space.
 A: Confusion 1 answer: A state-space is not connected with plotting in any manner. It is a (sometimes abstracted) mathematical space, as many dimensional as the number of dynamical variables of your system. In the case of the tent map, that dimension is 1, which means that the state-space is one dimensional (i.e. a line). An easy way to think about it is that the phase-space is where your system "lives": it contains all possible states you could find your system in. In the case of the tent map, a one dymensional system, a state is simply a number between 0 and 1. Thus the phase space is the line segment [0,1). 
Partitioning the phase space literally means: divide it into boxes. In the case of 1 dimension, those boxes are line segments. In your specific case of symbolic dynamics, the phase-space is partitioned into only 2 segments: [0, 0,5) and [0.5, 1). 
In general, partitioning is useful in many things: finding patterns, analyzing chaotic behavior, calculating entropies and/or dimensions of attracting sets. The difference with the symbolic representation and the time-series representation simply falls to the user: "What do I want to do with it?"
Comment: Notice that the plot $x_{n+1}$ versus $x_n$ simply plots the function of the equation of motion (in your case, a "tent").
Confusion 2 answer: Unfortunately the term "time-series" is not strictly defined and can be inderchanged to mean strictly one-dimensional time-series or multi-dimensional. A 1D time-series, is simply a series of numbers versus time. In many dimensions you can also refer to as time-series a group of 1D time-series. The main reason for this confusion is that the plural and singular of this word is the same.
A trajectory is a solution to a dynamical system of equations with some initial condition. In my experience this is equivalent to a multi-dimensional time-series that contains all the 1D time-series of all the variables of the system. A trajectory is also equivalent to a curve in the full-dimensional phase-space, in the case of a continuous dynamical system.
The orbit however is something different, at least in my knowledge: It is the projection of the trajectory in the real space coordinates. For example, if your system's variables were $(x, y, p_x, p_y)$, then the orbit would be $(x,y)$ (again as a function of time). An orbit is a curve in real-space which itself is (most of the time) a proper subset of the phase-space.
Extra: Visualizing 1D phase-spaces (edit):  Now, about visualizing a 1D phasespace. This is a bit tricky but it is very possible. For example, you might want to make parts of the line segment thicker and thicker, depending on how many orbit points are in this segment. A similar approach would be to use a colorbar, coloring differently line segments with different properties, e.g. more orbit points. 
More extra : Another way to visualize a 1D map is to use it's invariant density (also called invariant measure), which when plotted is a curve \rho(x) versus x, the system's "state" (which is simply a number in 1D).
A: About 1
Your example is correct, but the tent map is a 1-D system, and its state space is the interval $[0,1]$. State space is classically the same as the phase space: the space of all possible states of the system. Check this question for more.
The plot of $x_{n+1}$ as a function of $x_n$ is simply a way to visualize the equation that defines the 1-D map and its effect on points. It's not the state state of the tent map.
In the paper you link, the author, as far as I can tell, doesn't specify the equations that generate the results in the figures, so it's hard to comment. But, in general you can consider only one variable to define your symbolic dynamics, and that corresponds to rectangular partitions of the state/phase space.
About 2
Trajectory, or orbit: a sequence of positions, or curve in the phase space, $\vec{x}$ parametrized by $t$. In the case of the tent map, or any usual 1-D map, that is a sequence of points $x_n$ over the interval. It doesn't look like what you'd expect from a "curve", but that's what you get from a time-discrete system (though in more dimensions, these points might eventual build "proper" curves in the phase space).
Time series: that's the phase space coordinates as a function of $t$ or $n$ for a given trajectory.
